Applied Mathematics for Industrial Flow Problems
A research programme of the European Science Foundation.
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Millennium Prize Problems
The seven problems proposed by the Clay Mathematics Institute: P versus NP; Hodge Conjecture; Poincar Conjecture; Riemann Hypothesis; Yang-Mills Existence and Mass Gap; Navier-Stokes Existence and Smoothness; Birch and Swinnerton-Dyer Conjecture. Resources include articles on each problem by leading researchers.
Millennium Prize Problems Clay Mathematics Institute Dedicated to increasing and disseminating mathematical knowledge HOME | ABOUT CMI | PROGRAMS | NEWS EVENTS | AWARDS | SCHOLARS | PUBLICATIONS Millennium Problems In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven Prize Problems. The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solution over the years. The Board of Directors of CMI designated a $7 million prize fund for the solution to these problems, with $1 million allocated to each. During the Millennium Meeting held on May 24, 2000 at the Collge de France, Timothy Gowers presented a lecture entitled The Importance of Mathematics, aimed for the general public, while John Tate and Michael Atiyah spoke on the problems. The CMI invited specialists to formulate each problem. One hundred years earlier, on August 8, 1900, David Hilbert delivered his famous lecture about open mathematical problems at the second International Congress of Mathematicians in Paris. This influenced our decision to announce the millennium problems as the central theme of a Paris meeting. The rules for the award of the prize have the endorsement of the CMI Scientific Advisory Board and the approval of the Directors. The members of these boards have the responsibility to preserve the nature, the integrity, and the spirit of this prize. Paris, May 24, 2000 Please send inquiries regarding the Millennium Prize Problems to prize.problems@claymath.org . Poincar Conjecture In 1904 the French mathematician Henri Poincar, asked if the three dimensional sphere is characterized as the unique simply connected three manifold. This question, the Poincar conjecture, is a special case of Thurston's geometrization conjecture. The latter would give an almost complete understanding of three dimensional manifolds. Birch and Swinnerton-Dyer Conjecture Hodge Conjecture Navier-Stokes Equations P vs NP Poincar Conjecture Riemann Hypothesis Yang-Mills Theory Rules Millennium Meeting Videos Return to top Contact | Search | Terms of Use | Clay Mathematics Institute
Unsolved Mathematics Problems
Compiled by Steven Finch. Descriptions of some unsolved problems and numerous links to other collections.
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Most Wanted List
Elementary unsolved problems in mathematics, listed at the MathPages archive.
MathPages Wanted List MathPages Wanted List The twenty-five mathematical problems and questions listed below were first posted on the internet in 1995. Since that time, Problems 5, 7, 8, and 22 have been solved completely, and part of Question 12 has been answered. The other problems remain unsolved. The links in this list point to articles on the MathPages web site containing more background on each problem, and partial or related results. (1) If each "1" in the binary representation of the integer x signifies a point in the corresponding position on a linear lattice, and if x' denotes the binary digit reversal of x, prove or disprove that the equality xx' = yy' implies that x and y have the same multi-set of point-to point distances. Ref: Generating Functions for Point Set Distances Isospectral Point Sets in Higher Dimensions (2) Find an elementary proof that x^2 + y^2 and x^2 + 103y^2 cannot both be squares for non-zero integers x,y. Ref: Concordant Forms (3) Prove (or disprove) that the only solutions of ab = c (mod a+b) ac = b (mod a+c) bc = a (mod b+c) in positive coprime integers are (1,1,1) and (5,7,11). Ref: A Knot of Congruences Limit Cycles of xy (mod x+y) More Results on the Form xy (mod x+y) (4) Given any cyclical arrangement, A, of |G| elements (not necessarily distinct) of a group G, let f(A) denote the cycle consisting of the compositions of the cycles of A. If A consists of all |G| distinct elements of G, is it true that iterated application of f eventually leads to the identity cycle (i.e., the cycle consisting solely of the identity element of G)? Ref: Permutation Loops Cumulative Permutation Sequences String Algebra (5) In how many distinct ways can the integers 0 through 15 be arranged in a 4x4 array such that the bitwise OR over each row, column, and diagonal is 15, and the bitwise AND over each row, column, and diagonal is 0? Ref: 414298141056 Quarto Draws Suffice! Ref: Four Cubed SOLVED: 12 Dec 96. See the article 414298141056 Quarto Draws Suffice! for the answer to this combinatorial question, and a description of how it was solved. Special thanks to Steve Zook for being the first to settle one of the original 23 Most Wanted problems. Thanks also to Andrei Ivanov for independently confirming the result, and to Keith Lynch for useful discussions on this topic. (6) Is it true that ln(x)^2 | pi(x) - g(x) | is less than ---------- 4 where pi(x) is the number of primes less than x, and g(x) is the number of composite integers n less than 2x such that n plus the sum of its prime factors (counting multiplicities) exceeds 2x? Ref: Sum of Prime Factors(M) (7) Let M(n,k) denote the number of distinct monotone Boolean functions of n variables with k mincuts. Obviously M(n,0)=1 and M(n,1)=2^n. In addition, we have M(n,2) = (2^n)(2^n - 1) 2 - 3^n + 2^n M(n,3) = (2^n)(2^n - 1)(2^n - 2) 6 - 6^n + 5^n + 4^n - 3^n Is there an explicit formula for M(n,k) with k=4? Ref: Dedekind's Problem Progress on Dedekind's Problem SOLVED: 17 Sep 99. I've received an email from Vladeta Jovovic informing me that he, G. Kilibarda, and Z. Maksimovic of the University of Belgrade are preparing a paper in which they give explicit expressions for M(n,k) with k=4 to 10. See Progress on Dedekind's Problem . (8) The number 2^(k-1) + k is a prime for k = 1, 3, 7, 237, and 1885. Does anyone know of any other primes of this form? Ref: Sum of Prime Factors(M) ANSWERED: 10 Nov 99. Nuutti Kuosa has found that 2^(k-1) + k is a prime for k = 51381. For more details, see the note Sum of Prime Factors(M) . (9) For any positive integer N let m_k, k=1,2,...t denote all the solutions of m+SOPF(m)=N, where SOPF is the sum of prime factors (including multiplicities). Then define t tau(N) = N - SUM SOPF(m_i) i=1 Is tau(N) equal to zero for any integer N? Ref: Sum of Prime Factors(M) (10) Let c(N) denote the number of distinct configurations of N particles in static equilibrium on the surface of a sphere of radius R, assuming the particles repell each other with a force that varies as 1 (1+r^2). How does c(N) vary (if at all) with R? Ref: Background for Problem 10 (11) For equilibrium configurations (EC) of N particles on a sphere under the influence of an inverse-square repulsive force law, there is an EC for N=17 that contains an EC for N=7 as a subset. Also, there is an EC for N=22 that contains an EC for N=6 as a subset. Also, there is an EC for N=23 that contains an EC for N=5 as a subset. Are there any other (non-trivial) examples of this kind? Ref: Points On A Sphere (12) For any prime p let z(p) denote the number of distinct solutions of x^2 x^3 x^(p-1) 1 + x + --- + --- + ... + ------- = 0 (mod p) 2! 3! (p-1)! Is it true that the fraction of primes with z(p) = n asymptotically approaches a Poisson distribution, i.e., the fraction of primes p less than x with z(p)=n approaches 1 e*n! as x goes to infinity? Also, what is the smallest prime p such that z(p)=6? Ref: A Special Property of 151 PARTIALLY ANSWERED: 17 Jun 00. The second part of Question 12 has been answered. It asked for the smallest prime p such that the generalized exponential function has 6 roots modulo p. Don Reble has found that p = 11117 is the smallest such prime. Also, he evaluated all the primes less than 32768 and found the overall distribution in good agreement with the conjectured Poisson density. See the updated note on A Special Property of 151 for more details. (13) For primes p of the form 3k-1 the congruence x^p + y^p = (x+y)^p (mod p^2) has a solution with x,y,(x+y) all non-zero (mod p) iff p is one of the primes 59, 83, 179, 227, 419, 443, 701, 857, 887, 911, 929, 971, 977,...etc. Is there a simpler way of characterizing these primes? What is their density? Ref: On the Density of Some Exceptional Primes (14) The number 588107520 is expressible in the form (X^2 - 1)(Y^2 - 1) (where X,Y are integers) in five distinct ways. Is there a 6-way expressible number? Ref: Numbers Expressible As (a^2 - 1)(b^2 - 1) (15) For what integers k and n do there exist integers x,y,z such that |x|,|y|,|z| k^(1 n) and x^n + y^n + z^n = k? (With k=0 this is just the Fermat problem.) Ref: A Conjecture On The Fermat Function Ref: Sums of Three Cubes (16) With k=1,n=3 in problem (15) above, an infinite family of solutions is given by (1 +- 9m^3)^3 + (9m^4)^3 + (-9m^4 -+ 3m)^3 = 1 but this doesn't cover all the solutions with k=1,n=3. Are any (all?) of the other solutions given by an algebraic identity? Ref: Sums of Three Cubes (17) For any positive integer n let SOPF(n) denote the sum of the prime factors of n, including multiplicities. Is it true that every iteration of the form x - SOPF(ax+b) for constants a,b is ultimately periodic? Is the number of limit cycles finite for any given a,b? Sum of Prime Factors(M) (18) With SOPF(n) as defined in (17) above, are there infinitely many solutions of SOPF(x^2 + ax + b) = x for any integers a,b? Are there solutions of SOPF(x^2 +1217x + 370313) = x such that the quadratic is NOT the product of three primes? Is there a relation between the class number of b^2 - 4a and the number of solutions of SOPF(x^2+ax+b)=x? Is there a relation between the period of cycles of SOPF(x^2+ax+b) and the class number of b^2-4a? Sum of Prime Factors(M) (19) Given any circular arrangement of the n integers j through j+n-1, let S denote the sum of the qth powers of every sum of k contiguous numbers. Then let v(q,k,j,n) denote the number of distinct possible values of S for all possible arrangements. Can v(q,k,j,n) be expressed in closed form as a function of the indices? Which integer sequences are contained within this family? Which continuous functions (e.g., sin(x), exp(x)) are approximated by members of this family? Ref: The Dartboard Sequence (20) Suppose I announce a sequence of binary digits beginning with 1, such as "1 1 0 1 ..." representing the numbers 1, 3, 6, 13, and so on. Your objective is to stop me at some point and, by supplying one more binary digit, produce a number divisible by at least one of a given set T of "target" primes. For any given x 0, does there exist a set T of primes p x such that you are guaranteed a winning opportunity? What is the minimum number of elements of such a set? Ref: Binary Games (21) Let a(n) and b(n) denote integer sequences each satisfying the recurrence s[k] = 4s[k-1] + 9s[k-2] with the initial values {1,0,...} and {0,1,...} respectively. Find a composite integer N congruent to 2,5,6,7,8, or 11 (mod 13) such that a(N)=4 and b(N)=-1 (mod N). Ref: Pseudoprimes For x^2 - 4x - 9 Lucas and Perrin Pseudoprimes Symmetric Pseudoprimes (22) Given two integer sequences X = {x1,x2,...} and Y = {y1,y2,...}, let X = P(Y) signify that xn is the number of partitions of n into elements of Y. The two sequences X = {1,2,3,4,6,9,11,15,19,25,31,41,49,61,75,91,110,...} Y = {1,2,3,5,6,10,12,17,22,29,36,48,58,73,91,111,...} are duals of each other in the sense that X=P(Y) and Y=P(X). Do there exist three sequences X,Y,Z such that X=P(Y), Y=P(Z), and Z=P(X)? Ref: Cyclical Partition Sequences Enumerating All Cycles of Partition Sequences SOLVED: 19 Jan 97. See Enumerating All Cycles of Partition Sequences for Dan Ford's analysis of this problem, and some additional generalizations and questions. (23) For any positive integer n let tau(n) and sigma (n) denote the number and sum of the divisors of n, respectively. A number N is called "sublime" if tau(N) and sigma(N) are both perfect (in the ancient Greek definition). The only two known sublime numbers are 12 and 6086555670238378989670371734243169622657830773351885970528324860512791691264 Can anyone find another sublime number, or prove that no others exist? Can there exist an odd sublime number? Ref: Sublime Numbers (24) Prove or disprove that no sum of two or more consecutive positive nth powers equals an nth power for any n 3. Ref: Sums of Consecutive Nth Powers Equal to an Nth Power (25) Prove or disprove that there do not exist constants A,B and integers xi,yi (i=1,2,3,4) such that yi^2 = A xi^2 + B i = 1,2,3,4 and such that the four products (xi)(yi) are in arithmetic progression. Ref: No Four Rectangles in Line? Return to MathPages Main Menu
Unsolved Problem of the Week Archive
A list of unsolved problems published by MathPro Press during 1995.
Unsolved Problem of the Week Archive Welcome to the archive for the Unsolved Math Problem of the Week . Each week, for your edification, we publish a well-known unsolved mathematics problem. These postings are intended to inform you of some of the difficult, yet interesting, problems that mathematicians are investigating. We give a reference so that you can get more information about the topic. These problems can be understood by the average person. Nevertheless, we do not suggest that you tackle these problems, since mathematicians have been unsuccessfully working on these problems for many years. Should you wish to discuss aspects of these problems with others, one of the newsgroups, such as sci.math , might be the appropriate forum. 3-Sep-1995 Problem 36 : Primes of the form n^n+1 27-Aug-1995 Problem 35 : Must one of n points lie on n 3 lines? 20-Aug-1995 Problem 34 : Squares with Two Different Decimal Digits 13-Aug-1995 Problem 33 : Unit Triangles in a Given Area 6-Aug-1995 Problem 32 : Can the Cube of a Sum Equal their Product 30-Jul-1995 Problem 31 : Different Number of Distances 23-Jul-1995 Problem 30 : Sum of Four Cubes 16-Jul-1995 Problem 29 : Fitting One Triangle Inside Another 9-Jul-1995 Problem 28 : Expressing 3 as the Sum of Three Cubes 2-Jul-1995 Problem 27 : Factorial that are one less than a Square 25-Jun-1995 Problem 26 : Inscribing a Square in a Curve 18-Jun-1995 Problem 25 : The Collatz Conjecture 11-Jun-1995 Problem 24 : Primes Between Consecutive Squares 4-Jun-1995 Problem 23 : Thirteen Points on a Sphere 28-May-1995 Problem 22 : Triangles with Integer Sides, Medians, and Area 21-May-1995 Problem 21 : Sum of Seven Cubes 14-May-1995 Problem 20 : Primes of the Form n^2+1 7-May-1995 Problem 19 : Pushing Discs Together 30-Apr-1995 Problem 18 : Diophantine Equation of Degree 5 23-Apr-1995 Problem 17 : Lattice Points Covered by a Set 16-Apr-1995 Problem 16 : A Billiards Problem (revised) 9-Apr-1995 Problem 15 : Square Free Mersenne Numbers 2-Apr-1995 Problem 14 : A Series involving 1 n^3 26-Mar-1995 Problem 13 : Rational Distances to the Vertices of a Square 19-Mar-1995 Problem 12 : Graceful Trees 12-Mar-1995 Problem 11 : Odd Perfect Numbers 5-Mar-1995 Problem 10 : Egyptian Fractions 26-Feb-1995 Problem 9 : Tiling the Unit Square 19-Feb-1995 Problem 8 : pi+e 12-Feb-1995 Problem 7 : Magic Knight's Tour 5-Feb-1995 Problem 6 : Prime Fibonacci Numbers 29-Jan-1995 Problem 5 : Goldbach's Conjecture 22-Jan-1995 Problem 4 : Equichordal Points 15-Jan-1995 Problem 3 : The Rational Box 8-Jan-1995 Problem 2 : Twin Primes Conjecture 1-Jan-1995 Problem 1 : Catalan's Conjecture go to electronic publications page This page is maintained by MathPro Press ( unsolved@MathPro.com ).
Mathematical Problems
In various subjects, compiled by Torsten Sillke.
Mathematical Problems Mathematical Problems Graph Theory Stable sets - The number of stable independent sets of a graph Planar Graphs - Special embeddings of planar graphs Graph Parameter - The Rank and the Chromatic Number Cograph (P4 free Graphs) - Graphs with maximal rank Nordhaus type question - eigenvalues Number Theory Lucas-Lehmer series - Factors and Period length Partial anwers: K Brown 1 , K Brown 3 , Chapman Some Properties of the Lucas Sequence (2,4,14,52,194,...) of Kevin Brown Reputnics - when are they squares? (Chris Thompson) x^3 + y^3 + z^3 - Representations by sum of Cubes (Noam Elkies) FLT++ - x^r + y^s = z^t Combinatorics Hadamard Matrices - Maximizing Determinants HKL96 - M. Hudelson, V. Klee, D. Larman; Largest j-Simplices in d-Cubes: Some Relatives of the Hadamard Maximum Determinant Problem (1996) GKL - P. Gritzmann, V. Klee, D. Larman; Largest j-Simplices in n-Polytopes (unpublished) A Library of Hadamard Matrices - N. J. A. Sloane Grid-avoidance problems - A collection of problems Gray Codes - Gray Codes with few Tracks Distance Sets - Sets with equal Distances register swap - or the diameter of GL(n,F2) Combinatorial Games - Unsolved Problems (by R. K. Guy) Enumerations Autocorrelation - of Words, Range of the function ( Table ) Triangle Counting - in an regular n-gon Counting Problem - A Dodecahedral Counting Problem N-Queens - Bounds for the number of solutions List of References - Generating Function History The number of symmetric {1, -1} matrices with nonnegative row-sums and an odd problem Geometry Trapezoid Problem - a geometric construction problem Algorithms and Complexity Uncrossed Knight Paths - A NP question (Twixt) NP complete - Dominic Mazzoni's proof (an improved crossing gadget see section 5.2 of M. Cook's Still Lifes , local copy (2MB) Additionsketten - Ein NP-Problem Bit Count - how fast can you count your bits Recreational Mathematics Solitaire - The incredible bent strip problem Cube 3 - No 3 on a line, No 4 on a plane Resistance Cube - A new twist in the cube Tiling and Packing Tiling Problem - Tiling rectangles with X and I_n n-Bone Conjecture - Impossible Tetrahedral Tilings Word Problem - a free group relation (polyomino) Handed Pentominoes - rectanges with handed Pentominoes (Problem solved) Symmetric Packings - with an odd number of polyominoes Packing equal circles into a square Erich's Packing Center www.packomania.com Packing upto 50 equal circles in a square Repeated pattern of dense packings of equal disks in a square Crossless L3 rectangles Dissections Heptomino Dissection - Square a heptomino Hypercube Dissection - Dissections into Congruent Pieces Similar Triangles - Dissections into similar Triangles Contests Probability expectation - number of fixpoint of a random permutation Occupation - Likelihood Approximation Red Black Game - How Often Should You Beat Your Kids Expectation Linearity - Problems solved by Linearity Algebra Determinant - commuting block matrices ( Solution : Mathematical Magazin, 70:5 (Dec. 1997) 382) Analysis Polynomial zeros - Maximize the number of fixpoints Polynomial zeros Bound - Find a bound for the roots Resistance Cube - A new twist in the cube Circuity Problem - Max Circuity for Metrics Continued Fractions - multidimensional Teabag Volume Math-News FAQ - mathematics 0^0 - part of the FAQ 0^0 - discussion from sci.math brute force - intuition vs. brute-force (Aug 1996) Footballpool - "ternary" Golay (Sep 1996) Math Jokes - Great Mathematicians (Sep 1996) Finite Fields - References (Sep 1996) Numerical Coincidences - Exp(Pi*Sqrt(n)) (Sep 1996) Fractran - the Conway machine (Sep 1996) Fixed point theorem - the boarder is sufficient Concordant Primes - they exist (Sep 1997) little Fermat - an elementary proof (std) Euler Characteristic - topological vs. geometric (Sep 1996) www.ics.uci.edu ~eppstein junkyard euler - Euler's formula V-E+F=2 (15 proofs). Is the Self Dual Proof the BOOK proof? A Mathematical Theory of Origami Numbers and Constructions - Roger Alperrin (Dec 1999) Simple Divisibility Rules for the 1st 1000 Prime Numbers - C. C. Briggs (Jan 2000) Math-Sites Mathematical History www-groups.dcs.st-and.ac.uk ~history - a great site Mathworld mathworld.wolfram.com - Eric W. Weisstein's Treasure Troves Mathematical Constants http: pauillac.inria.fr algo bsolve constant constant.html - Steven Finch For the Borwein real number recognizer see - www.cecm.sfu.ca projects ISC For Sloane's integer sequence recognizer see - www.research.att.com ~njas sequences Robin Chapman's - Math pages www.maths.ex.ac.uk ~rjc rjc.htm Kevin Brown - Math pages www.seanet.com ~ksbrown Mental Calculations forum.swarthmore.edu k12 mathtips beatcalc.html Four Color Proof www.math.gatech.edu ~thomas FC fourcolor.html Statistics lib.stat.cmu.edu DASL DataArchive.html - which test do you need? Primes www.utm.edu research primes - primes in all forms. Donald E. Knuth - www-cs-staff.stanford.edu ~knuth index.html Herbert S. Wilf - www.cis.upenn.edu ~wilf - download "generatingfunctionology" Ilan Vardi - www.ihes.fr ~ilan - why pi exists Sci.Math FAQ - www.cs.unb.ca ~alopez-o math-faq math-faq.html e.g. Math Quotations are at - math.furman.edu ~mwoodard mquot.html Math Jokes - www.geocities.com eemichaelis framesversion index.html and Science Jokes - www.xs4all.nl ~jcdverha scijokes Mathematik Museum Giessen - www.math.de - under construction since a year Cut the Knot - www.cut-the-knot.com - monthly math articles Enrichment Mathematics - www.nrich.maths.org.uk - NRICH, from kids to adults Special plane curves - xahlee.org SpecialPlaneCurves_dir specialPlaneCurves.html of Xah Lee www.2dcurves.com of Jan Wassenaar www-groups.dcs.st-and.ac.uk ~history Curves Curves.html of the MacTutor History of Mathematics archive perso.club-internet.fr rferreol encyclopedie introduction.shtml of Robert Ferreol Bibliography The MathSciNet of the AMS ams.mathematik.uni-bielefeld.de This is free from an account at mathematik.uni-bielefeld.de www.ams.org mathscinet Searchable Electronic Journals from the AMS www.ams.org journals Numerical Recipies www.nr.com arXiv - front.math.ucdavis.edu Applied for Airlines Circuity Problem - Double Connection Rules Torsten Sillke Home | FSP Mathematisierung | Fakultt fr Mathematik | Universitt Bielefeld last changed 2000-04-14
E-MathSolutionS
Consulting on and solving mathematically formulated problems from simple equations to research projects. Based in Cambridge, MA, USA.
Help with Mathematically Formulated Problems from Any Field of Knowledge. __________________________________________________________________________________________ E-MathSolutionS.com No Magic. Just Solutions. __________________________________________________________________________________________ WELCOME to E-MathSolutionS.com- - - a site where You'll find the solution to the mathematical problem that wouldn't allow You to sleep at night. It could be a classical math problem from general, matrix or vector algebra, from geometry or stereometry, from planar or spherical trigonometry, from differential or integral calculus, etc. It could be a mathematical application - - - statistics, spectral analysis, signal processing, least-squares fitting, approximating and interpolating by polynomials or other functions, solving algebraic, transcendental or differential equations by various numerical methods, etc. It could be a single quadratic equation You've forgotten or had no time to learn how to solve. It could be a sophisticated research project in any field of fundamental knowledge or practical skills (physics, chemistry, pharmacology, engineering, geology, psychology, medicine, geophysics, industry, UFOlogy, etc.), providing it is (or can be) formulated in mathematical terms. No matter what is the problem, E-MathSolutionS.com is ready to provide math help, advice and solutions. Just come and ask. Good luck - - - Vadim THE PROCEDURE IS AS SIMPLE AS THIS: Step 1. Formulate Your Problem Details Step 2. E-mail Your Problem E-mail Step 3. Learn Turnaround and Price Details Step 4. Pay for Solution Payment Step 5. Get Your Solution Details Contact Information: access@E-MathSolutionS.com Terms and Conditions ___________________________________________________________________________________________________________
SINTEF Applied Mathematics
Contract research and development for industry and the public sector in the fields of technology, medicine and the natural and social sciences, in collaboration with the Norwegian University of Science and Technology (NTNU). Based in Oslo.
SINTEF Applied Mathematics [ SINTEF ] [ Contact us ] [ Search ] [ Norsk ] Departments: Other Information: Welcome to SINTEF Applied Mathematics The institute ... ...is your potential partner in any scientific or technical endeavour where computational methods or computing is crucial. Performing contract research, development and consultancy for industry and system vendors, SINTEF Applied Mathematics is involved in numerous activities ranging from basic research to product and software development and commercialisation. During the past decade we have worked systematically to build up object oriented computational software. Utilising these libraries and toolkits in project work, we deliver advanced and reliable client-tailored computational and modelling tools at a competitive cost. Here is a taste of our activity: Center of Excellence: Mathematics for Applications Wind forecasts for airports Paper delivery gets wireless help Virtual Globe - see also article in Gemini (in Norwegian) Airborne sightseeing via your PC For more comprehensive collections of projects, you are welcome to explore the web pages of our departments and groups. You are also welcome to take a look at our movie . SINTEF Applied Mathematics P.O.Box 124 Blindern N-0314 OSLO, Norway Phone: 47 22 06 73 00, Fax: 47 22 06 73 50 Visiting address: Forskningsveien 1, Oslo, Norway Vice President, Research: Tore Gimse Last modified: 20-February-2002 by knl
Auxetics Ltd.
Nonlinear analysis, design and modelling undertaken with nonlinear effects 'designed-in'. Based in Sheffield, UK.
Nonlinear systems design: Exploit the movement, spreading and focusing of energy to designed locations Nonlinear systems design: Exploit the movement, spreading and focusing of energy to designed locations Nonlinear effects can be 'designed-in' so that the unwanted energies can be, moved to new frequency locations or spread over a wide range of frequencies, where there is a need to suppress unwanted effects, such as noise, or vibrations at certain frequences or bands of frequencies. Our patented Energy Transfer Filters (ETF) enables Auxetics to provide these unique nonlinear solutions. Auxetics Ltd's mission is to further enhance and develop class-leading technologies based on linear and nonlinear dynamical systems identification, modeling, information and signal processing, analysis, design and synthesis methods, thanks to our extensive research which has culminated in our patented Energy Transfer Filters (ETF). Auxetics personnel have extensive experience of applying these technologies to a wide range of problems and are world-renowned experts in their respective fields. We provide a comprehensive range of services from feasibility studies to complete consultancy and design projects. Brief summaries of the technologies we can offer to solve your problems are listed below. About Auxetics Ltd || Personnel || Publications || Contact us Our Services include: Nonlinear Energy Transfer Filters Nonlinear Vibrations and Damping Nonlinear Materials Design Nonlinear Acoustics Nonlinear Modeling, System Identification and the NARMAX method Nonlinear Fatigue Nonlinear Frequency Response Methods Nonlinear Communication Systems Copyright Auxetics Ltd., 2004, Company No. 0409877, The Innovation Centre. 217 Portobello, Sheffield, S1 4DP, UK Telephone: +(44) (0) 114 222 4432 E-Mail: enquiries@auxetics.org.uk
Acrenet Ltd
Finnish applied mathematics consultancy services company: fuzzy logic, neural networks, numerical analysis, signal processing and matrix algebra.
Oy Acrenet Ltd - Solutions ForMaths.com Core Disciplines: Applied Mathematics and Statistics Neural Networks and Fuzzy Logic Information Systems Development Market and Media Research Copyright 2000-2005. Oy Acrenet Ltd applied mathematics communications research privacy dvb about us
Signal Science
Engineering services, research and development in Signal Processing, Computational Science and Error Analysis. Based in Sterling, VA, USA.
Home Home About Us Contact Us News Links Signal Science provides high level engineering services including: Proof of Concept Research and Development System Engineering Signal Processing Computational Science Error Analysis Object Oriented Software Engineering and ReEngineering Our target marketsare areas in the government, commercial, and financial sectors that require solutions to computationally intensive problems. Our approach is to understand our customer's needs and cost performance goals. We then createa customized approach to helpcustomers achieve their goals. We accomplish this byusingconceptdriven process oriented approaches.What this means isthat westart out by addressinga customer problem ata high level. We then look at variousapproaches to the solution of the problem and see what design patterns emerge.Our philosophy isthat problems naturally organize themselves into patterns that, once recognized,lead tohighly effective solutions. We then rapidly implement these solutions by bridging the technology gap between research and practice. Home | About Us | Contact Us | News | Links Copyright 2003 by Signal Science, LLC
Marvin - Institute for Computational Applications
A group of experts with academic backgrounds, solving problems involving mathematical, physical, numerical and computer sciences. Based in Ljubljana, Slovenia.
Marvin - Intitut za raunske aplikacije English | Slovenina Slovenina English (c) Marvin - intitut za raunske aplikacije 2004, vse pravice pridrane. (c) Marvin - Institute for Computational Applications 2004, all rights reserved.
The Mathematics Consultancy
Offering solutions in all areas of applied and pure mathematics. Based in Twickenham, UK.
The Mathematics Consultancy | Home | Site Index | About Us | Services | Partners | Contact us | Who we are... We are a company dedicated to help you with creating models, optimising processes and solving problems. We will bring you the very best in mathematical research, solutions, and models. Tailored mathematical services... We have experience in many areas of applied and pure mathematics. We work with industry, academics, and business analysts delivering mathematical services tailored to suit your needs. How your business can benefit... Businesses that employ mathematical methods have an advantage over their competitors. Business benefits are achieved through process and product efficiencies, business optimisation, market forecasting. Free Consultation for a limited period... We are always available to offer impartial advice and discuss any of your queries. Contact us now for a free consultation a with one of our key specialists. Business Solutions Business solutions and financial services for improving your profit margins. Environmental Forecasting Environmental forecasting techniques for industrial, consumer, and commericial projects. Computational Modelling Mathematical formulations using basic and complex computation techniques. Industrial Applications Industrial modelling to improve optimisation, processes, yields and quality. Financial Mathematics Risk anlysis, derivitive modelling and development and test of new forcasting models. Optical and Nano Solutions Nonlinear optics and nanomechanical device modelling. Oil Recovery and Civil Engineering Porous flow, oil recovery optimisation, stress analysis, coastal engineering. Process Optimisation Advanced linear programming for manufacturing cost flow, cost minimisation, and profit maximisation. Copywright The Mathematics Consultancy 2003. Richmond Bridge House, Twickenham, England. Tel +44(0)020 7022 2963.
Nexyad
Private research lab involved in engineering, studies and consulting: statistics, data analysis, signal processing, signal understanding, data mining, image understanding, vision, modelling knowledge-based systems, robotics. La Garenne Colombes, France.
Welcome applied maths that works (statistics, signal processing, vision, data analysis, control, data fusion, ...) click on the picture ... nexyad 2005
Mathematic.co.uk
Industrial mathematical consultancy, including detailed examples of numerical analysis.
Industrial Mathematical Solutions We will provide you with a quick, reliable assessment of any mathematical difficulties you may have, whether they are based in business, science, engineering, finance, modelling or any other area. Simply provide us with a clear definition of the problem faced, and an indication of the information available. We will then tell you whether the problem, as posed, is solvable. If it is not, we will try to work with you to present the problem in a form amenable to mathematical representation and solution. As examples of the kind of standard of solution which you can expect from us, we have prepared some original mathematical analyses of some problems for you to look at. These show the clarity of expression and efficiency of solution which we will apply to the tasks you set us. Perhaps these will even help you become aware of a wider selection of mathematical tools available. These are taken from a diverse range of fields and vary from the small-scale to the large. We are happy to help with any problem, however bizarre, commonplace, big or small.
Vrobel Consulting
Fractal data analyses and consultancy. Based in Kassel, Germany.
- - vrobel.com
The Colorado Kidd Research Organization
A research organization that provides the service of engineering consulting primarily concerned with mathematics. Urbana, Illinois, USA.
The Colorado Kidd | Research Organization Web Mail is a research organization that provides the service of engineering consulting primarily concerned with mathematics. Stuck with a problem? Just doesnt seem to work? Need a fresh approach? Like to improve efficiency? We provide a different, creative and innovative perspective to your engineering problem, mainly through effective and thorough mathematical analysis. Engineering Consulting. Mathematical Analysis. Creative Problem Solving. Close Teamwork . Founded November 24th, 1997 Urbana, Illinois 61802 United States of America U.S. Trademark Registration : 2351487 Serial : 75-605052 D-U-N-S : 05-368-4283 www.ColoradoKidd.com contact us: info@ColoradoKidd.com Complete Statistics Reset December 7, 2003 Would you like to change your name ? Send mail to info@ColoradoKidd.com with questions or comments. 1997 - 2005 All rights reserved. Last modified: March 10, 2005
Visual Math Institute
Nonprofit research institute devoted to mathematics, dynamical systems theory, chaos theory, applications, and education. Santa Cruz, CA, USA.
VMI The Visual Math Institute Ralph Abraham, Director || What's hot || FAQs || Sponsors || Publications || || Research || About the VMI || Documents || Other websites created by Ralph Abraham, maintained by the VMI: || Visual Chaos Project || Visual Euclid Project || Visual Kepler Project || || Webographics || Yarrowstalk ||
Rothwell Mathematics Colloquium
A centre of research in Rothwell, Leeds. Specialising in pure mathematics, and more particularly transformational geometry and cryptography. Includes a list of current projects and list of members.
CIRM
Centre International des Rencontres Mathmatiques. A conference centre for Mathematics, Theoretical Physics, Scientific Computing and Computer Algebra. Marseille, France.
C.I.R.M - Index For any question about a meeting, please use the following e-mail: colloque@cirm.univ-mrs.fr The CIRM: a center of conferences for high level scientific meetings in Mathematics, Theoretical Physics, Scientific Computing, Computer Algebra, etc... Mirror site of : EMIS IMU ICM98 C.I.R.M. 163, Avenue de Luminy Case 916, F-13288 MARSEILLE Cedex 09,France. Telephone : (33) 04 91 83 30 00. Fax :(33) 04 91 83 30 05
Russian Academy of Sciences, Moscow
Steklov Mathematical Institute.
Steklov Mathematical Institute General Information Staff Events Algorithmically Integrable Systems in Action Library
Centro Internacional de Matemtica
A non-profit Portuguese association with the aim of consolidating international contacts to foster mathematics research in Portugal and worldwide. Location, workshops, publications and associates.
CIM | Latest News: Workshop on Advances in Continuous Optimization - June 30 and July 1, 2006 Annual Scientific Council Meeting 2006 Working Afternoon SPM CIM - January 7, 2006 Research in Pairs at CRM Pisa 2006 Announcement: Two visiting positions will be available at the De Giorgi Center (2006-2007) (website and documents) About CIM News News from Associates Events Publications Research in Pairs Associates Scientific Council Administration Contacts My Account CIM (International Center for Mathematics) is a not-for-profit, privately-run association that aims at developing and promoting research in Mathematics. At present CIM has 31 associates, including 13 Portuguese Universities, the University of Macau, 15 Research Centres and Institutes, the Portuguese Mathematical Society (SPM) and the Portuguese Operational Research Society (APDIO). CIM was formally set up on the 3rd of December, 1993 and was launched as a national project to involve all Portuguese mathematicians. During the past years, CIM has organized several meetings in mathematics and many interdisciplinary conferences. As a result, CIM has become an important forum for national and international cooperation among mathematicians and other scientists. CIM is also a privileged place for the exchange of information among Portuguese researchers and scientists from Portuguese-speaking countries. CIM is managed by a Board of Directors elected by the associates in the General Assembly. The Board of Directors organizes the annual scientific program, which is submitted to and approved by the Scientific Council . This committee includes a number of national and international prestigious mathematicians that are invited by the Board of Directors and approved by the General Assembly. __________ Member of ERCOM - European Research Centres on Mathematics (C) CIM 2004
Institute of Fundamental Sciences: Mathematics Research
Covers a variety of topics from math in industry to topology. Located at the Massey University in Palmerston North, New Zealand.
Mathematics (Maths) Research, Institute of Fundamental Sciences, Massey University Home Institute of Fundamental Sciences SEARCH MASSEY LIBRARY | NEWS | EVENTS Future students | Current students | Extramural | Alumni | Staff | Programmes | Campuses | Departments | Research IFS Home Coming Events Study with IFS Chemistry Mathematics Physics Electronics Scholarships Prizes Student Staff Liaison Research at IFS News Chemistry Mathematics Physics IFS People IFS Specialist Services IFS Employment Opportunities IFS Bits newsletter Alumni NZMS newsletter Resources Chemistry Mathematics Physics College of Sciences Compliance Check Sheets (staff only) Mathematics Research Postgraduate Study in Mathematics Research Centres Centre for Mathematical Modelling Allan Wilson Centre for Molecular Biology and Evolution Centre for Mathematics in Industry Mathematics Research at Albany Campus Research Areas Chromatic Polynomials 1-Factor Graphs Non-Linear PDE's and Symmetries Mathematics Education Domain Decomposition Methods for Differential Equations Geometric Integration Differential Geometry Mathematical modelling Mathematical Phylogeny ( Allan Wilson Centre ) Facilities All Mathematics researchers have access to a range of computers and software packages. Personal computers in offices offer Matlab, Maple, , various compilers (Fortran, Pascal, etc), a Computational Fluid Dynamics programme (PHOENICS) and geothermal reservoir simulation programmes as well as standard word processing, spreadsheet and graphing software. Researchers have access to the HELIX supercomputer in the Allan Wilson Centre. Inquiries If you require any further general information about research in Mathematics at Massey University, please contact Kee Teo or for mathematical phylogeny, Mike Hendy. Chromatic Polynomials Kee Teo , Charles Little , Mike Hendy A simple graph consists of a set of objects called vertices and another set of objects called edges. Each edge is a pair of vertices and is said to join those vertices. Two vertices are said to be adjacent if they are joined by an edge. Given a set of colours, we assign colours to the vertices so that adjacent vertices receive distinct colours. The resulting colouring is said to be proper. The question arises as to how many proper colourings are possible with a prescribed maximum number of colours. It is easy to show that for any given graph the required number of proper colourings is given by a polynomial in the number of colours. This polynomial is called the chromatic polynomial of the graph, and has been extensively studied. Two graphs are chromatically equivalent if they share the same chromatic polynomial. In particular, a graph is said to be chromatically unique if it has the chromatic polynomial of no other graph. Such graphs are therefore characterised by their chromatic polynomials. The thrust of this project is to establish the chromatic uniqueness of certain families of graphs. 1-Factors in Graphs Kee Teo , Charles Little , Serguei Norine A 1-factor in a graph is a set S of edges such that each vertex is incident with just one edge of S. Our research concerns questions related to the enumeration of 1-factors. Non-Linear PDE's and Symmetries Dean Halford , Bruce van-Brunt , Marijcke Vlieg-Hulstman Our research is focussed on developing techniques for finding analytical solutions to non-linear partial differential equations (NL PDE's). In particular, we are interested in exploiting structures underlying certain classes of NL PDE's. These techniques include e.g. Backlund Transformations, Conservation Laws and Lie Symmetries. Mathematics Education Glenda Anthony , Peter Kelly , Gillian Thornley Current research involves two main areas: Research into issues related to tertiary teaching and learning. We are especially concerned with factors associated with student success and failure at first-year level, both internally and extramurally. Student motivation, learning styles, help-seeking behaviours and self-directed use of course material are all areas of interest. Recent research also concerns the participation of women in mathematics. Current research examines students' experiences in tertiary mathematics education, with a particular focus on majoring mathematics students and PhD students. The integration of technology (Computer Algebra Systems) into the teaching learning programmes is a continuing area of study and development. Research related to classroom teaching and learning. Most of this research involves teachers and mathematics educators who are completing postgraduate qualifications. Interest areas are diverse and include: the nature of proof; performance assessment, transition between primary and secondary school, language in mathematics, graphic calculators, probability and children's games, math newsletter, student writing of explanations and justifications, and programmes for talented students. Currently staff research is concerned with the relationship between understanding and memory in mathematics learning. Tammy Smith , Kee Teo , Bob Richardson The use of web technologies for mathematics teaching. Domain Decomposition Methods for Differential Equations Igor Boglaev One of our research interests in the numerical analysis of differential equations is the development of robust parallel algorithms for solving singularly-perturbed partial differential equations. The most effective approach for constructing the numerical algorithms is based on an iterative domain decomposition technique. Applications of the parallel algorithms are concerned with modelling of technological and engineering problems with boundary and interior layers. Numerical Methods for Differential Equations Dion O'Neale Geometric Integration Robert McLachlan Geometric integration is a new approach to simulating the motion of large systems. The new methods, inspired by chaos theory but driven by the demands of modern applications are faster, more reliable, and often simpler than traditional approaches. They are being used in areas as diverse as a possible celestial origin of the ice ages, the structure of liquids, polymers, and biomolecules, quantum mechanics and nanodevices, biological models, chemical reaction-diffusion systems, the dynamics of flexible structures, and weather forecasting. Although diverse, these systems have certain things in common that makes them amenable to the new approach. They all preserve some underlying geometric structure which influences the qualitative nature of the phenomena they produce. In geometric integration these properties are built into the numerical method, which gives the method markedly superior performance, especially during long simulations. In our research we are exploring all possible geometric or structural features that systems can have, the implications for their long-time dynamics, and how to design efficient numerical integrators that preserve these geometric properties. Differential Geometry Gillian Thornley , Matt Perlmutter Mathematical modelling Tammy Smith , Robert McKibbin Currently problems under investigation range from modelling hydrothermal eruptions to investigating the structure of hair and wool molecules. Mathematical Phylogeny (Allan Wilson Centre) Mike Hendy , Barbara Holland , Michael Woodhams , Bhalchandra Thatte The Allan Wilson Centre includes staff and students in the Institute of Molecular BioSciences, as well as interacting with other members of the IFS mathematics discipline group. The major focus of the interaction of the mathematicians and the biologists has been to develop an understanding of how to interpret historical information contained in biological sequence data. This has involved developing new algorithms, analysing existing algorithms, modelling the evolutionary processes and running simulations to test accuracy and efficiency. By interacting with active biologists, we have been able to keep the development targeted towards practical usage, and to focus on correcting for misleading signals which confound existing algorithms. It is a continuing challenge to assist biologists who are finding novel ways of examining the ever-increasing data-base of sequences now being stored. Some of the methods developed have been incorporated in computer packages which are now used world-wide. Other analytic investigations have led to interesting research results in combinatorics and graph theory. Contact Us | About Massey University | Sitemap | Disclaimer | Last updated: July 19, 2005 Massey University 2003 Page accessed [13866] times since 24 May 2004
IMSc
Institute of Mathematical Sciences. Chennai, India.
The IMSc Home Page Tel: +91-44-22541856, 2254 2588, 2254 2398, 2254 2397 Fax: +91-44-2254 1586 About us The Institute of Mathematical Sciences (IMSc) is a national institute for fundamental research in frontier disciplines of the mathematical and physical sciences: Theoretical Computer Science, Mathematics, and Theoretical Physics. NEWS Ph.D admission -- JEST 2006 Written test for pre-screened candidates for ILGTI Sys Admin post will be held at 10:00am on Monday Nov 14th 2005 IMSc Complex Systems School, 2-27 January, 2006 Mathematics Theoretical Computer Science Theoretical Physics Library Apply to us People Seminars Colloquia Eprint Archive Getting here Conferences Chennai Webmail Annual Report Tenders Computers Search WWW Search imsc.res.in 2004, The Institute of Mathematical Sciences webmaster
NZIMA
New Zealand Institute of Mathematics and its Applications. Auckland, New Zealand.
NZIMA Homepage New Zealand Institute of Mathematics its Applications (NZIMA) Kia ora! Welcome to the homepage of the New Zealand Institute of Mathematics its Applications (NZIMA) The NZIMA is one of New Zealand's seven Centres of Research Excellence , formed as a partnership between the University of Auckland (its host) and the NZ Mathematics Research Institute (Inc.) , with the aim of promoting mathematical research in New Zealand. Main Links About the NZIMA Advisory Board Affiliates Contact Details Directorate Funding Opportunities Governing Board International Visitors Other Activities Programmes Publications Upcoming Conferences Announcements The list of upcoming mathematical sciences events and conferences is now contained on a separate page. This can be accessed either by clicking here or by using the "Upcoming Conferences" link on the left. Maclaurin Fellows for 2005 are: Prof. Robert McLachlan (Massey University), for one year Prof. Martin Liebeck (Imperial College, London, UK) for six weeks (in January 2005) NZIMA Programmes for 2005 06 are as follows: Mathematical Models for Optimizing Transportation Services (starting in the first half of 2005). Programme Directors: Prof. Andy Philpott, Prof. David Ryan and Dr Matthias Ehrgott (University of Auckland). Hidden Markov Models and Complex Systems (starting mid-2005). Programme Director: Prof. David Vere-Jones (Victoria University of Wellington, and Statistics Research Associates). Geometric Methods in the Topology of 3-Dimensional Manifolds (starting in the first half of 2006). Programme Directors: Prof. David Gauld (University of Auckland), Dr Roger Fenn (University of Sussex) and Prof. Vaughan Jones (University of Auckland and University of California Berkeley). NZIMA's second annual report is now available - click here or on the "Publications" link opposite for more information. Institute Co-Director Marston Conder has been elected President-Elect of the Academy of the Royal Society of New Zealand The NZIMA is pleased to be able to help facilitate access by researchers in the mathematical sciences in NZ to high-performance computing platforms -- see under "Other Activities" opposite for further details about facilities, access and charging. Site visited 22912 times since 11 September 2002 Last updated on 18 April 2005 by webmaster@nzima.auckland.ac.nz
Institute of Industrial Mathematics
Created to enhance contacts between mathematicians, statisticians, and computer scientists at the University and researchers in industry. (Winnipeg, Manitoba, Canada)
University of Manitoba Institute of Industrial Mathematical Sciences: Welcome For Industry For Faculty For Students About IIMS People Affiliates Research Conferences Seminars Graduate Program Mathematical Biology Industrial Workshops High School Workshops Contact IIMS IIMS Home Fostering Collaborative Research With Industry In The Mathematical Sciences Upcoming Events An IIMS seminar titled "Uncovering the mysteries of disease deep inside the living brain: Another song and dance routine", will be held on Tuesday, November 22nd at 2:30 pm in Machray Hall 415. The speaker will be Dr. Melanie Martin of the Department of Physics at the University of Winnipeg. (This talk has been rescheduled from November 15th, when it was cancelled because of a heavy snowfall.) During the seminar, Dr. Martin will discuss the most recent developments in diagnosing and following diseases like Alzheimer's, stroke and Multiple Sclerosis, using Magnetic Resonance Imaging (MRI). She will talk about how MRIs work, what they can tell us and how the latest research is revealing the clues to treatment and cure. Current Recent News CAIMS2005! In June 2005, the IIMS hosted CAIMS2005 , the 26th Annual Meeting of the Canadian Applied and Industrial Mathematics Society. In the above picture, Bill Langford, President of CAIMS, is presenting a plaque to Abba Gumel, Director of the IIMS. Abba cochaired the organization of the meeting with Rob McLeod. Institute of Industrial Mathematical Sciences University of Manitoba, Winnipeg, MB Canada R3T 2N2 Phone:204-474-6724Fax:204-474-7602 Questions or Comments: iims@umanitoba.ca University of Manitoba , Winnipeg, MB Canada R3T 2N2 Phone:204-474-8880 Questions or Comments: www@umanitoba.ca
KIAS
School of Mathematics, Korea Institute for Advanced Study, Seoul.
WELCOME to KIAS | KIAS MOU The great success of modern science is due in large extent to the development of mathematics and analysis of models arising from natural and social phenomena. Research in the school of mathematics includes pure mathematics and applied mathematics such as algebra, number theory, algebraic geometry, topology, differential geometry, and analysis. ALGEBRA NUMBER THEORY ALGEBRAIC GEOMETRY GEOMETRY TOPOLOGY ANALYSIS KIAS 207-43 Cheongnyangni 2-dong, Dongdaemun-gu, Seoul 130-722, Korea TEL : +82 2 958 3711 FAX : +82 2 958 3770 WEBMASTER
Johann Radon Institute for Computational and Applied Mathematics
Part of the Austrian Academy of Sciences (AW): research in applied and computational mathematics.
Johann Radon Institute for Computational and Applied Mathematics Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences (AW) Home People Research Groups and Topics Publications Events Jobs Links Internal Location Contact Special Radon Semester 2005 Groebner Bases Special Semester 2006 Search Fields of activity of the institute Research program: For the topics of ongoing research both within the groups of the institute and in cooperation between these groups, see Research Groups and Topics Special programs: These are generally one-semester programs which are dedicated to varying topics from an applied science in which mathematical methods as represented in the institute are of interest, or to a methodological topic which is of importance to several problem classes from applied sciences. The aim will be that through intensive co-operation between representatives of the applied science and mathematicians, methods will be developed which will firstly further advance the applied science and secondly lead to interesting mathematical results and problems. At the beginning of each of these semesters, the topic shall be outlined and defined both from the applications and from the methods side in the course of an international meeting; a result of the meeting shall be a collection of topics of research on which work will be done in the following semester. In this work, both part of the institutes staff and guest scientists will be involved, who ideally will spend the whole semester at the institute. To a large extent, they will be representatives of the applied science, normally rather young Post-docs who are on leave from their institute, in some cases also specialists for mathematical methods which shall reinforce the methodological competence of the institute. The results of the co-operation shall be presented again about a year after the meeting in the course of a conference. As mentioned at the beginning, such special programs shall be carried out not only about applied topics which require diverse mathematical methods, but also about method classes with various application fields. Both types shall be represented approximately equally in long-term. For these special semesters to work efficiently, the core staff of the institute has to be in place. Thus, we expect to run the first special semester not before the fall semester of 2004. Informal proposals for topics are welcome. Mission Statement: The Johann Radon Institute for Computational and Applied Mathematics (RICAM) does basic research in computational and applied mathematics according to highest international standards obtains the motivation for its research topics also from challenges in other scientific fields and industry emphasizes interdisciplinary cooperation between its workgroups and with institutions with similar scope and universities world-wide cooperates with other disciplines in the framework of special semesters on topics of major current interest wishes to attract gifted postdocs from all over the world and to provide an environment preparing them for international careers in academia or industry cooperates with universities by involving PhD-students into its research projects promotes, through its work and reports about it, the role of mathematics in science, industry and society Director: Prof. Dr. Heinz W. Engl heinz.engl@oeaw.ac.at Phone: +43-(0)732-2468-9219 Fax: +43-(0)732-2468-5212 Mobile Phone: +43-(0)664-5209029 Deputy Directors: Prof. Dr. Ulrich Langer ulrich.langer@assoc.oeaw.ac.at Phone: +43-(0)732-2468-9168 Fax: +43-(0)732-2468-9148 Prof. Dr. Peter A. Markowich peter.markowich@assoc.oeaw.ac.at Phone: +43 1 4277 506-11 Fax: +43 1 4277 9 506 Advisory board (Kuratorium): Prof. Dr. Franco Brezzi, Universit degli Studi di Pavia Prof. Dr. Curt Christian, Universitt Wien Prof. Dr. Dr.h.c. Peter Deuflhard, Konrad-Zuse-Zentrum Berlin Prof. Dr. Peter M. Gruber, Technische Universitt Wien Prof.em. Dr. Dr.h.c.mult. Edmund Hlawka, Vienna Prof. Dr. Dr.h.c. Rolf Jeltsch, ETH Zrich Prof. Dr. Dr. h.c. Herbert Mang, Austrian Academy of Sciences Prof. Dr. Harald Niederreiter, Universtity of Singapore Prof. Dr. Dr.h.c. Helmut Neunzert, Universitt Kaiserslautern Prof. Dr. Olivier Pironneau, Universit Pierre et Marie Curie - Paris-6 Prof. Dr. Ludwig Reich, Universitt Graz Prof. Dr. William Rundell, Texas AM University Prof. Dr. Hans Troger, Technische Universitt Wien (Vice-Chairman) Hofrat Dr. Herbert Saminger, Government of Upper Austria Prof. Dr. Karl Sigmund, Universitt Wien (Chairman) Dr. Daniel Weselka, Federal Ministry for Education, Science and Culture The Institute is named after the famous Austrian mathematician Johann Radon (1887-1956) Medieninhaber: sterreichische Akademie der Wissenschaften Juristische Person ffentlichen Rechts (BGBl 569 1921 idF BGBl I 130 2003) Dr. Ignaz Seipel-Platz 2, 1010 Wien Diese Website dient zur Information ber die wissenschaftlichen Aktivitten der sterreichischen Akademie der Wissenschaften und setzt somit den gesetzlichen Auftrag um, die Wissenschaft in jeder Hinsicht zu frdern. View our server usage statistics This page was made with 100% valid HTML CSS - Send comments to Webmaster Today's date and time is 11 17 05 - 16:09 CET and this file ( index.html ) was last modified on 10 10 05 - 16:33 CEST
National Academy of Sciences of Ukraine
Institute of Mathematics, Kyiv.
Institute of Mathematics Ukrainian Your Mailbox login: password: News 23.05.2005. 27.04.2005. 17.03.2005. The Norwegian Academy of Science has decided to award the Abel Prize for 2005 to Peter Lax ... More 28.12.2004. 03.12.2004. Appeal of the researchers of the Institute of Mathematics to scientific community ... More 14.06.2004. Complete collection of Fushchych's papers 28.04.2004. The Norwegian Academy of Science has decided to award the Abel Prize for 2004 to Michael F. Atiyah and Isadore M. Singer ... More 20.04.2004. Math links page is updated ... More 13.02.2004. D.O. Hrave memorial plate is opened 13.02.2004. 70 years ago the Institute was founded. Best greetings to all our colleagues and friends. Webmaster
Slovak Academy of Sciences
Mathematical Institute; Extension in Kosice.
Mathematical Institute SAS (MU SAV) (Extension in Kosice) Mathematical Institute of the Slovak Academy of Sciences Extension in Kosice Slovenska verzia Headquarters: Mathematical Institute Slovak Academy of Sciences Stefanikova 49 814 73 Bratislava Slovak Republic Director: Prof. RNDr. Anatolij DVURECENSKIJ, DrSc. tel. fax (++421) (02) 52497-316 E-mail: mathinst@mat.savba.sk, dvurecen@mat.savba.sk Extension in Kosice: Mathematical Institute (MU SAV) Slovak Academy of Sciences Gresakova 6 040 01 Kosice Slovak Republic tel. fax (++421) (055) 622-8291 E-mail: musavke@saske.sk Head: Prof. RNDr. Jan JAKUBIK, DrSc. Principal Research Fellow E-mail: musavke@saske.sk Deputy: Roman FRIC, RNDr., DrSc., Associate Professor Principal Research Fellow E-mail: fric@saske.sk Secretary: Katarina STEFANCIKOVA E-mail: kstefan@saske.sk Research Fellows: Jan BORSIK, RNDr., CSc., Associate Professor Senior Research Fellow E-mail: borsik@saske.sk Vladimir DANCIK, RNDr., PhD. Research Fellow E-mail: vlado@saske.sk Peter ELIAS, RNDr., PhD. Research Fellow E-mail: elias@kosice.upjs.sk Jan HALUSKA, RNDr., CSc., Associate Professor Senior Research Fellow E-mail: jhaluska@saske.sk Emilia HALUSKOVA, RNDr., CSc. Research Fellow E-mail: ehaluska@saske.sk Galina JIRASKOVA, RNDr., CSc. Research Fellow E-mail: jiraskov@saske.sk Peter MIHOK, RNDr., CSc., Associate Professor Senior Research Fellow E-mail: mihok@saske.sk Miroslav PLOSCICA, RNDr., CSc. Senior Research Fellow E-mail: ploscica@saske.sk Miroslav REPICKY, RNDr., CSC., Associate Professor Senior Research Fellow E-mail: repicky@kosice.upjs.sk Peter VOJTAS, RNDr., DrSc., Professor Principal Research Fellow E-mail: vojtas@kosice.upjs.sk Marcel CELEC, Mgr., PhD Student E-mail: celec@saske.sk The Extention of Mathematical Institute in Kosice has been founded in 1978. Since the very beginning there is a close cooperation with mathematical departments at P. J. Safarik University and Technical University in Kosice. The research actvities focus mainly on algebra, graph theory, logic and set theory, topology, real analysis and applications. At present major grants from Slovak Scientific Grant Agency (VEGA) are awarded, namely 1. Generalizations of continuity of functions, vector integration and series, project leader J. Borsik 2. Algebraic structures related to ordering and graph theory, project leader M. Ploscica Most of the fellows are part-time lecturers at various universities. Also, they serve as members of scientific boards, editorial boards, PhD committees, and other scienific and educational committees. Further details about fellows, their activities, and their publications can be obtained from their personal web pages or can be requested by e-mail. Conferences: The 18th Summer Conference on Real Functions Theory 2004 Summer School on General Algebra and Ordered Sets 2003 (in honour of the 80th birthday of Professor Jan Jakubik) January 2004
Rolf Nevanlinna Institute
Finnish national research institute of mathematics, computer science and statistics, based at the University of Helsinki.
Rolf Nevanlinna Institute Department of Mathematics and Statistics Faculty of Science Faculty of Social Sciences Institute home Research Research foundation Doctoral thesis award Contact info Department of Mathematics and Statistics Rolf Nevanlinna Institute The Rolf Nevanlinna Institute is a research institute of Department of Mathematics and Statistics that concentrates on mathematics and statistics and their applications in various areas of science. We currently have three research groups in inverse problems, biomathematics and biometry. The research includes both theory and applications, bringing pure and applied mathematics and a wide spectrum of interdisciplinary scientific and industrial topics together. RNI has a wide collaboration network, including research contacts in industry and the public sector.
CIMPA
(International Centre for Pure and Applied Mathematics) Nice, France.
CIMPA Click here for English Bienvenue sur le site web du Centre International de Mathmatiques Pures et Appliques Cliquer ici pour la version franaise Le CIMPA (Centre International de Mathmatiques Pures et Appliques) est une association internationale (loi de 1901) cre Nice (France) en 1978. Son objectif est de promouvoir la coopration internationale au profit des pays en dveloppement, dans le domaine de l'enseignement suprieur et la recherche en mathmatiques et dans les disciplines connexes, informatique notamment. CIMPA Le Dubellay, 4 avenue Joachim - Bt. B, 06100 Nice, FRANCE Tl : (33) 4 92 07 79 30 Fax : (33) 4 92 07 05 02 E-mail : cimpa@unice.fr Prsident : Mario Wschebor - Directeur : Michel Jambu - Secrtariat : Agns Gomez , Jeanick Allanic Tutelles Ces Pages sont faites sous la responsabilit du CIMPA et hberges par la Cellule MathDoc Plan du site Pour toute remarque, crivez cimpa@unice.fr Copyright 1999 [CIMPA]. Tous droits rservs. Rvision : mercredi 24 aot 2005.
Centre for Mathematics and its Applications
Canberra, Australia.
ANU - Mathematical Sciences Institute (MSI) - Centre for Mathematics and its Applications Skip Navigation ANU Home | Search ANU Mathematical Sciences Institute (MSI) Centre for Mathematics and its Applications MSI Home People News Research Study " CMA Research reports Proceedings National Symposia Department Seminars Events Jobs MSI Intranet Internal pages Quick Links Search MSI Contact us ANU Staff page Centre for Mathematics and its Applications (CMA) The Centre for Mathematics and its Applications (CMA) has the primary function of a research institute in the mathematical sciences, fulfilling both national and international roles in that capacity. The CMA is Australia's representative in the International Mathematical Science Institutes (IMSI) , the international consortium which includes all the world's major mathematical science research institutes. The CMA also cooperates with the Australian Mathematical Sciences Institute (AMSI) . The CMA publishes a series of research monographs and has extensive series of mathematical and statistical research reports . The Head of the CMA is Professor Neil Trudinger . Recent achievements Sue Wilson and her colleagues in the Centre for Bioinformation Science (CBiS) investigated the reason that disease gene results are not always replicable. Her important modelling result shows that the cause may be due to researchers analysing each gene separately and ignoring the complex interactions among the genes. ANU scientists now are developing alternative genomic analyses of familial data. Neil Trudinger and Xu-Jia Wang solved an important conjecture concerning the existence of locally convex hypersurfaces of constant Gauss curvature with prescribed boundaries. Min-Chun Hong also settled a long-standing conjecture by proving that if an ellipsoid is sufficiently elongated then the equatorial map is energy minimizing. Alan McIntosh successfully completed his long-time research by solving the square root problem of Kato concerning elliptic differential operators, in collaboration with an international team. Amnon Neeman's 450-page book Triangulated Categories was published by Princeton University Press in their prestigious series, Annals of Mathematics Studies. National Research Symposia In its national role, the CMA organizes and sponsors a series of National Research Symposia in the Mathematical Sciences involving Australian and overseas researchers in mathematics, statistics and applications. The symposia provide a focal point for Australian researchers in an area of current interest in the mathematical and statistical sciences. They are often held in conjunction with the visit to Australia of leading international experts. Proceedings are often published. An invitation and more information about the National Research Symposia are available. Visitor program Each research program has a large number of visitors and extensive international contacts. These are of a substantial nature as here there is a wealth of powerful international collaborations. A large number of leading mathematicians and statisticians visit us from overseas as well as from within Australia. Our members regularly visit their colleagues overseas, and are invited to participate in international conferences. Indeed, the CMA has an exceptional profile for its size. Postgraduate Study There is an active program with supervision available for students wishing to study for the degrees of Doctor of Philosophy (PhD) and Master of Philosophy in the mathematical sciences. These are degrees by research, the PhD usually being of three years full-time study, or from four to six years part-time study. Candidates must present the results of their original research as a substantial thesis, although some course work may be required to bolster the student's background knowledge in preparation for research. Joint programs are encouraged with other areas of the University, in particular with computational science and bioinformatics. Contact Details Administrative Officer: Annette Hughes CMAadmin AT maths.anu.edu.au Phone: +61 2 6125 2897 (ANU internal: 52897) Administrator: Chris Wetherell CMAoffice AT maths.anu.edu.au Phone: +61 2 6125 0706 (ANU internal: 50706) Fax: +61 2 6125 5549 (ANU internal: 55549) Mailing: Centre for Mathematics and its Applications Bldg 27 The Australian National University ACT 0200 Australia Copyright | Disclaimer | Privacy | Contact ANU Page last updated: 17 October, 2005 Please direct all enquiries to: MSI webmaster Page authorised by: Dean, MSI The Australian National University - CRICOS Provider Number 00120C
Banff International Research Station
A joint venture between MSRI and PIMS to establish a center for research workshops.
The Banff International Research Station with the participation of Visitors Information General Information What's New BIRS Receives Record Funding from Alberta, the US and Mexico Multimedia and Mathematics at BIRS BIRS hosts Heads of G8 Research Councils Call for Proposals 2005 Banff International Research Station
TICMI
Tbilisi International Centre of Mathematics and Informatics, Georgia.
TICMI TBILISIINTERNATIONALCENTREOF MATHEMATICSANDINFORMATICS I. Vekua Institute of Applied Mathematics I. Javakhishvili Tbilisi State University Georgian Academy of Natural Sciences Structure ProvisionalStatute International Scientific Committee Announcements BulletinofTICMI LectureNotesofTICMI Research Projects: NATO Science Programme (Collaborative Linkage Grant NATO PST.CLG.976426 5437) I.Vekua 100 IUTAM Symposium on Relation of Shell, Plate, Beam, and 3D Models ISAAC Conference on "Analysis, Applications, and Computations" dedicated to the 100th birthday anniversary of Ilia Vekua
CRCIM
Czech Research Consortium for Informatics and Mathematics.
MITACS
Mathematics of Information Technology and Complex Systems. Canadian research network.
MITACS 2004 About MITACS Scientific Programs Events News Partnerships Member Resources Contact Us Subscribe Search En Franais Innovation Through Mathematical Sciences The Mathematics of Information Technology and Complex systems, is a Network of Centres of Excellence (NCE) for the Mathematical Sciences. MITACS is recognized worldwide as an effective new model for research development in the mathematical sciences - one that addresses the imperatives of research, education and technology transfer. The Network focuses on developing mathematical solutions, which address issues in the fastest growing sectors of the nation's economy. Internships Program Connections - September 2005 2005 Student Awards Program CAIMS-MITACS 2006 Joint Annual Conference
Feza Grsey Institute
stanbul. A joint institute of the Bogazici University and TUBITAK (Scientific and Technical Research Council of Turkey).
Feza Gursey Institute General Members Seminars Research Semesters Library Board of Consultants To Apply How to get here Previous Seminars Previous Research Semesters Istanbul Fizik Takvimi ITU-Istatistiksel Fizik Gunleri Links Gilgamesh PC Cluster Project. Feza Gursey Feza Grsey Institute Emek Mah.Rasathane Yolu No:68, 34684 engelky, Istanbul, TURKEY Tel:+90-216-308 94 32 (5 lines) Fax:+90-216-308 94 27 Announcements on Workshops and Schools Memorial Lecture for Prof.Dr. Fikret KORTEL webadmin@gursey.gov.tr Photo: Courtesy of Prof.Dr. Mehmet Erbudak
Tata Institute of Fundamental Research, Mumbai
School of Mathematics
School of Mathematics, TIFR SCHOOL OF MATHEMATICS Tata Institute of Fundamental Research Dr. Homi Bhabha Rd, Mumbai 400 005 India Phone: (+91) 22-22804545 Fax: (+91) 22-22804610, 22-22804611 "Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." Bertrand Russell, "The Study of Mathematics" Welcome from 212.235.208.157; the local time is November 28, 2005; 5:38 PM GMT+0530 International Conferences: Diophantine Equations in honour of T.N. Shorey and Representations of Real Reductive Groups in honour of R. Parthasarathy. You can dowload a paper on the work of T.N. Shorey in these formats: dvi , ps pdf Fast links Some useful links Postal Address and Phone FAX numbers Access to mail via secure webmail Set and stop a forward address for your email account. Indo-French cooperation program Websites of Mathematical Journals Another set of links to journals on line Latest Math Preprints from Los Alamos preprint server (you can access the main archive , or its Indian mirror at MatScience , or a nice interface for it at UC Davis ) TIFR Webserver AMS Webserver (American Mathematical Society) the School of Maths Website Maps of India How to reach TIFR Members in Mumbai (Bombay) TIFR Maths Bangalore server List of visitors Lectures Postdoctoral Positions Graduate Studies VSRP (Students Summer Visiting Program) NBHM Website Other servers of the School of Maths: Main webserver (this page) Secure webserver EMIS mirror IMU mirror Linux information NBHM School of Mathematics, TIFR. Send comments about this page to: webmaster at math.tifr.res.in Last modified Mon Aug 1 09:30:11 GMT+0530 2005
MPIM
Max-Planck-Institut fuer Mathematik, Bonn.
MPIfM MAX PLANCK INSTITUTE FOR MATHEMATICS MAX-PLANCK-INSTITUT FR MATHEMATIK Vivatsgasse 7, D-53111 Bonn, Germany Postal Address: P.O.Box: 7280, D-53072 Bonn Phone: (49) 228 402-0 Fax: (49) 228 402277 Local Math-Net Homepage: english --- german IMPRS for Moduli Spaces and their Applications General information How to find us Allgemeine Informationen Wegbeschreibung Max Planck Society for the Advancement of Science Max-Planck-Gesellschaft zur Frderung der Wissenschaften Practical information Useful addresses Application for membership Bonn information People The list of guests Permanent staff Departments Library Computer Public relations Administration Activities and Services Weekly activities Long-term activities MPI-Preprints MathSciNet --- If you have suggestions about this document, please send e-mail to --- Copyright 2004 Max-Planck-Institut fr Mathematik Impressum
IPAM
Institute for Pure and Applied Mathematics, Los Angeles, CA, USA.
IPAM - Home Page Home | People | Events | Programs | Visitor Info Home About Contact Us Newsletter Search News Photos Publications Links Directions to IPAM Welcome to the Institute for Pure and Applied Mathematics (IPAM). Please use the links in the navigation bar above (major categories) and along the left (items within a major category) to navigate our web site.For information on proposing a program, click here . Bridging Time and Length Scales in Materials Science and Bio-Physics September 12 - December 16, 2005 CIMMS Satellite Workshop at Caltech: Multiscale Modeling and Computation - Basic Theory and the Geosciences November 17 - 18, 2005 Position Available: IPAM Associate Director July 2006 - June 2008 Watch video from Graduate Summer School: Intelligent Extraction of Information from Graphs and High Dimensional Data Upcoming programs include: IPAM Long Programs Bridging Time and Length Scales in Materials Science and Bio-Physics . September 12 - December 16, 2005. CIMMS Satellite Workshop at Caltech: Multiscale Modeling and Computation - Basic Theory and the Geosciences . November 17 - 18, 2005. Culminating Workshop at Lake Arrowhead (By Invitation Only). December 11 - 16, 2005. Cells and Materials: At the Interface between Mathematics, Biology and Engineering . March 13 - June 16, 2006. Apply Online . Orientation session for long-term participants. March 13, 2006. Tutorials . March 14 - 17, 2006. Apply Register Online . Workshop I: Membrane Protein Science and Engineering . March 27 - 31, 2006. Apply Register Online . Workshop II: Microfluidic Flows in Nature and Microfluidic Technologies . April 18 - 22, 2006. Apply Register Online . Workshop III: Angiogenesis, NeoVascularization and Morphogenesis . May 8 - 12, 2006. Apply Register Online . Workshop IV: Systems Biology and Molecular Modeling . May 22 - 26, 2006. Apply Register Online . Culminating Workshop at Lake Arrowhead (by invitation only). June 11 - 16, 2006. Securing Cyberspace: Application and Foundations of Cryptography and Computer Security . September 11 - December 15, 2006. Apply Online . Tutorials . September 12 - 15, 2006. Apply Register Online . Workshop I: Number Theory and Cryptography - Open Problems . October 9 - 13, 2006. Apply Register Online . Workshop II: Locally decodable codes, private information retrieval, privacy-preserving data-mining, and public key encryption with special properties . October 23 - 27, 2006. Apply Register Online . Workshop III: Foundations of secure multi-party computation and zero-knowledge and its applications . November 13 - 17, 2006. Apply Register Online . Workshop IV: Special purpose hardware for cryptography: Attacks and Applications . December 4 - 8, 2006. Apply Register Online . Culminating Workshop at Lake Arrowhead (by invitation only). December 11 - 15, 2006. Random Shapes . March 12 - June 15, 2007. Apply Online . IPAM Short Programs Sequence Analysis Toward System Biology . January 9 - 13, 2006. Apply Register Online . Document Space . January 23 - 27, 2006. Apply Register Online . Heart Modeling: Image Acquisition. Segmentation, Modeling and Analysis . February 6 - 10, 2006. Apply Register Online . Swarming by Nature and by Design . February 27 - March 3, 2006. Apply Register Online . Affiliates' and Other Workshops UCLA-IPAM-NSF workshop on Thin Films and Fluid Interfaces . January 30 - February 2, 2006. IPAM Summer Programs (and related events) IPAM Research in Industrial Projects for Students "RIPS" 2006 . June 25 - August 25, 2006. Directions to IPAM Home [ People ] [ Events ] [ Programs ] [ Visitor Info ] [ About ] [ Contact Us ] [ Newsletter ] [ Search ] [ News ] [ Photos ] [ Publications ] [ Links ] Contact:
Smith Institute for Industrial Mathematics and System Engineering
The Smith Institute delivers solutions and technical services to companies through the application of mathematical modelling and analysis. The Institute manages the Faraday Partnership for Industrial Mathematics to promote industrial competitivness through improved collaboration between industry and the science base.
Smith Institute Printable Page Search Site Home About this Site Search Information News Positions Available About Us Services Solutions Faraday Partnership Institute Structure Contact Us Activities Events Projects Sectors Study Groups Home The Smith Institute helps companies to gain a competitive advantage in their products, processes and operations through the application of mathematical modelling and analysis. In a knowledge-driven economy, these skills provide cost-effective solutions to operational or design problems, and are also vital to the formulation of industrial strategy. The Institute provides consultancy, technical advice and research coordination, and works closely with leading university research groups through the Faraday Partnership for Industrial Mathematics. [more] news : 2005 10 18 The Knowledge Transfer Network for Industrial Mathematics On 18th October Lord Sainsbury, Minister for Science and Innovation, announced the migration of 19 Faraday Partnerships to form new Knowledge Transfer Networks (KTNs). The KTN for Industrial Mathematics will be managed by the Smith Institute, building on the successes of the Faraday Partnership for Industrial Mathematics through a range of new knowledge transfer activities. [more] news : 2005 10 17 Call for Mathematics CASE proposals EPSRC's call for proposals for Mathematics CASE is open until 1st December 2005. [more] news : 2005 08 30 Performance modelling Performance modelling covers a set of techniques that are targeted at two broad questions. In system design, how can the capacity best be dimensioned to match the expected load on the system and its required quality of service? And in operation, how are system resources best managed to ensure a smooth and cost-effective workflow? [more] [read more news...] job : 2005 08 23 CASE Studentship in evolutionary search strategies for facial composites A fully-funded CASE Ph.D studentship is available within the School of Physical Sciences at University of Kent, Canterbury, from October 1st 2005. [more] Homepage | About the Institute | Contact Us
IPM - Institute for Studies in Theoretical Physics and Mathematics
Tehran, Iran.
Institute for Studies in Theoretical Physics and Mathematics (IPM) Copyright 1999-2005 Institute for Studies in Theoretical Physics and Mathematics (IPM) Best View in 1024x768 - 16k Color
FIM
Forschungsinstitut fuer Mathematik, ETH, Zurich, Switzerland.
FIM - Institute for Mathematical Research People | Guests | Information for Guests Contact | Sitemap | Help Search Weekly Bulletin | Special Activities | Nachdiplom Lectures Preprints | ETH Monographs | Links ETH Zrich - D-MATH - FIM Institute for Mathematical Research FIM is part of the European Research Centres On Mathematics ( ERCOM ) Wichtiger Hinweis: Diese Website wird in lteren Versionen von Netscape ohne graphische Elemente dargestellt. Die Funktionalitt der Website ist aber trotzdem gewhrleistet. Wenn Sie diese Website regelmssig benutzen, empfehlen wir Ihnen, auf Ihrem Computer einen aktuellen Browser zu installieren. Weitere Informationen finden Sie auf folgender Seite . Important Note: The content in this site is accessible to any browser or Internet device, however, some graphics will display correctly only in the newer versions of Netscape. To get the most out of our site we suggest you upgrade to a newer browser. More information 2005 Mathematics Department | Imprint | November 11, 2005
CMAP
Centre de Mathmatiques Appliques, Palaiseu, France.
Centre de Mathematiques APpliquees English version Le laboratoire Directeur: Vincent Giovangigli Prsentation Pland'accs Enseignement Organigramme Publications Personnel Liens Recherche rapide: Logiciels: Xd3d Lastwave EGLib Intranet CMAP Webmail CMAP Webmaster Rapport d'activit 2004 Les Equipes de recherche Modlisation et inversion en lectromagntisme (resp. H. Ammari) Fluides et calcul parallle (resp. F. Nataf) Alatoire, finance et statistique (resp. N. El Karoui). Thorie du signal (resp. S. Mallat) Optimisation de forme et matriaux (resp. G. Allaire) Gnie logiciel et visualisation scientifique (resp. J.F. Colonna). Sminaires Analyse numrique Modles Stochastiques Colloquium Sminaire Bachelier Groupe de travail Modles Stochastiques en finance Problmes inverses Groupes de recherche GDR MOMAS, MOdlisations MAthmatiques et Simulations numriques lies aux problmes de gestion des dchets nuclaires. Rseau Europen (RTN Marie Curie) MULTIMAT CMAP Ecole polytechnique, 91128 Palaiseau cedex, France - Tel: +33 1 69 33 41 50, Fax: +33 1 69 33 30 11
Biomathematics and Statistics Scotland
Research, consultancy and training for agricultural and biological research organisations.
BioSS Home Page BioSS undertakes research, consultancy and training in mathematics and statistics as applied to agriculture, the environment, food and health. Our primary remit is: to support the research programme of the Scottish Executive Environment and Rural Affairs Department and its sponsored institutes, through specialist advice and training, and to provide research in statistics and biomathematics. We also have strong links with the private sector offering high-level computational skills to clients. About half our staff are based on the King's Buildings science campus of Edinburgh University, while other groups are based at research organizations in Aberdeen, Ayr and Dundee. Biomathematics Statistics Scotland, James Clerk Maxwell Building, The King's Building, Edinburgh, EH9 3JZ Tel: 0131 650 4900 - Fax: 0131 650 4901 enquiries Biomathematics Statistics Scotland 2004. All rights reserved.
ISM
Institut des sciences mathmatiques, Montral, Canada.
Institut des sciences mathmatiques INSTITUT DES SCIENCES MATHEMATIQUES The Institut des sciences mathmatiques (ISM) is a consortium of the six Qubec universities that offer a Ph.D. programme in mathematics ( Concordia , Laval , McGill , the Universit de Montral , UQAM and the Universit de Sherbrooke ). Relying on a large community of university researchers, the ISM coordinates both the material and intellectual resources of its member departments to form a critical mass, making Montral and Qubec a North-American centre for training and research in mathematics. Twareque Ali from Concordia University currently directs the Institute. Information for: students postdoctoral fellows professors cegep teachers Member Departments: Concordia McGill Universit de Sherbrooke Universit de Montral UQAM Laval Contact us: Institut des sciences mathmatiques Universit du Qubec Montral Case postale 8888, succursale Centre-Ville Montral (Qubec) Canada H3C 3P8 email: ism@math.uqam.ca Civic address: 201, ave Prsident Kennedy, PK-5213 Montral, Qubec, H2X 3Y7 tel.: (514) 987-3000, extension 1811 fax: (514) 987-8935
Chennai Mathematical Institute, Chennai
Formerly SPIC Mathematical Institute.
Chennai Mathematical Institute We have moved to our new campus from October 2005. Home About People Teaching Activities Admission Details Library Contact Upcoming Events OPC '06 Events News IOI 2005 Convocation '05 For CMI-ites E-Mail Web-Forum Google Search Chennai Mathematical Institute is a centre of excellence for teaching and research in the mathematical sciences. Founded in 1989 as part of the SPIC Science Foundation, it has been an autonomous institute since 1996. The research groups in Mathematics and Computer Science at CMI are among the best known in the country. The Institute has nurtured an impressive collection of PhD students. In 1998, CMI took the initiative to bridge the gap between teaching and research in India by starting BSc and MSc programmes in Mathematics and allied subjects. Students who have graduated from CMI have gone on to join leading institutions throughout the world. CMI occupies a unique position in Indian academia, attracting substantial funding from both corporate and government sources. CMI's vision is to build on its early success and develop into a well-rounded academic institution, in the tradition of the best universities around the world. Search WWW Search cmi.ac.in
Institute for Mathematical Sciences
Based at the National University of Singapore.
Institute for Mathematical Sciences (NUS) Quick Links... ------------------------ About IMS Introduction Photos Location Official Openning ------------------------ People Management Board SAB Institute Staff Former SAB MB SAB short CVs Photos ------------------------ Programs Activities Current up-coming Past Activities ------------------------ Membership Application Forms ------------------------ Publications Lecture notes series IMS preprint series Published papers IMS newsletter Tutorials lecture notes ------------------------ Visitor Info Accommodations Getting to IMS ------------------------ Contact Us Feedback ------------------------ Related Links Local links Mathematical Institutes ------------------------ Figuring Out Life: NUS - Karolinska Joint Symposium on Application of Mathematics in Biomedicine (28 - 29 Nov 2005) Call for Pre-Proposals for Thematic Programs
Centre for Mathematical Physics and Stochastics (MaPhySto), Aarhus
Department of Mathematical Sciences, University of Aarhus.
Network in Mathematical Physics and Stochastics - MaPhySto MaPhySto The Danish National Research Foundation: Network in Mathematical Physics and Stochastics Funded by The Danish National Research Foundation Welcome to the web-pages of MaPhySto - the Danish National Research Foundation Network in Mathematical Physics and Stochastics. The Scientific Director of the Network is Arne Jensen . From the following links you may learn more about the Network and its activities. About MaPhySto Events People Publications News ( Latest ) Vacancies Visit MaPhyStos old pages . It is also possible to search our web-pages . MaPhySto Department of Mathematical Sciences University of Aarhus Ny Munkegade, DK-8000 Aarhus C, Denmark Email: matarne@math.aau.dk maphysto@imf.au.dk URL: www.maphysto.dk Phone: (+45) 9635 8846 Fax: (+45) 9815 8129
MRC
Mathematics Research Centre, Warwick, England, UK.
Warwick Mathematics Institute - MRC Search University | Contact Us | A-Z Index | GeneralInformation Admissions Undergraduate Postgraduate Research Events People University Maths Home MMIV webadmin@maths Mathematics Research Centre General Information Scientific Programmes Current Events Past Events Visitor Host Page (internal only) Administrative Contacts Director - Professor Miles Reid FRS Phone: 44 + (0)24 7652 3491 E-Mail: miles@maths.warwick.ac.uk Secretary - Ms Yvonne Collins Phone: 44+(0)24 7652 2681 E-Mail: mrc@maths.warwick.ac.uk Secretary - Ms Hazel Higgens Phone: 44 +(0)24 7652 8317 E-Mail: mrc@maths.warwick.ac.uk MRC Houses Phone Numbers House 1 +44 (0)24 76 419338 House 2 +44 (0)24 76 419242 House 3 +44 (0)24 76 418072 House 4 +44 (0)24 76 419076 House 5 +44 (0)24 76 418277 Flat 6a +44 (0)24 76 418571 Flat 6b +44 (0)24 76 418512 Address: Mathematics Research Centre University of Warwick Coventry CV4 7AL - UK Fax: +44 (0)24 7652 3548 Phone: +44 (0)24 7652 8317 E-Mail: mrc@maths.warwick.ac.uk How to get to Warwick About the MRC
RIMS
Research Institute for Mathematical Sciences, Kyoto, Japan.
Research Institute for Mathematical Sciences Japanese About RIMS [9 6] How to get to RIMS Staff Visitors [11 1] Graduate School Library Joint Research Service Computer Facilities Workshops(in Japanese) Workshops Seminars(in Japanese) [11 4] RIMS Preprint [11 8] Publications of RIMS Kkyroku(in Japanese) 21COE Program "Formation of an International Center of Excellence in the Frontiers of Mathematics and Fostering of Researchers in Future Generations" International Project Research 2005 Mathematics of the Navier-Stokes Equations and its Applications Mathematical Aspects and Applications of Nonlinear Waves ( October 26 - October 28, 2005 ) Mathematical Aspects of Complex Fluid and Its Application ( November 16 - November 18, 2005 ) Kyoto Conference on the Navier-Stokes Equations and their Applications ( January 6 - January 10, 2006 ) Fluid dynamics for mixing, chemical reaction and combussion ( January 11 - January 13, 2006 ) 2006 Arithmetic Algebraic Geometry 2006 Theoretical Effectivity and Practical Effectivity of Grbner Bases 2007 Mirror Symmetry and Topological Field Theory 1999 - 2004 Project Research RIMS | Workshops | Seminars | Staff | Preprints | What's new | Other Info MSJ | JSIAM | JPS | IPSJ Don't miss | IMU | EMIS | ICM 2006 Research Institute for Mathematical Sciences Kyoto University Kyoto, 606-8502 JAPAN FAX: +81-75-753-7272 Last modified: November 8, 2005
Sobolev Institute of Mathematics
Novosibirsk, Russia.
English Web- ... . 500. . .... .. . 1957. XX. ... , , . : (1912-1999) (1912-1986) (1928-1976) (1911-1973) (1909-1967) (1908-1989) (1921-1981) . . , : , ; ; , ; ; ; . , , . 2004 54 , 8 , 4 . 7 , .. , .. , .. , .. , .-. . . , - .. ... 2004 - . - . () . 17.11.2005 . , 18 003.015.02 ... . .. 15.11.2005 . , 18 11-00. - ... . .. 11.11.2005 . 15 2005 . 9.00 . " ". .. 09.11.2005 . . 310 01.11.2005 . C 1 3 2006 . XI - . .. 31.10.2005 . . 294 1 2 " HGNET ". : - (, , ), . . , .239. ... ..,4 630090 . : (383)333-28-92 : (383)333-27-93 : (383)333-09-96 : (383)333-25-93 : (383)333-25-98 e-mail: im@math.nsc.ru 2004, ... ,
Steklov Institute at St.Petersburg
The mathematics institute of Russian Academy of Sciences, consisting of 11 laboratories - mathematical physics, geometry and topology, number theory, algebra, mathematical analysis, mathematical physics, mathematical problems of physics, mathematical problems of statistical physics, statistical methods, representation theory and computing mathematics, mathematical problems of geophysics.
Petersburg Department of Steklov Institute of Mathematics
Institute of Mathematics
Mathematical Conference Center in Bedlewo, Poland.
Institute of Mathematics
Harish-Chandra Research Institute
(Formerly Mehta Institute of Mathematics and Mathematical Physics). Allahabad, India.
Harish-Chandra Research Institute HRI AboutHRI Mathematics Physics Visitors Opportunities Address Search Harish-Chandra Research Institute News in the month 2005-10 (2005-10-21) Teichmller Year : Jan 2005-Jan 2006 is a special year on "Teichmuller Theory and Moduli Problems". The International Workshop will take place in January 2006. Read more about the Teichmller year at HRI . (2005-10-21) Lie Theory Conference : International Conference on "Infinite Dimentional Lie Algebras and its applications" will be held at HRI between December 12, 2005 and December 17, 2005. Read more about the conference at HRI . (2005-10-07) Symposium on "String Theory: Basic Notions and Recent Developments" was held on October 1-2, 2005. This is a satellite symposium on the occasion of the platinum jubilee of the National Academy of Sciences, India. For the talks presented at the symposium click here . (2005-08-04) SERC Preparatory school will be held in HRI from November 6 -- 26, 2005. Read more about Preparatory SERC school at HRI . (2005-08-04) 9th Discussion Meeting on Harmonic Analysis will be held in HRI between 17th - 19th October 2005. Read more about 9th Discussion Meeting on Harmonic Analysis at HRI . (2005-03-15) Interviews for admission to the graduate programmes in mathematics : The list of JEST 2005 candidates who have been selected for an interview, for admission to the graduate programmes in mathematics at HRI, has been announced. Read more about the mathematics interviews at HRI . The Institute Harish-Chandra Research Institute (HRI) is an institution dedicated to research in mathematics, and in theoretical physics. It is located in Allahabad, India, and is funded by the Department of Atomic Energy, Government of India. Read more about HRI. Mathematics at HRI The mathematics group at HRI conducts research in algebra, algebraic geometry, analysis, Lie algebras, number theory, and topology. Read more about mathematics at HRI . Physics at HRI Physics at HRI involves research in theoretical physics, specifically, in astrophysics, condensed matter theory, particle physics, and string theory. Read more about physics at HRI . Visitors HRI welcomes visitors in mathematics, physics, and allied sciences. We have a strong visitor programme, and, in addition, we have exchange programmes with several institutions in India and abroad. If you are interested in visiting HRI, please contact mathvisit at mri dot ernet dot in for mathematics, and physvisit at mri dot ernet dot in for physics. Meanwhile, read more about visiting HRI . Information An accompanying document contains information on conferences, seminars, and other activities in mathematics . Also available is information on conferences, and other activities in physics . Information on facilities such as the Institute library catalogue (accessible only from within the HRI network), and the computing environment at HRI are also available in the facilities section of this Web site. Webmail If you are a member of the Institute, you can use Webmail from outside HRI by visiting either the external ERNET Webmail server , or the external BSNL Webmail server . To use Webmail from within HRI, please visit either the internal ERNET Webmail server , or the internal BSNL Webmail server . HRI AboutHRI Mathematics Physics Visitors Opportunities Address Search Harish-Chandra Research Institute Chhatnag Road , Jhusi Allahabad 211019 , India Phone: +91(532)2667510, 2667511, 2668311, 2668313, 2668314 Fax: +91(532)2567748, 2567444 About this page ; HRI Webadmin webadmin at mri dot ernet dot in Updated: 2005-10-21T13:50:25Z
MPI for Mathematics in the Sciences (MIS)
Max-Planck-Institute for mathematics in the sciences, Leipzig.
Max Planck Institute for Mathematics in the Sciences Max Planck Institute for Mathematics in the Sciences The Institute Home AbouttheInstitute Research Groups People IMPRS Research Involvements Weekly talks Conferences Publications Public Relations Open positions WheretofindtheMPI Guest House Impressum Secure login (SSH2) MPI MIS Webmail WWW Other Max-Planck-Institutes Partner Institute for Computational Biology Department of Mathematics and Computer Science IZBI (Interdisciplinary Centre for Bioinformatics) University of Leipzig City of Leipzig Mathematical Sites Physics Sites Chemistry Sites Biology and Biomathematics Sites MPG PhD Student Network New arrivals FrankPasemann (01.11.) Fraunhofer Institut (Germany) Today's lectures (17.11.2005) 15:15, room A 01 Lutz Habermann: Symplectic Yang-Mills theory, Ricci tensor and connections (see Abstract) 16:45, room A 01 Marco Khnel: On harmonic maps and complex structures Next conference 22nd GAMM-Seminar Leipzig on Large Scale Eigenvalue Computations Leipzig, January 19 - 21, 2006 Next lectures 22.11.2005, 11:15, room A 01 Luca Mugnai: On a conjecture of De Giorgi concerning the approximation of the Willmore Functional 23.11.2005, 11:00, room A 01 Arleta Szkola: Algorithmic Complexity in Classical and Quantum Information Theory - Part I 23.11.2005, 14:00, room A 01 Rainer Siegmund-Schultze: Algorithmic Complexity in Classical and Quantum Information Theory - Part II (see Abstract) 23.11.2005, 16:15, room G 10 Ellen Kuhl: Continuum biomechanics - pantha psiloni (see Abstract) 24.11.2005, 15:15, room Universitt Leipzig, Felix-Klein-Hrsaal (Raum 4-24) Hubert Schwetlick: Altes und Neues ber Newton-Techniken bei Eigenwertproblemen (see Abstract) Info about the Editor: Impressum Webmaster, E-Mail: webmaster mis.mpg.de Last updated: 07. November 2005
Alfrd Rnyi Institute of Mathematics
Hungarian Academy of Sciences, Budapest.
Rnyi Institute The Institute General information Contact Directions Positions Studia Documents (in Hungarian) vegzseb Library Math Links People Research divisions Management Research fellows Young researchers, post-docs, graduate students Staff Honorary and external members Visitors Activities Conferences (new old) Seminars (in Hungarian) European projects FIST BudAlgGeo DiscConvGeo PHD PhD program Announcements Knots, contact structures and foliations Nov 17-20 FoIKS 2006 Feb 14-17 Conference on Lattice Theory 2006 Jun 6-9 For Members WebMail: Read your E-mail Intraweb Information on research grants (in Hungarian) Information on OTKA grants (in Hungarian) Intzmnynk orszgos s nemzetkzi hlzati kapcsolatt az NIIF Program biztostja
Razmadze Mathematical Institute
Georgian Academy of Sciences.
Razmadze Mathematical Institute Mathematical research in Georgia began immediately after the opening of Tbilisi State University (1918). The initiators of the research were the first Georgian scientists in mathematics A.Razmadze , N.Muskhelishvili , A.Kharadze and G.Nikoladze . On October 8, 1933 a research institute of mathematics, physics and mechanics with N.Muskhelishvili as director was set up under Tbilisi State University. On October 1, 1935, at the initiative of N.Muskhelishvili and his closest associates V.Kupradze and I.Vekua , the mathematics and mechanics section of the above-mentioned institute was transformed into a mathematical research institute under the auspices of the Georgian Branch of the USSR Academy of Sciences. The institute was included into the Georgian Academy of Sciences after the latter was founded in February, 1941. In 1944 the institute was granted the name of A. Razmadze. Along with N.Muskhelishvili , V.Kupradze and I.Vekua , to the first generation of mathematical researchers of the institute belong A.Bitsadze , V.Chelidze , G.Chogoshvili , D.Dolidze , L.Gokieli, A.Gorgidze, A.Kalandia , I.Kartsivadze , A.Kharadze , B.Khvedelidze , D.Kveselava, G.Lomadze, L.Magnaradze , K.Marjanishvili, Sh.Mikeladze , A.Rukhadze, N.Vekua , A.Walfisz . Their fundamental results on the mathematical theory of elasticity [ 28, 48-56, 71, 72, 74, 75, 77, 80, 91, 97-100 ], singular integral equations [ 73, 76, 78, 79, 81, 82, 101-103 ], complex and real analysis [ 7, 8, 31-33, 93, 95, 96 ], differential equations and mathematical physics [ 2-5, 44-47, 70, 83, 90, 92, 94 ], topology [ 10-14 ], the theory of numbers [ 104-108 ] and computational mathematics [ 60-68 ] won the institute a high scientific reputation worldwide. At various times, with institute collaborated the outstanding foreign scientists P. S. Aleksandrov, S.Bergman, S.N.Bernstein, M. V. Keldysh, M. A. Lavrentyev, V. I. Smirnov and S.L.Sobolev. The first director of the institute was V.Kupradze (1935-1941). Throughout 1941-1976 the institute almost unintermittently was headed by N.Muskhelishvili . From September, 1951 to April, 1952 the acting director of the institute was V.Chelidze . In 1976-1989 the institute was directed by N.Vekua , and since 1989 its director has been I.Kiguradze . The institute's deputy directors for scientific research were I.Vekua (1940-1941), A.Gorgidze (1941-1954), G.Manjavidze (1954-1977) and T.Burchuladze (1977-1989). The present deputy director for scientific research is V.Kokilashvili who has been holding this post since 1989. At various times, the research departments of the institute were headed by S.Bergman, S.Bernstein, V.Chelidze , G.Chogoshvili , T.Gegelia , A.Kalandia , R.Kapanadze , A.Kharadze , E.Khmaladze, B.Khvedelidze , V.Kupradze , L.Magnaradze , G.Mania , M.Mikeladze, Sh.Mikeladze , N.Muskhelishvili , T.Shervashidze , O.Tsereteli , I.Vekua , N.Vekua , A.Walfisz . Presently, the institute has 9 departments: Department of Algebra (head H. Inassaridze ) Department of Geometry and Topology (head T. Kadeishvili ) Department of Mathematical Analysis (head V. Kokilashvili ) Department of Differential Equations (head I. Kiguradze ) Department of Mathematical Physics (head R. Duduchava ) Department of the Theory of Elasticity (head R. Bantsuri ) Department of Theoretical Physics (head A. Tavkhelidze ) Department of Probability Theory and Mathematical Statistics (head T. Toronjadze ) Department of Scientific Information (head M. Balavadze ) The scientific secretary of the institute is N. Partsvania . The institute has the Scientific Certification Board (chairman I. Kiguradze ) which awards the academic degrees of Doctor and Candidate of Physical and Mathematical Sciences in the following specialities: mathematical analysis, differential equations, mathematical physics. Since 1937 the institute has been publishing the scientific journal "Proceedings of A.Razmadze Mathematical Institute" . The institute is the founder of two international scientific journals: "Georgian Mathematical Journal" and "Memoirs on Differential Equations and Mathematical Physics" . As from 1997, the Symposium on Differential Equations and Mathematical Physics (DEMPh) is annually held at the institute - 2003 Georgian Fonts
Argonne National Laboratory
Mathematics and computer science division.
Welcome to MCS Mathematics and Computer Science Division Search: ( Details ) Research in the MCS Division at Argonne National Laboratory is funded principally by the Mathematical, Information, and Computational Sciences Division , Office of Advanced Scientific Computing Research , Office of Science of the U.S. Department of Energy . Our mission is to increase scientific productivity in the 21st century by providing intellectual and technical leadership in the computing sciences -- computer science, applied computational mathematics, and computational science. Open Positions MCS has a variety of positions available. This site is updated frequently. What's New The University of Chicago is spearheading a $148 million expansion of the TeraGrid . Hans Kaper has received a $1 million grant from DOEs Office of Science to study a multiscale approach to self-organization of microtubules. The research is funded under the Office of Sciences Multiscale Mathematics program. MPICH2 has received an RD 100 award. The software, developed by Bill Gropp, Rusty Lusk, Rob Ross, Rajeev Thakur, and Brian Toonen, is a high-performance, portable implementation of community standards for the message-passing model of parallel computation. For more information, see the Argonne Web site. MCS computer scientist Rob Ross has received the Presidential Early Career Award for Scientists and Engineers in recognition of his contribution to the advancement of science. The Presidential Award is the highest honor bestowed by the U.S. government on outstanding scientists and engineers who are beginning their independent careers. Details are online. Mark Hereld and Rick Stevens have won a Best Paper Award at the PROCAMS Workshop. Their paper is entitled Pixel Aligned Warping for Multiprojector Tiles Displays. Virtual meetings have come of age with the Access Grid, according to a recent article in the UCAR quarterly. Globus Toolkit v. 4 has been released. GT4.0 is the product of close to two years' work by a distributed team, many of whom work for Argonne. This work has produced a software system with first-class usability, performance, documentation, and functionality. For more information, see the Web site. Paul Fischer of Argonne National Laboratory and his colleagues Fausto Cattaneo and Aleksandr Obabko at the University of Chicago have received a DOE INCITE award of 2 million hours of supercomputing time to study how stars and solar systems form. Fischer and his group will use the award to develop large-scale simulations on the Seaborg system at the National Energy Research Scientific Computing Center (NERSC). More Research From virtual reality to multimedia projects, from logical reasoning programs to Grand Challenge Applications -- these pages will introduce you to the many exciting research projects going on at MCS. Also take a quick overview of our expanding efforts in Computational Biology. Resources Here you will find descriptions of and links to our various "laboratories" and research centers, including the Distributed Systems Laboratory, Futures Lab, Laboratory for Advanced Numerical Software, Optimization Technology Center, and Regional Climate Center. For a history of computing at Argonne, see our Web site. For an overview of computational science activities, see our Web page. For information about the Jazz computer, start with the Web site on the Laboratory Computing Resource Center. You might also be interested in our work on petaflops computing. Computing The division's computing environment, consisting of nearly 2,000 different computers, is based on a network of diverse workstations including Linux, Windows, and PCs for administrative and database purposes; portable computing devices (laptops); home computing devices; and local area network communications gear. We also operate 3 large Linux clusters for high-end computational science and computer science research. These Web pages are intended principally for MCS Division members. New users should start here for instructions about getting an account or for information on our software and machines. Here too you may find information about and answers to FAQs. People Our division has more than 80 staff members, including postdoctoral researchers, students, and visitors. This alphabetical listing will help you contact someone here quickly. Collaboration The key to many new discoveries in science and engineering is collaboration. Here you will find links to numerous joint projects we are working on and the people we are working with. Software We are strongly committed to developing and diffusing our software into the user community. Here we provide a listing of our software products, together with information about documentation, new releases, and source files. Publications Our primary "product" -- in addition to software -- is articles in journals, books, conference proceedings, and technical reports. For easy reference we have abstracted and archived (in postscript form) our publications of the past several years. Information These pages include recent accomplishments, press announcements, and employment opportunities at Argonne and MCS. Also included here are directions and a map to help you if you are visiting MCS. Suggestions or comments on these pages are welcome. [ MCS | Research | Resources | People | Collaboration | Software | Publications | Information ] Last updated on August 22, 2005 Disclaimer Security Privacy Notice webmaster@mcs.anl.gov
PIMS
Pacific Institute for the Mathematical Sciences, Canada.
Home - Pacific Institute for the Mathematical Sciences Home PIMS Home About PIMS Contact Us PIMS Offices Activities Seminars BIRS Collaborative Research Groups Scientific Industrial Education Proposals and Nominations Announcements Publications and Videos PIMS only PIMS Postdoc Day On Saturday, October 29, PIMS-UBC hosted the annual PIMS Postdoc Day, which was jointly organized with the University of Washington VIGRE Program. Participants at the PIMS Postdoc Day The event provided much needed information for the professional development of postdocs; topics which were discussed at length included teaching and mentoring research connections industrial connections job applications interview skills Read more... Pacific Rim Mathematical Forum, Mini-Symposium and Reception On October 14-15, 2005, the Pacific Institute for the Mathematical Sciences and the Mathematical Sciences Research Institute (Berkeley) jointly hosted the Pacific Rim Mathematical Forum at the Banff International Research Station. The goal of this meeting was to lay the groundwork for establishing a network of mathematical centres throughout the Pacific Rim. Read more... PIMS Industrial Forum 2006 The Pacific Institute for the Mathematical Sciences is pleased to announce the dates for the 9th PIMS Graduate Industrial Mathematical Modelling Camp (June 21-25, 2006) and the 10th PIMS Industrial Problem Solving Workshop (June 26-30, 2006). Both events will take place at Simon Fraser University. Read more... Canada-Mexico Meeting The Mexican Mathematical Society (Sociedad Matematica Mexicana, SMM) will be hosting the first joint CMS-SMM special session at its annual meeting in Mexico City, on October 25, 2005. The scheduled events include a plenary lecture by Gordon Slade (UBC) on Critical Oriented Percolation as well as invited lectures by Thomas Salisbury (York Fields, CMS President-Elect) and Alejandro Adem (PIMS Deputy Director). The session will also include lectures by Mexican colleagues. Read more... Recent Additions PIMS Graduate Industrial Math Modelling Camp 2005 Proceedings more... PIMS Magazine, Fall 2005 Issue more... Video of Ben Green's lecture on Arithmetic Progressions on Primes more... Videos from 2004 PIMS SFU Computing Science Distinguished Lectures 2004 more... JUMP: Summer Training Sessions at PIMS more... Read more... [ Back ] Recent Publications and Videos 2005 GIMMC Proceedings PIMS Magazine Fall 2005 Issue Dan Rudolph, PIMS Dist. Chair Videos IAM-PIMS-MITACS Distinguished Colloquia 2004-05 Videos PIMS 2003-04 Annual Report PIMS Education Day Videos Pi in the Sky BIRS News BIRS Receives Record Funding from Alberta, the US and Mexico Multimedia and Mathematics at BIRS BIRS 2006 Calendar BIRS 2005 Pacific Institute for the Mathematical Sciences. All rights reserved. Terms and Conditions Credits Contact Us
Newton Institute
The Isaac Newton Institute for Mathematical Sciences, Cambridge, UK.
Isaac Newton Institute for Mathematical Sciences homepage News Visitor List Mailing Lists Search Feedback Useful Links Sir David Wallace appointed as next Director from 1 October 2006 The Isaac Newton Institute for Mathematical Sciences is a national and international visitor research institute. It runs research programmes on selected themes in mathematics and the mathematical sciences with applications in a very wide range of science and technology. It attracts leading mathematical scientists from the UK and from overseas to interact in research over an extended period. Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, U.K. Tel.: +44 1223 335999, Fax.: +44 1223 330508 Email: info@newton.cam.ac.uk Seminars This Week Next Week Monday Seminars Workshops and Events Seminars on the Web INI Activity in the UK Research Meetings Elsewhere Current programmes 1. Pattern Formation in Large Domains 2. Global Problems in Mathematical Relativity Participant Information General Information Booklet Closure Periods How to get to the Institute Housing Information Library and Resources Computing
Erwin Schrdinger Institute for Mathematical Physics
Vienna, Austria.
ESI The Erwin Schrdinger International Institute for Mathematical Physics Supported by: member of: and
The Organic Mathematics Project
Based at the Centre for Experimental and Constructive Mathematics (partly supported by Simon Fraser University, the Centre de recherches mathmatiques (CRM), the TeleLearning Research Network, the University of Western Ontario, and by Waterloo Maple) involving the development of WWW-based and Maple-assisted tools for document annotation and math activation.
Organic Mathematics: Technical Issues The Organic Mathematics Project - Mission Statement - Project Context The OMP is a collaborative research project involving researchers at the Centre for Experimental and Constructive Mathematics (partly supported by Simon Fraser University and the CECM, by the Centre de recherches mathmatiques (CRM), the TeleLearning Research Network , by University of Western Ontario , and by Waterloo Maple ) and in-part involves the development of WWW-based and Maple-assisted tools for document annotation and math activation. Portions of this project are supported by Waterloo Maple Software Corp. , MathActive, Inc. and the Canadian Mathematical Society . The Organic Mathematics Workshop is the culmination of the first phase of the Multi Modal Mathematics Plexus project. This project is funded over a 3.5 year period as part of the Network Centres of Excellence - Telelearning based out of Simon Fraser University . Organic Mathematics Workshop Dec. 12 - 14, 1995 Description Some details of the three day workshop on Electronic and Virtual Mathematics Workshop Schedule A draft schedule of the talks and activities Proceedings These proceedings are considered a work in progress Project members and sponsors A list of individuals and sponsoring organizations Invited speakers These authors spoke at the Workshop and contributed papers to the Proceedings Software Development Generic Maple Form Interface A Web-based cgi-bin script interface to a standard Maple engine Sample Activated Mathematics Text Activated text which uses developed technologies How to Use Activated Mathematics The problem of Pascal's Triangle Inverse Symbolic Calculator An interactive interface to a large collection of numerical reverse engineering algorithms Related Documents and Events The History and Philosophy of Mathematics A descriptive collection of references and resources that define the context within which experimental mathematics (and these Proceedings) appear. Included are several articles written by invited authors. Technology Transfer to Port Moody Secondary School The Organics Mathematics project will be moving to the real world in the coming months, experiments with the techology in a high school context in cooperation with SchoolNet . Special Session on Computational and Experimental Mathematics at the CMS 50th Anniversary Winter Meeting Dec. 9-11, 1995 International Conference on the Learning Sciences A Call for Papers July 24-27, 1996
MSRI
Mathematical Sciences Research Institute, Berkeley, CA, USA.
Mathematical Sciences Research Institute - Home Page ACTIVITIES AT MSRI Calendar Programs Workshops Summer Graduate Workshops Seminars Events Announcements Past Projects Math Circles BAMO PROPOSALS APPLICATIONS Application Materials Visa Information Propose a Program Propose a Workshop Policy on Diversity ALUMNI DEVELOPMENT MSRI Alumni Archimedes Society Why Give to MSRI Ways You Can Give to MSRI Donate to MSRI Planned Gifts Frequently Asked Questions ABOUT MSRI Mission and Governance Staff Member Directory Contact Us Directions For Visitors Pictures Library Computing Scientific Graphics Project COMMUNICATIONS Streaming Video Lectures MSRI in the Media The Emissary Newsletter Outlook Electronic Newsletter Subscribe to Newsletters Books, Preprints Videotapes SUPPORT SPONSORS Federal Support Corporate Affiliates Sponsoring Publishers Foundation Support Academic Sponsors SITE MAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SEARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SHORTCUT: Choose a Destination... Calendar Programs Workshops Summer Grad Workshops Seminars Events Announcements Residence Program Math Circles BAMO Application Materials Visa Information Propose a Program Propose a Workshop Policy on Diversity MSRI Alumni Archimedes Society Why Give to MSRI Ways to Give to MSRI Donate to MSRI Planned Gifts FAQ Mission Governance Staff Member Directory Contact Us Directions For Visitors Pictures Library Computing SGP Video Lectures MSRI in the Media Emissary Newsletter Outlook Newsletter Subscribe to Newsletters Books, Preprints, etc. Federal Support Corporate Affiliates Sponsoring Publishers Foundation Support Academic Sponsors Upcoming Workshops November 14,2005 to November 18,2005 Optimal Mass Transport and its Applications November 18,2005 to November 22,2005 Flavors of Groups November 28,2005 to December 02,2005 Geometric and Analytical Aspects of Nonlinear Dispersive Equations December 05,2005 to December 09,2005 Probability, Geometry and Integrable Systems January 09,2006 to January 20,2006 Stringy Topology in Morelia Events Friday, February 24 The Art of Problem Solving Announcements MSRI Closed for Maintenance May 30th, 2006 through June 9th, 2006 Now Available: Survey of Putnam Exam Top Finishers, by Steve Olson IMPORTANT: MSRI Planned Web Site and E-mail Outages - Nov. 20, Nov.28-Dec 2. VMath Special Production - New Horizons in Undergraduate Mathematics: Grbner Bases with Bernd Sturmfels MSRI is Moving Back Mailing Information and Driving Directions MSRI Launches a New Web Site MSRI's Capital Campaign Please Support Chern Hall and MSRI's Capital Campaign for Building Expansion and Renovation The Right Spin Live Construction Web Cam Banff International Station 2005 2006 Calendars are now available Support MSRI Give Now to the Annual Fund and Join the Archimedes Society
Institut Mittag-Leffler
The Royal Swedish Academy of Sciences.
Institut Mittag-Leffler General Information Background, history, programs, funding. Present program (2005 fall): Wave Motion List of visitors Seminars Future programs 2006 spring: Algebraic Topology 2006 2007: Moduli Spaces 2007 fall: Stochastic Partial Differential Equations 2008 spring: Complex Analysis of Several Variables Grants 2006 2007 Scholarship grants for recent PhDs and advanced graduate students. (Deadline January 31, 2006) Call for proposals for the year 2008 2009 (Deadline February 1, 2006.) EPDI postdoc program The European Postdoctoral Institute grants postdoc scholarships in a European setting. Information for Visitors Please note: This information is directed to persons who have been invited to participate in a program at the Institute. Participation in the programs is by invitation only. Publications The Institute publishes the Acta Mathematica and the Arkiv fr matematik . Both journals are partially available on-line. RIP program Shorter stays at the Institute, outside the main program, by Scandinavian mathematicians. Specialarbeten i matematik fr gymnasiet. Staff Computers Preprints Library Gallery Links Internal Institut Mittag-Leffler, Auravgen 17, S-182 60 Djursholm, Sweden, Tel. +46 8 622 05 60, Fax. +46 8 622 05 89
Mathematisches Forschungsinstitut
Oberwolfach, Germany.
MFO This webpage uses frames, a feature that your browser does not support. You may contact for general information and for technical issues.
LaCIM
Laboratoire de combinatoire et d'informatique mathmatique, Universit du Qubec, Montral, Canada.
LACIM: Laboratoire de Combinatoire et d'Informatique Mathmatique You must use a browser that can display frames to see this page.
Institute for Mathematics and its Applications (IMA)
The IMA is a federally funded US institute which promotes the use of mathematics in other fields both in academia and in industry.
Welcome to the Institute for Mathematics and its Applications (IMA) About the IMA What's Happening Programs Activities Publications Visitor Information Contact Information People Application Forms Program Feedback Talk Materials Program Solicitation Room Reservations Join our Mailing Lists Search The IMA was founded by and receives major support from the National Science Foundation Division of Mathematical Sciences and the University of Minnesota to carry out a crucial interdisciplinary mission . It also receives support and direction from its Participating Institutions and Participating Corporations . Hot Topics Summer Programs Math Matters Industrial Programs New Directions PI Programs Seminars Annual Program: Imaging , September 1, 2005-June 30, 2006 Workshop: Integration of Sensing and Processing , December 5-9, 2005 Upcoming Events: IMA MCIM Industrial Problems Seminar: David Arathorn (General Intelligence Corporation), The Map-seeking Method: From Cortical Theory to Inverse-Problem Method , November 18, 2005 Variational PDE Based Image Reconstruction and Processing Seminar : Changfeng Gui (University of Connecticut), A New Level Set Method for Image Segmentation Without Reinitialization Math Matters - Public Lecture: Philip Holmes (Princeton University), Does Math Matter to Brain Matter? , December 8, 2005 Inverse Problems Seminar : Ahmed Tewfik (Department of Electrical Engineering, University of Minnesota), Adaptive Radar Detection and Imaging , December 12, 2005 News: IMA seeks Associate Director starting August 2006 job announcement NSF announces record funding for the IMA The IMA has been awarded a $19.5 million renewal grant by the National Science Foundation for the period 2005-2010, the largest single research investment in mathematics ever made by NSF. NSF math director William Rundell made the announcement at the IMA on July 20. Press coverage: U of M Star Tribune SIAM News MPR UPI IMA featured on Science and Society . On August 31, IMA Director Douglas Arnold discussed the IMA and interdisciplinary mathematical research on World Talk Radio. Windowsmediastream Podcast(mp3) Air Force Research Laboratory has joined the IMA as a Participating Institution. Medtronic has joined the IMA as a Participating Corporation. Future Annual Programs: Applications of Algebraic Geometry , September 2006-June 2007 Mathematics of Molecular and Cellular Biology , September 1, 2007-June 30, 2008 Comments or questions: web(at)ima.umn.edu Privacy The University of Minnesota is an equal opportunity educator and employer. 2005 Regents of the University of Minnesota. All rights reserved.
IHES
Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France.
Institut des Hautes Etudes Scientifiques
ICMS
International Centre for Mathematical Sciences, Edinburgh, UK.
ICMS Front Page Click here if these pages are trapped inside another site's frames. Welcome to the International Centre for MathematicalSciences. Call for proposals 2006 07 ICMS Welcomes proposals in all aspects of mathematical sciences and in interdisciplinary areas with significant mathematical content. ICMS aims to react rapidly to proposals and can arrange smaller workshops in 6-8 months from acceptance. See Call for Proposals for full details. Forthcoming deadlines: 30 November 2005 for proposals on which a decision will be made in March 2006 31 March 2006 for proposals on which a decision will be made in July 2006 Workshop Reports 2005 ICMS 14 India Street, Edinburgh EH3 6EZ, UK Tel: +44 (0) 131 220 1777 Fax: +44 (0) 131 220 1053 enquiries@icms.org.uk Access at 14 India St 14 India Street is the birthplace of James Clerk Maxwell and is owned by the James Clerk Maxwell Foundation . Future Events | Travel Information | Call for Proposals | Publications Previous Events | Useful Links | About ICMS | Front Page SITE NEWS - November 2005 ICMS has a new logo. Consequently the style of this website isunder review. The appearance of some pages will change over the next few months prior to a completely new site being launched. Site created 22 October 1999 Address http: www.icms.org.uk If you have any comments on, or problems with this site please email Madeleine Shepherd at ICMS, who designed and constructed this site.
IBM T. J. Watson Research Center
Headquarters of the research division of International Business Machines corporation. Research focuses primarily on physical and computer sciences, semiconductors, systems technology, mathematics and information services, applications solutions.
IBM Research | Watson Research Center | Home Page Country region [ change ] Terms of use All of IBM Home Products Services solutions Support downloads My account IBM Research Home Watson Research Visitor Information History Cambridge Feedback Related Links Careers at Research Other Labs Local Education Outreach Project List IBM Journal of R D IBM Systems Journal EPA Performance Track Report for Yorktown Watson Research Center Welcome To The T.J. Watson Research Center The Watson Research Center -- Yorktown The Watson Research Center -- Hawthorne The Watson Research Center -- Cambridge The IBM Thomas J. Watson Research Center is the headquarters for the IBM Research Division -- the largest industrial research organization in the world with eight labs worldwide. Established in 1961, the Watson Research Center is located in Westchester County, New York and Cambridge, Massachusetts and spans three sites and four buildings -- the main laboratory in Yorktown Heights, two buildings in Hawthorne, and one building in Cambridge. An approximate 1,790 people are employed between these four facilities. The research focuses primarily on IT hardware (ranging from exploratory work in the physical sciences to semiconductors and systems technology); software (including areas as diverse as security, programming, mathematics and speech technologies); and services, with a focus on applying them to transform businesses in a wide range of industries. Yorktown Info Directions Hawthorne Industry Solutions Lab Info Directions Cambridge Info Directions Lodging Dining Entertainment Hospitals Transportation EPA Performance Track Report for Yorktown More pictures of: Yorktown Hawthorne Cambridge About IBM | Privacy | Legal | Contact
IAS
School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA.
School of Mathematics - Home 1 Einstein Drive Princeton, New Jersey 08540 US Tel: 609-734-8100 Fax: 609-951-4459 math@math.ias.edu Home Activities Park City Seminars Programs Women and Mathematics Administration Mission Contact Us How to Apply Local Info People Faculty Members Staff Publications Annals of Mathematics Video Lectures QFT Services Computing Library Links Webmail Special Programs Theoretical Computer Science and Discrete Mathematics School of Mathematics 75th Anniversary Celebration Women and Mathematics Programs Geometry and Arithmetic,A Conference on the Occasion of the Sixty-First Birthday of Pierre Deligne (October 17, 2005 - October 20, 2005) Lie Groups, Representations and Discrete Matematics Workshop (November 14, 2005 - November 18, 2005) Holomorphic Curves Focus Group (May 22 - June 19, 2005) Special Years Bloch-Kato Conjecture (2004-2005 Academic Year) Lie Groups, Representations and Discrete Mathematics (2005-2006 Academic Year) Algebraic Geometry (2006-2007 Academic Year) Seminars SEE AGENDA Various Speakers 11 14 2005 (9:30am - 4:50pm) S-101 NO SEMINAR 11 14 2005 (11:15am - 12:15pm) Copyright 2005 Institute for Advanced Study webmaster@math.ias.edu
Fields Institute
Toronto, Ontario, Canada.
Fields Institute for Research in Mathematical Sciences Promoting research, education and training, and cooperation with business November17,2005 Home About Us Programs Activities Proposals Applications Prizes Honours People Contacts Mailing List Audio Slides Publications Sponsors Fundraising Information for Visitors Mathematical Institutes Societies Search Royal Canadian Institute, Science Lectures - Jean Taylor (November 20) Coxeter Lecture Series - Lai-Sang Young (November 23-25) Thematic Program on Renormalization and Universality in Mathematics and Mathematical Physics (August-December 2005) To see all our activities, visit our Calendar of Events Phone: 416-348-9710 Business Fax: 416-348-9714 Members Fax: 416-348-9385 The Fields Institute for Research in Mathematical Sciences 222 College Street Toronto, Ontario M5T 3J1 Canada E-Mail: geninfo@fields.utoronto.ca
Euler Institute
Euler International Mathematical Institute, St. Petersburg, Russia.
EIMI: main page Euler International Mathematical Institute contact information History Previous Events Current Events Forthcoming Events Euler Institute is a member of ERCOM Welcome to St.Petersburg! The Euler IMI is a part of Petersburg Department of Steklov Institute of Mathematics
IRMA (Strasbourg)
Institut de Recherche Mathmatique Avance, Strasbourg, France.
IRMA Strasbourg Institut de Recherche Mathmatique Avance de Strasbourg IRMA - UMR 7501 CNRS ULP - 7 rue Ren Descartes - 67084 Strasbourg Cedex, FRANCE Tl : (33) [0]3 90 24 01 29 - Fax : (33) [0]3 90 24 03 28 Sites miroirs Mirror sites MathSciNet American Mathematical Society MathSciNet Mirror Site En collaboration avec l'Universit Louis Pasteur et son Service commun de Documentation ( SCD-ULP ), 34 Bd de la Victoire, BP 1037, 67070 Strasbourg Cedex. Tl. : (33) 03 88 45 02 45 Zentralblatt MATH Database Heidelberger Akademie der Wissenschaften, FIZ Karlsruhe, Springer Verlag, European Mathematical Society Zentralblatt MATH Database Mirror Site EMIS European Mathematical Society European Mathematical Information Service Mirror site
DIMACS Center for Discrete Mathematics
Center for Discrete Mathematics and Theoretical Computer Science, Rutgers, New Jersey. Fosters theoretical computer science publications, research, discussion and workshops.
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) A collaborative project of: Rutgers, The State University of New Jersey Princeton University ATT Labs - Research Bell Labs Telcordia Technologies NEC Laboratories America Affiliate Members: Avaya Labs Georgia Institute of Technology HP Labs IBM Research Microsoft Research Rensselaer Polytechnic Institute Stevens Institute of Technology What's New at DIMACS Surf the DIMACS Site Alphabetical Index of Site Workshop Calendar Contacting the Center Call for Suggestions Search the DIMACS Web Pages
Centrum voor Wiskunde en Informatica (CWI)
National Research Institute for Mathematics and Computer Science. The National Research Institute for Mathematics and Computer Science Amsterdam.
Welcome to the Centrum voor Wiskunde en Informatica CWI, founded in 1946, is the national research institute for Mathematics and Computer Science in the Netherlands. CWI performs frontier research in mathematics and computer science and transfers new knowledge in these fields to society in general and trade and industry in particular. News Faster and cheaper adaptation of source code with new language technology Better methods to guarantee software quality
CRM (Montreal)
Centre de recherches mathmatiques, Montreal, Canada.
Centre de recherches mathmatiques - page d'accueil Bottin | Nous joindre | Archives | Messagerie Web | WebSSH | Abonnez-vous | English propos du CRM Activits scientifiques par chronologie par catgorie Chaires Aisenstadt Publications Bourses postdoctorales Prix Renseignements pour les visiteurs Partenaires Bulletins, rapports annuels et communications mathmatiques Mathmatiques, arts et socit Intranet Anne thmatique en analyse en thorie des nombres 2005-2006 Programmes thmatiques courts 2005-2006 Programme multidisciplinaire et industriel 2005-2006 Programme national sur les structures de donnes complexes (activits au CRM) Autres ateliers et activits coles d't Colloque CRM-ISM en mathmatiques Colloque de statistique CRM-ISM-GERAD Proposition de projets webmestre
Centre de Recerca Matemtica (CRM)
Mathematical research center in Barcelona. Centre d'investigaci creat per tal de millorar la recerca matemtica a Catalunya. El Centre de Recerca Matemtica (CRM) t com a objectiu proporcionar als matemtics catalans un institut de recerca que pugui contribur a la millora de la recerca en matemtiques a Catalunya.
Centre de Recerca Matemtica
Banach Center
Stefan Banach International Mathematical Center, Warsaw, Poland.
Stefan Banach International Mathematical Center
ATT Labs
ATT Laboratories, Murray Hill, NJ.
ATT Labs Research ATT Home | ATT Labs ATT Labs - Research Who We Are About Us Innovators In the Community Our People Join Us What We Do Research Areas Projects Software Tools Licensing Portfolio Patents Collaboration Universities Industry News and Information In the News Publications Site Feedback Site Search Enter keyword: ATT Labs Research is one of the premier telecommunications laboratories in the world in terms of excellence in the frontiers of science, invention of new communications concepts and tools, and incubation of new services. It builds on over 80 years of leadership across all areas of communications and related fields, and continues to produce innovative technology that differentiates ATTs services and operations. Some recent innovations include: Software and Design Techniques supporting ATT's CallVantage VoIP Service Traffic Analysis System for IP Networks of ATT, customers, and a cross system carrier mode messaging and alerting service Very large scale (hundreds of terabytes) data management technology provided by Daytonatm Recent Technological Contributions from ATT Labs Research 2005 Creation of ATT Traffic Analysis Service (TAS) tools addressing 24x7 network-wide IP traffic analysis and leveraging Daytonatm scalable data warehouse technology. 2005 Successful field trials ofpre-standard WiMax equipment supporting broadband fixed wireless access to ATT customers. 2005 Creation of innovative IP multicast network management tools to support industry-leading proactive and reactive management for ATTs emerging IP multicast services. 2005 Daytonatm provides the data management for Hawkeye, a call detail database, where it is managing over 1.92 trillion records and 312 terabytes of information. more ... Featured Project Featured Publication The `Robust Yet Fragile' Nature of the Internet 2005 Winner Announced! Winter Top Ten Contest Hawkeye wins Grand Prize for largest normalized volume database! Data management for Hawkeye is provided by Daytona tm . ATT Wins Technology Emmy! ATT received theTechnology Emmy for their leadership role in launching Telstar, the first intercontinental satellite transmission. Other Featured Projects ATT VoiceTone EDGE - Enablement and Debugging of Growing Enterprises Grid Computing Technology Interactive Software for Statistical Visualization Featured Publications Analysis and Design of ATT's Global PNNI Network Authenticated System Calls Bandwidth Constraints Models for MPLS Traffic Engineering Optimizing Network Performance in Replicated Hosting The Impact of Web Services Integration on Grid Performance Recent Patents Hybrid Fiber Twisted Pair Local Loop Network Service Architecture Subscription-Based Priority Interactive Help Services On The Internet System For Bandwidth Extension Of Narrow-Band Speech Legal Notice Copyright 2005 ATT All rights reserved.
AML (Courant)
Applied Mathematics Laboratory, Courant Institute, New York University, NY.
AML Welcome to the Courant Institute Applied Mathematics Laboratory AML Crew Summer 2005 Kathleen Mareck Bin Liu Jin-Qiang Zhong Tom Bringley Erica Kim Steve Childress Mike Shelley Lionel Rosellini Jun Zhang Gonzague de la Hautiere Karishma Parikh Sunny Jung About the AML A Research Projects Sampler People Support AML Seminar AML Publications and Reports AML ViSLab The Applied Mathematics Laboratory is supported by the Department of Energy and the National Science Foundation. Contact: AML, Courant Institute, 251 Mercer St., New York, N.Y., 10012 phone: 212-9983239 fax: 212-9953639 email: amlatcims.nyu.edu
(Netherlands) Euler Institute for Discrete Mathematics and its Applications (EIDMA)
Discrete Algebra and Geometry, Coding Theory, Information Theory and Cryptology, Combinatorial Optimization and Algorithms, Graph Theory.
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International Mathematical Science Institutes
An international consortium of research institutes in the mathematical sciences that run thematic research programs and have large visitor programs. IMSI meetings are normally held in conjunction with major international meetings and provide a forum for developing cooperation among the member institutions.
Fields Institute - Aboutus IMSI ABOUT THE FIELDS INSTITUTE November17,2005 Home About Us Programs Activities Proposals Applications Prizes Honours People Contacts Mailing List Audio Slides Publications Sponsors Fundraising Information for Visitors Mathematical Institutes Societies Search International Mathematical Science Institutes IMSI is an international consortium of research institutes in the mathematical sciences that run thematic research programs and have large visitor programs. IMSI meetings are normally held in conjunction with major international meetings and provide a forum for developing cooperation among the member institutions. IMSI meetings have been held at the International Congress of Mathematicians (ICM) in Zurich (1994), ICIAM meeting in Hamburg (1995) and ICM in Berlin (1998). Recent IMSI gatherings were scheduled at the meeting of the European Mathematical Society in Barcelona 2000 and the ICM in Beijing in 2002 . The next IMSI meeting will be at the ICM in Madrid in 2006 . American Institute of Mathematics (Palo Alto, California, USA) American Mathematical Society (AMS) (Providence, Phoda Island, USA) Australian Mathematical Sciences Institute (AMSI) (Victoria, Australia) Banff International Research Station for Mathematical Innovation and Discovery (Banff, BC, Canada) Centre de Recerca Matemtica (CRM) (Bellaterra, Spain) Centre de Recherches Mathmatiques (CRM) (Montral, Qubec, Canada) Centre for Mathematics and its Applications (Canberra, Australia) Center for Scientific Computation And Mathematical Modeling (CSCAMM) (College Park, MD, USA) Centre International de Rencontres Mathmatiques (CIRM) (Marseille, France) Clay Mathematics Institute (Cambridge, Massachusetts, USA) DIMACS (AT T Labs) (Rutgers University, Piscataway, New Jersey, USA) Erwin Schrdinger Institute for Mathematical Physics (Vienna, Austria) Euler Institute (Affiliation of Steklov Institute in St. Petersburg) (St. Petersburg, Russia) EURANDOM (Eindhoven, The Netherlands) The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada) Forschungsinstitut fr Mathematik (FIM) (Zurich, Switzerland) Institute for Advanced Study (Princenton, New Jersey, USA) Institute for Mathematical Sciences (IMS) (Singapore) Institute for Mathematics and its Application (IMA) (Minneapolis, Minnesota, USA) Institute for Pure and Applied Mathematics (IPAM) (Los Angeles, California, USA) Institute of Mathematical Sciences, The Chinese University of Hong Kong (Hong Kong) Institute of Mathematical Statistics (Beachwood, Ohio, USA) Institut de Mathmatiques de Luminy (IML) (Marseille, France) Institut des Hautes tudes Scientifiques (IHS) (Bures-sur-Yvette, France) Institut Henri Poincar (Paris, France) Institut Mittag-Leffler (Djursholm, Sweden) Instituto Nacional de Matemtica Pura e Aplicada (IMPA) (Rio de Janiero, Brazil) International Banach Center (Warsaw, Poland) International Center for Mathematical Sciences (ICMS) (Edinburgh, UK) International Centre of Theoretical Physics (ICTP) (Trieste, Italy) International Mathematical Union (IMU) (Princeton, New Jersey, USA) Isaac Newton Institute for Mathematical Sciences (Cambridge, England) Istituto Nazionale di Alta Matematica (INdAM) (Rome, Italy) Johann Radon Institute for Computational and Applied Mathematics (RICAM) (Linz, Austria) MaPhySto - Centre for Mathematical Physics and Stochastics (Aarhus, Denmark) Mathematical Biosciences Institute (MBI) (Columbus, Ohio, USA) Mathematical Sciences Research Institute (MSRI) (Berkeley, California, USA) Mathematisches Forschungsinstitut Oberwolfach (Oberwolfach-Walke, Germany) Max-Planck-Institute for Mathematics in the Sciences (Leipzig, Germany) Max-Planck-Institut fr Mathematik (Bonn, Germany) Nankai Institute of Mathematics (Tianjin, China) New Zealand Institute of Mathematics and its Applications (NZIMA) (Auckland, New Zealand) Pacific Institute for the Mathematical Sciences (PIMS) (Vancouver, BC, Canada) Research Institute for Mathematical Sciences (RIMS), Kyoto University (Kyoto, Japan) Statistical and Applied Mathematical Sciences Institute (SAMSI) (North Carolina, USA) Steklov Mathematical Institute of the Russian Academy of Sciences (Moscow, Russia) Tata Institute of Fundamental Research (Mumbai, India) Warwick Mathematics Institute, University of Warwick (Coventry, England) Weierstrass Institute for Applied Analysis and Stochastics (Berlin, Germany)
Romanian Academy
Institute of Mathematics.
Institute of Mathematics of the Romanian Academy INSTITUTE OF MATHEMATICS "SIMION STOILOW" OF THE ROMANIAN ACADEMY Research Workgroups Events Exchanges Links Regional Cooperations Conferences Contact Webmaster
AIM
American Institute of Mathematics, Palo Alto, CA.
AIM: American Institute of Mathematics Information: Background News Governance Sponsors and Affiliates Staff ARCC: Upcoming Workshops Outlines of open problems Participant resources AIM Activities: Preprints submitting Conferences Projects REU Resources Outreach talks posters Library reprints books 5-year Fellowship Visitors American Institute of Mathematics 360 Portage Ave Palo Alto, CA 94306-2244 Contact Information Directions to AIM Associate Director AIM seeks a person to serve as Associate Director of the AIM Research Conference Center(ARCC) for a two-year position beginning mid-2006. Please see the announcement for details. Gaps between primes The next ARCC workshop will take place November28to December2. Other upcoming workshops: The Modeling of Cancer Progression and Immunotherapy , December12toDecember16. Moduli spaces of knots , January3toJanuary7. Random analytic functions , January16toJanuary20. The property of rapid decay , January23toJanuary27. ARCC: AIM Research Conference Center Design for Morgan Hill Facility View of the main entrance: Visit the ARCC homepage for more information.
CECM
Centre for Experimental and Constructive Mathematics, Simon Fraser University, Canada.
Welcome to CECM AboutUs Awards Computing Facilities Contact Info Employment Logo VisitorInformation Members Permanent Members Associate Members Visitors Students Past Members Partners Industry Agencies Events CECM Computer Algebra Discrete Math Number Theory IRMACS MITACS PIMS Research Symbolic Computation Complexity Issues Comp. Classical Analysis Comp. Number Theory Mathematical Visualization Interfaces QuickLinks Computer Algebra Discrete Math Number Theory Computing Facilities Department of Math Canadian Math Society IRMACS MITACS PIMS TextSize: Upcoming Events CECM Colloquium October 14th, 2005 at 3:30pm in K9509. Dominique Orval, Ecole Polytechnique de Montreal, will speak on The interpretation of nonlinear interior methods as damped Newton methods. schedule Computer Algebra Seminar The computer algebra group meetings are regularly scheduled on Fridays. schedule Discrete Mathematics Seminar The discrete mathematics seminars are regularly scheduled on Tuesdays. schedule CECM Day 05 Computational Mathematics August 3, 2005 at the IRMACS Centre. CECM Day info Centre for Experimental and Constructive Mathematics Shrum Science Building P8495 Simon Fraser University 8888 University Drive Burnaby BC V5A 1S6 Canada p604.291.5617 f604.291.5614 Best viewed with Safari , Firefox and other standards compliant browsers.
ArXiv Front: SG Symplectic Geometry
Symplectic geometry section of the mathematics e-print arXiv.
SG Symplectic Geometry Thu 17 Nov 2005 Search Submit Retrieve Subscribe Journals Categories Preferences iFAQ SG Symplectic Geometry Calendar Search Authors: All AB CDE FGH IJK LMN OPQR ST U-Z New articles (last 12) 17 Nov math.SG 0511418 On some completions of the space of Hamiltonian maps. Vincent Humilire (CMLS). SG ( AP ). 16 Nov math.SG 0511385 Locally holomorphic maps yield symplectic structures. Robert E. Gompf . 10 pages. Comm. Anal. and Geom. 13 (2005), 511-525. SG . 9 Nov math.SG 0511194 Symplectic connections. Pierre Bieliavsky , Michel Cahen , Simone Gutt , John Rawnsley , Lorenz Schwachhofer . SG ( DG ). 2 Nov math.SG 0511015 On deformations of Hamiltonian actions. Lucio Bedulli , Anna Gori . 12 pages. SG . Cross-listings 8 Nov math.AG 0511163 Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform. Tamas Hausel . 8 pages. AG ( CO RT SG ). 7 Nov math.GT 0511101 A note on symplectic rational blow--downs. Andras I. Stipsicz , Zoltan Szabo . 20 pages. GT ( SG ). 7 Nov math.GT 0511097 The Bennequin number, Kauffman polynomial, and ruling invariants of a Legendrian link: the Fuchs conjecture and beyond. Dan Rutherford . 17 pages. GT ( SG ). 2 Nov math.DG 0511013 Reduction and submanifolds of generalized complex manifolds. Izu Vaisman . 33 pages. DG ( SG ). Revisions 17 Nov math.SG 0210103 Toward a topological characterization of symplectic manifolds. Robert E. Gompf . 32 pages. J. Symp. Geom. 2 (2004), 177-206. SG . 11 Nov math.SG 0505561 The Maslov index as a quadratic space. Teruji Thomas . 20 pages. SG ( RT ). 10 Nov math.SG 0506615 Homological orthogonality of "symplectic" and "lagrangian"- corrected version. Nik. A. Tyurin . 4 pages. SG . 1 Nov math.SG 0510172 Finiteness of Hofer-Zehnder symplectic capacity of neighborhoods of symplectic submanifolds. Guangcun Lu . 30 pages. SG ( DG ). Recent Calendar 2005 111+95 November 4+4 October 8+5 September 9+8 August 12+6 July 8+7 June 13+18 May 13+12 April 9+6 March 14+11 February 11+7 January 10+11 2004 146+138 December 12+14 November 14+17 October 11+13 September 15+10 August 10+3 July 13+8 June 9+14 May 16+8 April 16+10 March 14+14 February 7+13 January 9+14 2003 117+124 December 15+5 November 7+13 October 18+14 September 14+8 August 12+8 July 12+15 June 4+9 May 6+14 April 9+8 March 10+12 February 6+7 January 4+11 2002 101+125 December 7+7 November 9+11 October 12+14 September 6+13 August 8+13 July 9+9 June 6+8 May 8+8 April 10+12 March 7+11 February 7+8 January 12+11 2001 74+99 December 6+7 November 7+8 October 10+7 September 3+5 August 3+12 July 8+6 June 10+10 May 4+10 April 6+8 March 4+10 February 5+7 January 8+9 2000 59+120 December 8+5 November 5+14 October 10+16 September 5+14 August 6+11 July 5+8 June 1+7 May 4+11 April 4+7 March 2+7 February 6+12 January 3+8 1999 45+75 December 3+10 November 6+7 October 2+5 September 4+3 August 2+3 July 6+6 June 5+5 May 4+12 April 5+5 March 2+11 February 3+4 January 3+4 1998 21+60 December 1+14 November 4+7 October 5+8 September 4+4 August 1+6 July 3+9 June 1+3 May 1 April 0+4 March 0+1 February 1+2 January 0+2 1997 0+3 November 0+1 June 0+1 February 0+1 1995 0+2 October 0+1 August 0+1 1994 0+3 September 0+1 June 0+2 1993 1+7 December 1+3 August 0+1 June 0+1 March 0+1 January 0+1 1992 0+2 October 0+2 Total: 675+853 articles (primary+secondary) Search Author Title ID Anywhere Cat MSC articles per page Show Help AC AG AP AT CA CO CT CV DG DS FA GM GN GR GT HO KT LO MG MP NA NT OA OC PR QA RA RT SG SP ST Authors: All AB CDE FGH IJK LMN OPQR ST U-Z Home Search Submit Retrieve Subscribe Journals Categories Preferences iFAQ - for help or comments about the Front - for help about submissions or downloading arXiv articles
ArXiv Front: MG Metric Geometry
Metric geometry section of the mathematics e-print arXiv.
MG Metric Geometry Thu 17 Nov 2005 Search Submit Retrieve Subscribe Journals Categories Preferences iFAQ MG Metric Geometry Calendar Search Authors: AB CDE FGH IJK LMN OPQR ST U-Z New articles (last 12) 14 Nov math.MG 0511306 Multiple planar coincidences with N-fold symmetry. Michael Baake (Bielefeld), Uwe Grimm (Milton Keynes). 13 pages. MG ( CO ). 14 Nov math.MG 0511289 Energy and length in a topological planar quadrilateral. Sa'ar Hersonsky . 9 pages. MG ( CO ). 8 Nov math.MG 0511171 Theory of valuations on manifolds, IV. Further properties of the multiplicative structure. Semyon Alesker . 40 pages. MG . 8 Nov math.MG 0511147 Planar coincidences for N-fold symmetry. Peter A. B. Pleasants (Brisbane), Michael Baake (Bielefeld), Johannes Roth (Stuttgart). 38 pages. J. Math. Phys. 37 (1996) 1029-1058. MG ( CO ). 4 Nov math.MG 0511071 The one-sided kissing number in four dimensions. Oleg R. Musin . 10 pages. MG ( CO ). Cross-listings 10 Nov math.GR 0511231 A logarithm law for automorphism groups of trees. Sa'ar Hersonsky , Frederic Paulin . 10 pages. GR ( MG ). 1 Nov math.GT 0510666 On a Conjecture of Milnor about Volume of Simplexes. Ren Guo , Feng Luo . 8 pages. GT ( MG ). 28 Oct math.FA 0510612 Approximating orthogonal matrices by permutation matrices. Alexander Barvinok . 18 pages. FA ( MG ). 27 Oct math.FA 0510547 Nonembeddability theorems via Fourier analysis. Subhash Khot , Assaf Naor . FA ( MG ). Revisions 17 Nov math.MG 0505065 The Brascamp-Lieb inequalities: finiteness, structure, and extremals. Jonathan Bennett , Anthony Carbery , Michael Christ , Terence Tao . MG ( CA ). 8 Nov math.MG 0406305 Polygons in buildings and their refined side lengths. Michael Kapovich , Bernhard Leeb , John J. Millson . 18 pages. MG ( DG ). 26 Oct math.MG 0509512 Theory of valuations on manifolds, III. Semyon Alesker , Joseph H. G. Fu . 33 pages. MG ( DG ). Recent Calendar 2005 66+65 November 5+1 October 5+9 September 3+7 August 10+5 July 4+4 June 12+10 May 7+5 April 3+7 March 7+8 February 7+3 January 3+6 2004 100+64 December 11+5 November 7+8 October 9+5 September 4+8 August 5+6 July 13+5 June 14+5 May 7+6 April 10+4 March 7+3 February 7+7 January 6+2 2003 64+53 December 8+4 November 7+6 October 2+3 September 5+9 August 8+5 July 5+3 June 6+3 May 7 April 3+2 March 2+5 February 4+8 January 7+5 2002 44+45 December 8+5 November 4+7 October 5+2 September 2 August 4+2 July 3+2 June 2+3 May 4+5 April 2+7 March 6+6 February 1+3 January 3+3 2001 31+21 December 3+2 November 4+1 October 6+2 September 1+5 August 4+1 July 2+1 June 3+5 May 1+1 April 1+1 March 3 February 2+1 January 1+1 2000 20+39 December 0+3 November 2+8 October 5+3 September 1+5 August 2+6 July 0+1 June 2+2 May 1+4 April 0+2 March 0+3 February 4+1 January 3+1 1999 25+18 December 1+2 November 1 October 1+2 September 2+4 August 1+1 July 2+1 June 6 May 4+1 April 3+3 March 2 February 2+1 January 0+3 1998 20+11 December 0+4 November 10 October 1 September 2+3 August 1 July 1+1 June 2+2 May 0+1 April 2 February 1 1997 5+7 December 1+2 October 0+1 July 2 June 1 March 0+2 January 1+2 1996 7+2 September 1 August 1 May 4+1 March 1 January 0+1 1995 8+2 November 1 October 1 September 0+1 August 2 May 2 March 1 January 1+1 1994 2+5 November 0+2 October 0+2 May 1 April 1 March 0+1 1993 3 October 1 July 1 February 1 1992 10+2 November 1 October 3+1 April 1 January 5+1 Total: 405+334 articles (primary+secondary) Authors AB A Akian, Marianne Alon, Noga Aramyan, Rafik Arora, Sanjeev Alesker, Semyon Andreev, P. D. Arcozzi, Nicola Asanov, G. S. Alexandrov, Victor B Baake, Michael Barthe, Franck Bernig, Andreas Borcea, Ciprian Bailey, Benjamin Aaron Barvinok, Alexander Bernstein, Joseph Bosse, Hartwig Baillif, Mathieu Batakis, Athanasios Betke, Ulrich Boutin, Mireille Balacheff, Florent Batal, Yair Bezdek, Karoly Bowen, Lewis Ball, Keith Bazylevych, L. E. Blekherman, Grigoriy Brandenberg, Rene Balser, Andreas Beguin, Francois Block, Florian Bronnimann, Herve Banaszczyk, W. Belolipetsky, Michael Bogaevsky, Ilya A. Brudnyi, A. Barany, Imre Benjamini, Itai Bokowski, Juergen Brudnyi, Yu. Barbot, Thierry Bennett, Jonathan Boll, David W. Buliga, Marius Barre, Sylvain Bergman, George M. Bonk, Mario Burger, Marc Bartal, Yair CDE C Camenga, Kristin A. Chang-jian, Zhao Chouikha, A. Raouf Conway, John Cameron, P. J. Charney, Ruth Christ, Michael Corneli, Joseph Candes, Emmanuel Chebotarev, P. Yu. Cohn, Henry Cortes, C. Carbery, Anthony Chen, Wenxiong Connelly, Robert D Davydkin, Ivan Deza, Antoine Donovan, Jerry Dubejko, Tomasz Derevnin, Dmitriy Deza, M. Dougherty, Randall Dumitrescu, Adrian Develin, Mike Deza, Michel Dragomir, Sever Silvestru Dutour, M. Devillers, Olivier Dilworth, S. J. Dranishnikov, Alexander Dutour, Mathieu E Eastwood, Michael Elkies, Noam Erdahl, Robert Eremenko, A. Ebbers-Baumann, Annette Eppstein, David Erdahl, Robert M. Everett, Hazel Edmonds, Allan L. FGH F Felikson, A. Fillastre, Francois Fradelizi, Matthieu Friedrichs, Olaf Delgado Felikson, Anna Foertsch, Thomas Frettloh, Dirk Fu, Joseph H. G. Ferguson, Samuel P. G Gang-song, Leng Giacomini, Hector Goodman--Strauss, Chaim Grima, C. I. Gao, Fuchang Giannopoulos, Apostolos A. Goodman-Strauss, Chaim Grimm, Uwe Gardner, Richard J. Gichev, V. M. Gordon, Yehoram Grishukhin, Viacheslav Gaubert, Stephane Glickenstein, David Graham, R. L. Groetschel, Martin Geoghegan, Ross Goldengorin, Boris Graham, Ronald Grotschel, Martin Getmanenko, Alexander Goodman, Jacob E. Graham, Ronald L. Grune, Ansgar H Hajja, Mowaffaq Hemkemeier, Boris Hindawi, Mohamad A. Hougardy, Stefan Hales, Thomas C. Henkel, Oliver Hodgson, Craig D. Howard, Ralph Harrell, Evans M. Henk, Martin Hoffman, Chris Hruska, G. Christopher Hasto, Peter A. Heppes, Aladar Holmsen, Andreas Hug, Daniel Hausel, Tamas Hersonsky, Sa'ar Holton, Charles Hurtado, F. Heitzig, Jobst Hillar, Christopher J. Holt, Paul Huson, Daniel H. IJK I Iozzi, Alessandra Iseri, Howard Ivanshin, P. N. J Jazar, Mustapha Joseph, Dieter Joswig, Michael K Kahn, Jeff Kenyon, Richard Kloeckner, Benoit Kubis, Wieslaw Kaibel, Volker Kharchenko, Alexander Koldobsky, Alexander Kumar, Abhinav Kalai, Gil Klartag, B. Kolountzakis, Mihail N. Kuperberg, Greg Kapovich, Michael Kleiner, Bruce Konyagin, Sergei Kuperberg, Krystyna Karlsson, Anders Klein, Rolf Kopteva, Natalia Kuperberg, Wlodzimierz Kemper, Gregor Kleitman, Daniel J. Koufany, Khalid Kutateladze, S. S. Kennedy, Tom LMN L Laba, Izabella Lazard, Sylvain Lenz, Daniel Lubachevsky, Boris D. Lafont, J.-F. Leeb, Bernhard Lindenbergh, Roderik Lubachevsky, R. L. Graham B. D. Lafont, Jean-Francois Lee, George Lindenbergh, Roderik C. Ludwig, Monika Lagarias, J. C. Lee, James R. Linial, Nathan Lutwak, Erwin Lagarias, Jeffrey C. Lee, Jeong-Yup Litvak, A. E. Lutz, Frank H. Lalin, Matilde Leger, J. C. Lonke, Yossi Lytchak, Alexander Lang, Urs Leger, Nicholas Lubachevsky, B. D. Lytschak, Alexander M Makai Jr., Endre Matousek, Jiri Mendel, Manor Mineyev, Igor Malesevic, Branko J. Maurey, Bernard Mendel, Nathan Linial. Manor Moody, Robert V. Maley, F. Miller McCuan, John Mescheryakov, E. A. Morbidelli, Daniele Mallows, C. L. McLaughlin, Sean Meyer, Mathieu Morgan, Frank Mallows, Colin L. McMullen, Peter Miller, Ezra Morozov, O. S. Marquez, A. Meckes, Mark W. Millson, John J. Moukadem, Nazih Martin, Greg Mednykh, Alexander Milman, V. D. Murakami, Jun Martini, Horst Melleray, Julien Milman, Vitali D. Musin, Oleg R. Martyushev, E. V. N Nagano, Koichi Nguyen-Khac, V. NguyenVan, K. Nowak, Piotr W. Naor, Assaf NguyenKhac, V. Norbury, Paul Nussbaum, Roger D. Nazarov, A. I. Nguyen-Van, K. OPQR O Ontaneda, Pedro Orden, David P Pach, Janos Pak, Igor Petrov, F. V. Pollack, Ricky Paffenholz, Andreas Panteleeva, E. Pfeifle, Julian Poonen, Bjorn Paiva, Juan Carlos Alvarez Pasechnik, Dmitrii V. Pfetsch, Marc E. Prassidis, S. Pajor, A. Peixoto, M. M. Pleasants, P. A. B. Praton, Iwan Pajor, Alain Penrose, Roger Pleasants, Peter A. B. Pretorius, Lou M. R Radin, Charles Richter-Gebert, Jurgen Roskies, Julie Rudelson, Mark Ranestad, Kristian Rieck, Yo'av Rote, Guenter Rybnikov, Konstantin Reisner, Shlomo Rivin, Igor Rote, Gunter Rybnikov, Konstantine Reitzner, Matthias Robbins, David P. Roth, Johannes Rylov, Yuri Repetowicz, Przemyslaw Roberts, James W. Rubin, Matatyahu Rylov, Yuri A. Rhea, Darren L. Rocha, A. C. Rubinstein, Michael Ryshkov, Serge\ui S. ST S Sadun, Lorenzo Schulte, Egon Shtogrin, M. I. Steinhurst, Benjamin Sandoval-Villalbazo, A. Schutt, Carsten Siersma, D. Stillinger, F. H. Santos, F. Semmes, Stephen Siersma, Dirk Streinu, Ileana Santos, Francisco Sergeev, Sergei Simanyi, Nandor Sturmfels, Bernd Schewe, Lars Servatios, Brigitte Sing, Bernd Sulanke, Rolf Schick, David Meintrup Thomas Servatius, Herman Smith, Warren D. Sullivan, John M. Schlichenmaier, Thilo Shabot-Marcos, N. Sodin, Sasha Sutherland, S. Schlumprecht, Thomas Shamis, E. V. Sosov, E. N. Svrtan, Dragutin Schmitt, Peter Shchepin, Evgeny Sottile, Frank Swanepoel, Konrad Schoenfeld, Eric Shen, Zhongmin Soufi, Ahmad El Swanepoel, Konrad J. Schramm, Oded Shinya, Hisanobu Spencer, Joel Szarek, Stanislaw J. Schroeder, Viktor Shor, Peter W. Spietz, Lafe Szucs, Andras Schuermann, Achill Shtogrin, M. Springborn, Boris A. T Tao, Terence Theobald, Thorsten Torquato, S. Tumarkin, P. Tapp, Kristopher Thurston, William P. Toth, Gabor Fejes Tumarkin, Pavel Tessera, R. Timorin, Vladlen Tsolomitis, Antonis Tyszka, Apoloniusz Tetenov, Andrew U-Z U Ushijima, Akira V Valenzuela, J. van Manen, Martijn Veljan, Darko Vitale, Richard A. Vallentin, Frank Varilly, Anthony Vershik, A. M. Volenec, Vladimir Valtr, Pavel Vaughn, Rick Vitale, R. A. von Renesse, Max-K. van Manen, M. Veerman, J. J. P. W Walsh, Cormac Werner, Elisabeth Wilks, A. R. Wills, Joerg M. Wang, Yang Whiteley, Walter Wilks, Allan R. Winchester, Adam Webster, Corran Whitesides, Sue X Xie, Xiangdong Y Yan, C. H. Yang, Deane Yetter, David N. Yu, Josephine Z Zamfirescu, Tudor Zeghib, Abdelghani Zelke, Mariano Ziegler, Gunter M. Zarichnyi, M. M. Zeghig, Abdelghani Zhang, Gaoyong Zvavitch, Artem Search Author Title ID Anywhere Cat MSC articles per page Show Help AC AG AP AT CA CO CT CV DG DS FA GM GN GR GT HO KT LO MG MP NA NT OA OC PR QA RA RT SG SP ST Authors: All AB CDE FGH IJK LMN OPQR ST U-Z Home Search Submit Retrieve Subscribe Journals Categories Preferences iFAQ - for help or comments about the Front - for help about submissions or downloading arXiv articles
Pick's Theorem
Describes a theorem for finding an area of simple lattice polygons. Includes proof and Java applet simulation.
Pick's Theorem: An Interactive Activity Username: Password: Sites for teachers Sites for parents Terms of use Awards Interactive Activities CTK Exchange Games Puzzles Arithmetic Algebra Geometry Probability Eye Opener Analog Gadgets Inventor's Paradox Did you know?... Proofs Math as Language Things Impossible My Logo Math Poll Cut The Knot! MSET99 Talk Other Math sites Front Page Movie shortcuts Personal info Reciprocal links Privacy Policy Guest book News sites Recommend this site Sites for teachers Sites for parents Wholesale Shopping Health Information Online Student Loan Help Networking Software Management Training Courses Cut The Knot! An interactive column using Java applets by Alex Bogomolny Pick's Theorem May 1998 Georg Alexander Pick, born in 1859 in Vienna, perished around 1943 in the Theresienstadt concentration camp. [ 9 ] First published in 1899, the theorem was brought to broad attention in 1969 through the popular Mathematical Snapshots by H. Steinhaus. The theorem gives an elegant formula for the area of simple lattice polygons, where "simple", as usual, only means the absence of self-intersection. Polygons covered by the theorem have their vertices located at nodes of a square grid or lattice whose nodes are spaced at distance 1 from their immediate neighbors. The formula does not require math proficiency beyond middle grade school and can be easily verified with the help of a geoboard . Pick's Theorem Let P be a lattice polygon. Assume there are I(P) lattice points in the interior of P, and B(P) lattice points on its boundary. Let A(P) denote the area of P. Then A(P) = I(P) + B(P) 2 - 1 The most illuminating proof comes from [15]. The applet below is an online variant of the common geoboard. To create a vertex click next to a lattice node. It dose not matter which node you choose. You'll be able to drag an existent vertex to any other node later on. Edges are added automatically. The new node is always inserted between the very first and last vertices. Intersecting edges are shown in red. (The applet uses an adaptation of a scan conversion algorithm from [13]. The book is replete with ideas. It just appears that this one was not worked out completely.) With Pick's theorem one may determine area of a (polygonal) portion of a map. On a transparent paper draw a grid to scale and superimpose the grid over the map. Count the number of nodes inside and on the boundary of the map region. Apply Pick's formula with the selected scale. More importantly, there are links to several other beautiful results. Pick's formula is equivalent to the celebrated Euler's formula [7]. It also implies the basic property of the Farey Series. The Farey series FN of order n is the ascending sequence of irreducible fractions m n between 0 and 1 whose denominators do not exceed N. A fraction m n belongs to FN iff 0 m n N, gcd(m,n) = 1, where gcd(m,n) is the greatest common divisor of m and n. For example, F5 is 0 1, 1 5, 1 4, 1 3, 2 5, 1 2, 3 5, 2 3, 3 4, 4 5, 1 1 Farey series is characterized by two wonderful equivalent properties . G.H. Hardy who prided himself in not having done anything "useful", found it worthwhile to include three different proofs of the basic property of the Farey series in his and E.M. Wright's book. (This is a classical work with the Index located not at the end of the book but, in a contemporary manner, elsewhere on the Web. You'll have to look hard to find it there as the url has been changing.) The sequence of denominators of terms in the Farey series is palindromic . The proof may not be immediately obvious. But, as is often the case, having a bigger picture proves useful. The Farey series are embedded into the Stern-Brocot tree for which this property comes almost for free. The area measurement application of Pick's theorem I mentioned above comes from the real world experience. Grnbaum and Shepard quote D.W.DeTemple who attended a presentation on application of mathematics in the forest industry: Although the speaker was not aware that he was essentially using Pick's formula, I was delighted to see that one of my favorite mathematical results was not only beautiful, but even useful. I am rather curious whether the forester shared in the delight. There is no surprise in that mathematics is useful. Even G.H.Hardy will be remembered in part because of the Hardy-Weinberg law which became centrally important in the study of many genetic problems [6]. I am charmed by the title of an undergraduate text, Applied Abstract Algebra (R.Lidl and G.Pilz, Springer-Verlag, 1997, 2nd edition.) What would Hardy say? The goal of course is to pass the delight on. References A.H.Beiler, Recreations in The Theory of Numbers , Dover, 1966 M.Bruckheimer and A.Arcavi, Farey Series and Pick's Area Theorem, The Mathematical Intelligencer v 17 (1995), no 4, pp 64-67. J.Cofman, Numbers and Shapes Revisited , Clarendon Press, 1995 J.Conway and R.Guy, The Book of Numbers , Copernicus, 1996 H.S.M.Coxeter, Introduction to Geometry , John Wiley Sons, NY, 1961 Encyclopdia Britannica W.W.Funkenbusch, From Euler's Formula to Pick's Formula Using an Edge Theorem, The Am Math Monthly v 81 (1974), pp 647-648 R. Graham, D. Knuth, O. Patashnik, Concrete Mathematics , 2nd edition, Addison-Wesley, 1994. B.Grnbaum and G.C.Shepard, Pick's Theorem, The Am Math Monthly v 100(1993), pp 150-161 G.H.Hardy, A Mathematician's Apology , Cambridge University Press, 1994. G.H.Hardy and E.M.Wright, An Introduction to the Theory of Numbers , Oxford University Press, Fifth Edition, 1sshu996 R.Honsberger, Ingenuity in Mathematics , MAA, 1970 T.Pavlidis, Algortihms for Graphics and Image Processing , Computer Science Press, 1982 H.Steinhaus, Mathematical Snapshots , Dover, 1999 D.E.Varberg, Pick's Theorem Revisited, The Am Math Monthly v 92(1985), pp 584-587 On Internet Euler's Formula, Proof 10: Pick's Theorem Farey Sequence Farey Sequence Farey Series From the sci.math newsgroups Geoboards in Classroom Pick's Theorem Pick's Theorem Pick's Theorem on three dimensional regular rectangular solids Stern-Brocot Tree Pick's Theorem Exercises for the Geoboard Pick's Theorem, A Proof Farey Series Pick's Theorem Applies to the Farey Series Stern-Brocot Tree Applications of Pick's Theorem Copyright 1996-2005 Alexander Bogomolny 15490008 Search: All Products Apparel Baby Beauty Books DVD Electronics Home Garden Gourmet Food Personal Care Jewelry Watches Housewares Magazines Musical Instruments Music Computers Camera Photo Software Sports Outdoors Tools Hardware Toys Games VHS Computer Games Cell Phones Keywords: Google Web CTK Latest on CTK Exchange probability Posted by tennyson 6 messages 03:13PM, Aug-29-05 The number 0.142857..... Posted by Zakatos 2 messages 10:54AM, Nov-05-05 SSA Postulate Posted by Allison 1 messages 02:23PM, Nov-16-05 Gift Exchange probability Posted by Owen 10 messages 11:27PM, Nov-16-05 A self-referencing sentence in us ... Posted by Paul R. 1 messages 09:42PM, Nov-16-05 Complex number solutions Posted by Owen 2 messages 10:16AM, Nov-14-05
Sliceform Models by John Sharp
Three-dimensional objects constructed by slotting together a series of planar cross sections made from card. Downloadable templates in PDF, PS and Word formats.
Maths Year 2000 Sliceforms Section 1 - Introduction What are Sliceforms? Ten minute Sliceforms More advanced Sliceforms Who invented them? Sliceforms book and poster Section 2 How to make and assemble the models Section 3 - Sliceform templates and download instructions Adobe Acrobat Word 6 EMF and EPS files Section 4 - Sliceform Gallery A Sliceform tetrahedron A Sliceform zonohedron A Sliceform hyperbolic paraboloid A Sliceform super-egg These Sliceforms are copyright 1999-2000 John Sharp. You are allowed to use the templates to make models for your own use and for teaching purposes, but not to use any of the models for commercial purposes.
Zome Edu
Educational toys used to teach geometry and model building to create squares, spheres, stars and other regular shapes with sticks and connecting pieces. Products include kits and lesson plans.
Zome - Educators Customer Service FAQ Shopping Cart Zome is used in over 6,000 schools and institutions across the US. It is used in grades 1 through the university level to help teach algebra, scale, number sense, symmetry, proportion, geometry, DNA structure, trigonometry, and more. Zome is based on a version of the 61-zone structural system which includes purely mathematical models from tilings to hyperspace projections, as well as molecular models of quasicrystals and fullerenes, and architectural space frame structures. Zome's balls (nodes) and struts let you build models within the 61-zone system which lifts the study of these objects off the flat page and literally puts it into the hands of students. In classrooms across the globe, teachers are taking advantage of the unique benefits Zome brings to the study of math, art, science, and technology. The high-quality modeling of Zome makes it easier and more enjoyable for students to learn, and increases their capacity for abstract reasoning through tactile and visual feedback. Students understand new concepts better, and find them more interesting. Zome products consist of... Individual Parts: Purchase quantities of a particular Zome piece. Kits: Choose a kit providing a large assortment of different Zome construction parts or a kit which has been thoughtfully put together to allow the construction of models which focus on a particular math or science topic. Literature: Zome lesson plans are available free as a download from this website or purchasable in printed form. A learning DVD is available as well as various books convering a wide range of science and math application. Bundles: Multiple kits grouped together or a kit and literature grouped together as a bundle. Bundles group together like themed products and are offered at a cost less than that of purchasing each item seperately. Customer Service Order Tracking FAQ Zome Retailers Privacy Policy Affiliate Program Product Registration Contact Us About Zometool, Inc. Copyright2005Zometool, Inc.All Rights Reserved.
Regular Polytopes
Derivation of volume equations for regular polygons, polyhedra, and polytopes, with images.
Regular Polygons Contents I. Geometric Properties II. Diagonals, Graphs, and Euler's Formula References Supplements and Links Regular n-gon Calculator: Contains implementations of many of the formulas derived below. Regular polygons Mathematica notebook Regular Polygons Project Regular Polyhedra Regular Polytopes I. Geometric Properties Below is a table and diagram summarizing the variables used in the explanation. Angles are measured in degrees throughout so that this information will be accessible to students who have not worked in radians. The results can be converted to radians using the conversion 180 = radians. A area enclosed by polygon a apothem (radius of inscribed circle) dk length of kth diagonal d diameter of circumscribed circle m length of median n number of sides p perimeter r radius of circumscribed circle s length of side smallest angle formed by a diagonal and a side angle formed by consecutive sides angle formed by consecutive radii (the central angle) The perimeter of a regular n-gon with side length s is p = ns. To find the area, we divide the polygon into n congruent triangles by drawing segments from the center to each vertex, as illustrated for the septagon to the right. The sum of the central angles is 360 since it represents the interior angle of a circle, so = 360 n. The area of each triangle is as 2 and there are n triangles, so A = nas 2 is the area of a regular n-gon with side length s and apothem a. This can be simplified to A = ap 2; this formulation provides a basis for generalization to higher dimensions. The sum of the interior angles of a triangle equals 180, so 180 = + ; this gives after substition for . Then the sum of the angles of a (regular) polygon is n = 180(n 2). In the same triangles used above, the law of sines gives which we can use to find s in terms of the radius r: (1) where d = 2r is the diameter of the circumscribed circle. By the Pythagorean theorem , Substituting from equation (1) gives which we solve for a to obtain (2) (This expression can also be found using the law of sines as for (1) .) Substituting this into A = nas 2 for a, we find (3) When n is odd, m = r + a. Substituting for a and r yields (4) Finally we will find the lengths of diagonals in a regular polygon. From a given vertex, draw a segment to each of the n 1 other vertices, and call the lengths of these segments (in order) d1, d2, ..., dn 1, as illustrated below. d1 = dn 1 = s because these lengths correspond to sides of the polygon. Define as the angle between any two consecutive segments di and di + 1. All these angles are the same because they subtend the same arc length of the circle. Specifically, by the angle in circle theorems , = 2 = 180 n. (This can also be obtained from the equation 180 = + 2, which is the sum of the angles in the triangle formed by two consecutive sides and a d2 diagonal.) For k = 2, 3, ..., n 2, let k be the angle opposite of diagonal k in the triangle formed a side, diagonal k, and diagonal k 1. Then 2 = and n 1 = , and in general By the law of sines in the kth triangle, Solving for dk, we find (5) To summarize, the following equations hold for regular polygons (n 3). II. Diagonals, Graphs, and Euler's Formula A diagonal is defined to be a line connecting two nonconsecutive vertices of a polygon. All polygons with the same number of sides have the same number of diagonals, so the number of diagonals of a regular n-gon will be the number of diagonals of any polygon with n sides. To derive an expression for the number D(n) of diagonals in an n-gon, we will count the total number of line segments that connect two vertices, including the sides, and then subtract n from the result to take care of the sides. Starting at a given vertex, there are n 1 lines to be drawn to other vertices. The next vertex already has a line drawn to the first vertex, so only n 2 line segments can be drawn. The third vertex can be connected to only n 3 vertices, and so on, until we reach the last vertex, which already has a line segment to every other vertex. The total number of line segments drawn is Subtracting n gives a count of (6) diagonals. If V is the number of intersection points ("vertices"), E is the number of segments ("edges"), and F is the number of regions ("faces") including the region that is the exterior of the polygon, then Euler's formula is Using Euler's formula, then, if we know the number of intersection points and the number of segments, then the number of regions is F = E V + 2 (or E V + 1 if we only want to count interior regions). Let From Poonen and Rubinstein's paper, the number of interior intersections made by the diagonals of a regular n-gon is and the number of regions is For a regular 24-sided polygon, these formulas tell us that the diagonals make 7321 interior intersections, 9024 regions, and 16344 segments. References B. Poonen and M. Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon , SIAM J. Discrete Math., v.11 (1998), p. 135156. Relevant links: Regular Polygon on MathWorld
Guy's Polyhedra Pages
Vertex figures, filling, faceting diagrams, stellation, defining polytopes through generators, trimethoric and trisynaptic polyhedra, space-filling polyhedra, lost stellations of the icosahedron, and links.
Polyhedra Index Page Guy's polyhedra pages Papers, current research and random resources. Contents What's new General theory of polytopes and polyhedra Stellation and facetting Space-filling polyhedra Other resources What's new 21 May 2005. Icosahedral precursors Renamed yet again from "Icosahedral templates", to reflect another change in terminology. Some useful clarifications, including a slight reordering of my new rules and updated commentary. Still unfinished. Part IV of the series on stellating the icosahedron and facetting the dodecahedron. 9 April 2005. Polytopes: degeneracy and tidiness Precursors renamed from 'generators'. Other minor updates. Geometrical untidiness is distinguished from from topological degeneracy. Some types of untidiness and degeneracy are discussed. Features located at infinity can have two opposing images. Unfinished. 5 April - 12 May 2005. Ditela, polytopes and dyads A new name for line segments completes the pantheon of names for polytopes in any number of dimensions. Revised. 22 Mar 2005. Polytopes, duality and precursors Precursors renamed from 'generators' (elsewhere I used yet another term, 'templates'). First half reworked. The rest may never make it home. Unfinished. 6 March 2005. Vertex figures Added the complete vertex, general vertex figure, apeirohedra, ref. to Holden. References renumbered. Bold fonts resolved. Mathematicians have used different definitions of the vertex figure for different purposes: this essay brings some order to the chaos. 8 Jan 2005. Some lost stellations of the icosahedron Update to my page collecting all the lost stellations which I have come across so far. 29 July 2004. Tidy dodecahedra and icosahedra Updated following correspondence with N J Bridge. "Faceting" now "facetting". Other minor edits. Part III in a series of five or more papers on stellating the icosahedron and facetting the dodecahedron. 18 July 2004. Facetting diagrams General tidy-up. "Faceting" changed to "facetting". First stable version. The facetting diagram may be used to find facettings of a polyhedron. The diagram and its use are explained, using as examples the Platonic polyhedra and a few others. The dual relationship between facetting and stellation is noted. 17 July 2004. Filling polytopes Updated but still unfinished. In polytope theory, filling is shown to be of fundamental importance. Traditional theory ignores filling, and so is incomplete. General theory of polytopes and polyhedra Ditela, polytopes and dyads A new name for line segments completes the pantheon of names for polytopes in any number of dimensions. Vertex figures Mathematicians have used different definitions of the vertex figure for different purposes. They may be classified as intrinsic, dependent or polar types. The face of the dual polyhedron is in general reciprocal to a polar type of vertex figure. Nonconvex and asymmetric situations are examined. The complete vertex leads to a more general definition. Polytopes, duality and precursors A new definition of polytopes as set pairs is presented, based on the inclusion of vertex figures. Dual or reciprocal polytopes are understood as differing manifestations of the same precursor. A notation suited to the simpler and more regular cases is described. Unfinished. Polytopes: degeneracy and tidiness Geometrical untidiness is distinguished from from topological degeneracy. Some types of untidiness and degeneracy are discussed. Features located at infinity can have two opposing images. Unfinished. Filling polytopes In polytope theory, filling is shown to be of fundamental importance. Traditional theory ignores filling, and so is incomplete. This essay is incomplete too. Trimethoric (and trisynaptic) polyhedra As in Mathematics and Informatics Quarterly 2 2001, Vol. 11, p.p. 71-74. Trimethoric and trisynaptic polyhedra represent two previously unrecognised classes. Is there a self-dual hendecahedron? Thanks to those who to told me yes, there is one called the canonical form. But are there also any "non-canonical" solutions? Stellation and facetting Facetting diagrams The facetting diagram may be used to find facettings of a polyhedron, whereby pieces are cut away to reveal new faces and edges, but not new vertices. Stellating the icosahedron and facetting the dodecahedron - index page Intro and links to the rest. The main papers are listed below: In search of the lost icosahedra As in The Mathematical Gazette 86, July 2002, p.p. 208-215. Towards stellating the icosahedron and faceting the dodecahedron As in Symmetry: Culture and Science, Vol. 11, 1-4, 2000, p.p. 269-291. Tidy dodecahedra and icosahedra Icosahedral precursors Unfinished. Space-filling polyhedra The Archimedean honeycomb duals As in The Mathematical Gazette 81, July 1997, p.p. 213-219. A remarkable family of 14 polyhedra. The text is reproduced by kind permission. Five space-filling polyhedra As in The Mathematical Gazette 80, November 1996, p.p. 466-475. Pretty, but not earth-shattering. The text is reproduced by kind permission, with some revision and additions. A 3-D quasicrystal structure? A possible candidate for a 3-dimensional, aperiodic crystal structure. Weaire-Phelan Bubbles The closest to ideal bubbles yet found. Other resources Stardust Polyhedron kits and puzzles. Links to other polyhedron websites Steelpillow Guy's Home Page Email me
Numericana - Polyhedra Polytopes
Enumeration, cartesian coordinates of vertices, naming, and counting the shapes.
Polyhedron, Polyhedra, Polytopes, ... - Numericana home | index | units | counting | geometry | algebra | trigonometry functions | calculus analysis | sets logic | number theory | recreational | misc | nomenclature history | physics Final Answers 2000-2005 Grard P. Michon , Ph.D. Polyhedra Polytopes This set of articles started with a simple question ( below ) about hexahedra, which may still be found [in abbreviated form] at its original location in our main Geometry Page ... Hexahedra . The cube is not the only polyhedron with 6 faces. Enumeration of polyhedra : Tally of polyhedra with n faces and k edges. Counting Polyhedra (1): Table, comments, references... Counting Polyhedra (2): Larger table! The 5 Platonic solids : Cartesian coordinates of the vertices. Some special polyhedra may have a traditional (mnemonic) name. Polyhedra in certain families are named after one of their prominent polygons. Deltahedra have equilateral triangular faces. Only 8 deltahedra are convex. Naming Polyhedra : Not an easy task... Polytopes are the n-dimensional counterparts of 3-D polyhedra. A simplex of touching unit spheres may allow a center sphere to bulge out. Regular Antiprism : Height and volume of a regular n-gonal antiprism. Related Links (Outside this Site) Dr. George W. Hart 's site with its Encyclopedia and Pavilion of Polyhedrality . Vladimir Bulatov has a large collection of polyhedra with pretty VRML models . Detailed enumeration of polyhedra by Steven Dutch , Ph.D. Polyhedra (and Five space-filling polyhedra ) by Guy Inchbald . Crystallographic Polyhedra (Java applets) in Dr. Steffen Weber's homepage . Polyhedra Polytopes ( Zonohedra = Hypercube Shadows ) by Russell Towle . Puzzles Polyhedra by Jorge Rezende ( University of Lisbon ) Platonic and Archimedean Solids | Platonic and Archimedean Solids Isohedra | Fair Dice (Ed Pegg Jr.) | Dice (Klaus . Mogensen) Polyhedra (3D), Polychora (4D), Polytopes (nD) (Jerry of Nashville, TN. 2000-11-18 ) What [polyhedron] has six faces? A polyhedron with 6 faces is called a hexahedron. The cube is your most obvious hexahedron, however that's certainly not the only one. Disregarding geometrical distortions and considering only the underlying topology, there are 7 distinct hexahedra: Name of Hexahedron Edges Nodes Triangular Dipyramid 9 5 Pentagonal Pyramid 10 6 Tetragonal Antiwedge 10 6 Hemiobelisk 11 7 Hemicube 11 7 Cube 12 8 Pentagonal Wedge 12 8 The above triangular dipyramid has 5 vertices and 9 edges. It's the dual of a triangular prism, and looks like two tetrahedra "glued" on a common face. The pentagonal pyramid has 6 vertices and 10 edges; it's a pyramid whose base is a pentagon. Like all pyramids, the pentagonal pyramid is self-dual. The above three hexahedra are the only ones which exist in a versionwhere all 6 faces are regular polygons. The tetragonal antiwedge is the least symmetrical of all hexahedra; its only possible symmetry is a 180 rotation. Thisskewed hexahedron has the same number of edges and vertices as the pentagonal pyramid. Its faces consist of 4 triangles and 2 quadrilaterals. Sucha solid is obtained from two quadrilaterals that share an edge [the hinge] but do not form a triangular prism. After adding two edges to complete the two triangles whose sides are adjacent to the hinge, we are left with a nonplanar quadrilateral and must choose one of its 2 diagonals as the last edge of the polyhedron. Only one choice gives a convex polyhedron. Loosely speaking, there are two types of tetragonal antiwedges which are mirror images of each other; each is called an enantiomer, or enantiomorph of the other. The tetragonal antiwedge is thus the simplest example of a chiral polyhedron. Inparticular, any other hexahedron can be distorted into a shape which is its own mirror image, and the tetragonal antiwedge may thus unambiguously be called the chiral hexahedron. Eachenantiomer is self-dual; atetragonal antiwedge and its dual have the same chirality. The other types of hexahedra are more symmetrical and simpler to visualize. One of them may be constructed by cutting off one of the 4 base corners of a square pyramid to create a new triangular face. This hexahedron has 7 vertices and 11 edges. Itsfaces include 3 triangles, 2 quadrilaterals and 1 pentagon. Itcould also be obtained by cutting an elongated square pyramid (the technical name for an obelisk) along a bisecting plane through the apex of the pyramid and the diagonal of the base prism, as pictured at right. Forlack of a better term, we may therefore call this hexahedron an hemiobelisk. Also with 7 vertices and 11 edges, there's a solid which we may call a hemicube (or square hemiprism), obtained by cutting a cube in half using a plane going through two opposite corners and the midpoints of two edges. Its 6 faces include 2 triangles and 4 quadrilaterals. With 8 vertices and 12 edges, the cube (possibly distorted into some kind of irregular prism or truncated tetragonal pyramid) is not the only solution: Consider a tetrahedron, truncate two of its corners and you have a pentagonal wedge. It has as many vertices, edges and faces as a cube, but its faces consist of 2 triangles, 2 quadrilaterals and 2 pentagons. We can build a pentagonal wedge with 2 regular pentagons and 2 equilateral triangles, so that all edges but one are equal. The one "exceptional" edge is the longest side in the two trapezoidal faces. What's its length? Well, look at the wedge "from the side" (so both pentagons project into a line) you see two similar isosceles triangles. The base of the smaller is a regular edge seen perpendicularly (and therefore at its real size), whereas the base of the larger triangle is the length we're after. Theratio of similitude is simply the ratio of the height of a regular pentagon to the distance from a side to an adjacent vertex, namely 1+sin(p 5) sin(2p 5) = (1+5) 2 . A number known as the golden ratio, which happens to be the ratio of the diagonal to the side in a regular pentagon. The longest edge in our solid is thus 1.6180339887498948482... as long as any other. Inother words, both trapezoidal faces are congruent to the diagonal section of a pentagonal face (pictured at right). How many polyhedra have a given number of faces and or edges? We maintain on this site an authoritative table with the latest data. Let's just say it'snot easy to count these things. Polyhedra which are mirror images of each other are not counted as distinct. In the above article , we counted 7 types of hexahedra, but there would be 8 if we tallied separately both chiralities of the tetragonal antiwedge. There's only one tetrahedron: There are two types of pentahedra: the triangular prism and the square pyramid. There are 7 hexahedra (see previous article ), 34 heptahedra, 257 octahedra, 2606 enneahedra, 32300 decahedra, 440564 hendecahedra, 6384634 dodecahedra, 96262938 tridecahedra, 1496225352 tetradecahedra, etc . BenO ( Ben Ocean , 2001-07-28, via e-mail) I'm looking for the x,y,z coordinates of the vertices in the 5 Platonic Solids, preferably in such a way that they precisely nest within each other. (I want to recreate and manipulate these solids in AutoCAD.) The relevant data is available in topical pages from reliable sources like: PaulBourke , Ron Knott , VBHelper , etc. Regular Tetrahedron A B C D x = 2 2 -4 0 y = 23 -23 0 0 z = -2 -2 -2 32 Shown at right are the Cartesian coordinates of the vertices of a regular tetrahedron ABCD centered at the origin. These may be scaled and or rotated. As given, this tetrahedron has: A side equal to 43. A height equal to 42. A circumscribed sphere of radius 32. An alternate set of Cartesian coordinates for a smaller regular tetrahedron (of side 22, inscribed in a sphere of radius 3) would consist of every other vertex in a cube of side 2 (see below) centered at the origin. This highlights some geometrical relationships and dims others: A = [+1,+1,+1], B = [+1,-1,-1], C = [-1,+1, -1], D = [-1,-1,+1] In such a regular tetrahedron, two vertices are seen from the center at an angle known as the tetrahedral angle (which is very familiar to chemists) whose cosine is -1 3 and whose value is109.47122... The dihedral angle between a pair of faces is supplementary to that angle; its cosine is 1 3 and its value is about70.52878 (this may be called a cubic angle, for a reason which follows). Expressed in radians, three times this angle minus a flat angle (p) gives the value [in steradians, sr] of the solid angle at each corner of the tetrahedron, namely 0.55128559843... This is about 4.387% of the solid angle of a whole sphere (4p). Some astronomers prefer to use the square degree as a unit of solid angle (asquare degree equals p2 1802sr); the solid angle at the corner of a regular tetrahedron is 1809.7638632... square degrees ( or 6515150 square minutes ). We may choose 3 coordinates from the set {-1,+1} in 8 different ways and these correspond to the coordinates of the 8 vertices of a cube of side 2 centered at the origin (and inscribed in a sphere of radius3). Seen from the center of a cube, the angular separation between corners is either a flat angle (180 between diametrically opposed vertices), a tetrahedral angle of cosine -1 3 (about 109.47 between the opposite corners in a face), or a cubic angle whose cosineis1 3 and which is supplementary to a tetrahedral angle (about 70.53 between adjacent corners). The solid angle at each corner of a cube is clearly p 2, namely 1 8 of a whole sphere(4p). Similarly, there are 6 ways of choosing 3 coordinates from the set {-1,0,+1} so that only one of them is nonzero. These correspond to the coordinates of the 6 vertices in a regularoctahedron of side 2 centered at the origin (and inscribed in a sphere of radius1). From the center of a regular octahedron, two vertices are seen separated either by a right angle (90) or by a flat angle (180). The volume of a regular dodecahedron is (15+75) 4 times the cube of its side. Thedihedral angle has a cosine of -1 5 and a value of about 116.565 (2002-06-14) Is there a systematic way to name [some] special polyhedra? Only up to a point. The most "generic" way is to use for polyhedra the same naming scheme as for polygons , by counting the number or their faces: Thus, a tetrahedron has 4 faces, a pentahedron has 5, a dotriacontahedron (also called triacontakaidihedron) has32faces. However, counting just faces is not nearly enough to describe a polyhedron, even from a topological standpoint. In some cases, a nonstandard counting prefix is traditionally used for certain very specific polyhedra. For example, the dotriacontahedron shown at right is an Archimedean solid unambiguously known as an icosidodecahedron (literally, a polyhedron with 20+12 faces) because it includes 20 triangular faces and 12 pentagonal ones. Theicosidodecahedron could also be called a pentagonal gyrobirotunda but that name would masks its much greater symmetry compared to the related pentagonal orthobirotunda (J34). For the same reason, a special name has been given to the cuboctahedron, which might otherwise be called a triangular gyrobicupola. If the icosidodecahedron had not claimed the title for the above reason, the name could have been given to another Archimedean solid with 32 faces, the so-called truncated dodecahedron (which has 20 triangular faces and 12 decagonal ones). It wasn't... The notoriety of the icosidodecahedron has made it tempting for some (knowledgeable) people to use the nonstandard icosidodeca prefix (instead of dotriaconta or triacontakaidi) to name other unrelated things (like a 32-sided polygon ). Resist this temptation... The general situation is similar to the naming of chemical compounds . Certain families can be identified and a systematic naming can be introduced among such families. The next article gives the most common such examples. (2000-11-19) What types of polyhedra are named after a polygon? Take a regular polygon (an hexagon, say) and construct a polyhedron by considering an identical copy of that hexagon in a parallel plane. Join each vertex of the hexagon to the corresponding vertex in its copy and you obtain what's called an hexagonal prism. Instead, you may twist the copy slightly and join each vertex to the two nearest vertices of the copy. What you obtain is an hexagonal antiprism. In such families, the polyhedron is named using the adjective corresponding to the name of the polygon it's built on (e.g., "hexagonal"). There are many other families besides prisms and antiprisms for which this pattern applies. For example, if you cut a prism with a plane containing some edge of either base polygon (but not intersecting the other), this "half" prism is called a wedge (it includes the base polygon and its featured edge). Cutting an antiprism the same way, you obtain an antiwedge (the simplest chiral polyhedron is the tetragonal antiwedge, an hexahedron ). Alternately, if the cutting plane contains only a vertex, instead of an edge, the polyhedron you obtain by cutting a prism is best called an hemiprism. Some polyhedra based on an n-sided polygon Name Vertices Edges Faces Remarks pyramid n+1 2n n+1 One n-gon, n triangles dipyramid n+2 3n 2n 2n triangles deltohedron (see note below) 2n+2 4n 2n 2n quadrilaterals (a cube is a triangular deltohedron) prism 2n 3n n+2 Two n-gons, n quadrilaterals (a cube is a square prism) antiprism 2n 4n 2n+2 Two n-gons, 2n triangles cupola 3n 5n 2n+2 One 2n-gon, one n-gon, n quadrilaterals, n triangles orthobicupola gyrobicupola 4n 8n 4n+2 Two n-gons, 2n quadrilaterals, 2n triangles cupolapyramid 3n+1 7n 4n+1 One n-gon, n quadrilaterals, 3n triangles rotunda 4n 7n 3n+2 One 2n-gon, one n-gon, n pentagons, 2n triangles orthobirotunda gyrobirotunda 6n 12n 6n+2 Two n-gons, 2n pentagons, 4n triangles rotundapyramid 4n+1 9n 5n+1 One n-gon, n pentagons, n quadrilaterals, 3n triangles cupolarotunda (ortho- gyro-) 5n 10n 5n+2 Two n-gons, n pentagons, n quadrilaterals, 3n triangles hemiprism 2n-1 3n-1 n+2 Two n-gons (sharing one vertex), 2 triangles, n-2 quadrilaterals wedge 2n-2 3n-3 n+1 Two n-gons (sharing one edge), 2 triangles, n-3 quadrilaterals (left) antiwedge (right) antiwedge 2n-2 4n-6 2n-2 Two n-gons, 2n-4 triangles Only 4 vertices with 3 edges. Note: An n-gonal deltohedron is what a regular n-gonal dipyramid becomes if you twist its upper pyramidal cone 1 2n of a turn with respect to the lower one: The intersection of the two "cones" becomes a solid whose faces are quadrilaterals [see figure at right]. Do not confuse these deltohedra with the deltahedra mentioned next ... What are deltahedra? [Not to be confused with the above deltohedra.] Deltahedra are simply polyhedra whose faces are all equilateral triangles (apolyhedron whose faces are triangles which are not all equilateral is best called an irregular deltahedron). A deltahedron [or an irregular deltahedron] has necessarily an even number of faces (2n faces, 3n edges, and n+2 vertices). A noteworthy fact is that there are only 8 convex deltahedra (disallowing coplanar adjacent faces). Namely: 4 faces 6 faces 8 faces 10 faces Regular tetrahedron (4 faces). Triangular Dipyramid (6 faces). Regular octahedron or "Square Dipyramid" (8 faces). Pentagonal Dipyramid (10 faces). Snub Disphenoid (12 faces), J84. [The Snub Disphenoid was originally called "Siamese dodecahedron" byFreudenthal and vanderWaerden, who first described it in 1947.] Triaugmented Triangular Prism (14 faces), J51. Gyroelongated Square Dipyramid (16 faces), J17. Icosahedron or "Gyroelongated Pentagonal Dipyramid" (20 faces). 12 faces 14 faces 16 faces 20 faces All told, the convex deltahedra include 3Platonic solids (tetrahedron, octahedron and icosahedron). 3dipyramids (triangular, square and pentagonal). 3Johnson solids (J17, J51, and J84). This adds up to 8, instead of 9, because the regular octahedron happens to be counted twice (as a Platonic solid and a square dipyramid)... What are some [other] named polyhedra? It is quite challenging to enumerate all convex polyhedra whose faces are regular polygons. This includes first the 5 Platonic solids and the 13 Archimedean solids (only 2 of which are chiral, the snub cube and snub dodecahedron). Although first discovered by Archimedes, the original classification was lost and only twelve Archimedean solids were known during the Renaissance. The thirteenth (the chiral snub dodecahedron) was added by Kepler when he reconstructed the full Archimedean classification, in1619. The vertices of Archimedean solids are all equivalent and Kepler observed that this property was also shared by two infinite families of convex polyhedra with regular faces: the prisms and antiprisms. The final touch was added in 1966, when Norman W. Johnson gave the full classification of 92 other such polyhedra whose vertices are not all equivalent. These are now called Johnson solids . Only 5 of the 92 Johnson solids are chiral [gyroelongated triangular bicupola (J44), gyroelongated square bicupola (J45), gyroelongated pentagonal bicupola (J46), gyroelongated pentagonal cupolarotunda (J47), and gyroelongated pentagonal birotunda (J48)]. For a solid to be Archimedean, all its vertices must have the same arrangement of faces around them, but this is not sufficient. Acounterexample is the elongated square gyrobicupola (J37), also called pseudo-rhombicuboctahedron (pictured atleft), which is not Archimedean. To describe these and other common polyhedra, some systematic nomenclature is useful. Inparticular, any polyhedron gives rise to many other types whose names include one or more of the following adjectives: Elongated: By replacing (one of) the largest m-sided polygon, with an m-gonal prism (that polygon may not be a face of the polyhedron, but an "internal" polygon with apparent edges). This adds m vertices, 2m edges, and m faces. Thesimplest example, shown at right, is the elongated tetrahedron (J7), which is an heptahedron. Gyroelongated: By replacing (one of) the largest m-sided polygon, with an m-gonal antiprism (that polygon is usually not a face of the polyhedron, but an "internal" polygon with apparent edges). This adds m vertices, 3m edges, and 2m faces. Gyroelongation can be performed in two different ways (often leading to different chiral versions of the same polyhedron). Snub: Snubbing is an interesting chiral process which, roughly speaking, amounts to loosening all faces of a polyhedron and rotating them all slightly in the same direction (clockwise or counterclockwise), creating 2 triangles for each edge and one m-sided polygon for each vertex of degree m. Apolyhedron and its dual have the same snub(s)! If a polyhedron has k edges, its snub has 5k edges, 2k vertices and 3k+2 faces. Truncated: By cutting off an m-gonal pyramid at one or more (usually all) of the vertices. This add (m-1) vertices, m edges and 1 face for each truncated vertex. Augmented: By replacing one or more of the m-sided faces with an m-gonal pyramid, cupola, or rotunda. etc. ... A certain number of other terms are available to describe certain interesting special cases: Cingulum (Latin: girdle; cingere to gird). Fastigium (Latin: apex, height). Sphenoid (Greek: wedge). etc. ... The rest of the nomenclature used in the context of Johnson solids, is best described in the words of Norman W. Johnson himself: If we define a lune as a complex of two triangles attached to opposite sides of a square, the prefix spheno- refers to a wedgelike complex formed by two adjacent lunes. The prefix dispheno- denotes two such complexes, while hebespheno- indicates a blunter complex of two lunes separated by a third lune. The suffix -corona refers to a crownlike complex of eight triangles, and -megacorona, to a larger such complex of 12 triangles. The suffix -cingulum indicates a belt of 12 triangles. There is an important relation among different types of polyhedra which is of primary importance when enumerating them . Each polyhedron has a dual which is obtained essentially by interchanging the role of vertices and faces. A polyhedron and its dual have the same number of edges but each edge is seen in the dual as connecting two faces instead of two vertices. Duality is a topological transformation which has only an indirect geometrical equivalent. The duals of Archimedean polyhedra are called Catalan solids. The names of chiral polyhedra are starred (*). Platonic Solids F E V Duals Tetrahedron 4 6 4 Tetrahedron Cube 6 12 8 Octahedron Octahedron 8 12 6 Cube Dodecahedron 12 30 20 Icosahedron Icosahedron 20 30 12 Dodecahedron Archimedean Solids F E V Duals (Catalan Solids) Truncated Tetrahedron 8 18 12 Triakis Tetrahedron Cuboctahedron 14 24 12 Rhombic Dodecahedron Truncated Cube 14 36 24 Triakis Octahedron Truncated Octahedron 14 36 24 Tetrakis Cube Rhombicuboctahedron 26 48 24 Deltoidal Icositetrahedron Great Rhombicuboctahedron 26 72 48 Disdyakis Dodecahedron Icosidodecahedron 32 60 30 Rhombic Triacontahedron Truncated Icosahedron 32 90 60 Pentakis Dodecahedron Truncated Dodecahedron 32 90 60 Triakis Icosahedron Snub Cube* 38 60 24 Pentagonal Icositetrahedron* Rhombicosidodecahedron 62 120 60 Deltoidal Hexacontahedron GreatRhombicosidodecahedron 62 180 120 Disdyakis Triacontahedron Snub Dodecahedron* 92 150 60 PentagonalHexacontahedron* The truncated icosahedron (the shape of a traditional soccer ball) is now more commonly known as a buckyball ever since it was found to be the structure of a wonderful new molecule, now called fullerene (C60) in honor of the famous American architect R. Buckminster ("Bucky") Fuller (1895-1983), who created and advocated geodesic domes in the late 1940s. The buckyball is one of 4 Archimedean solids without triangular faces. The other three are the truncated octahedron (atleft), the great rhombicosidodecahedron (atright) and the great rhombicuboctahedron. The 4 Archimedean polyhedra illustrated so far are simplicial (i.e., only 3 edges meet at each vertex). There are 3 others such simplicial polyhedra, illustrated next, which happen to be obtained (like the buckyball and truncated octahedron above) by truncating a Platonic solid. This leaves 4 nonchiral Archimedean solids for which 4 edges meet at every vertex. Finally, 5 edges meet at every vertex of the two chiral Archimedean polyhedra: The full classification of these 13 solids is believed to have been discovered by Archimedes of Syracuse (c.287-212BC), but the work was lost and the thirteenth of them (the Snub Dodecahedron, whose two chiralities are pictured above right) was apparently forgotten until Kepler (1571-1630) reconstructed the whole set, in 1619. How do polyhedra generalize to dimension 4 or more? The equivalent of a polyhedron in dimension 4 is called a polychoron (plural polychora). Polychora are discussed extensively on beautifully illustrated pages proposed by George Olshevsky and Jonathan Bowers . Although the introduction of the term polychoron is fairly recent, it seems now generally accepted, as there's no serious competition (the etymology of "polyhedroid" is poor and misleading). Theterm was coined in the 1990's by George Olshevsky , whose earlier proposal of "polychorema" (plural: "polychoremata") was unsuccessful. Olshevsky's new proposal had the early support of Norman W. Johnson, after whom the 92 "convex regular-faced solids" are named (Johnsonsolids). Apolychoron is bounded by 3-dimensional faces, called cells. The four-dimensional equivalent of the Euler-Descartes formula is a topological relation which relates the number of vertices (V), edges (E), faces (F), and cells (C) in any polychoron enclosing a portion of hyperspace homeomorphic to a 4D open ball (provided edges, faces and cells are homeomorphic, respectively, to 1D, 2D and 3D open balls): V - E + F - C = 0 In an unspecified number of dimensions, the counterpart of a 2D polygon, a3Dpolyhedron, or a 4D polychoron is called a polytope. The boundary of an n-dimensional polytope consists of hyperfaces which are (n-1)-dimensional polytopes, joining at hyperedges, which are (n-2)-dimensional polytopes. (All the hyperfaces of an hyperface are thus hyperedges.) Thisvocabulary is consistent with the well-established term hyperplane to designate a vector space of codimension1 (in a hyperspace with a finite number n of dimensions, a hyperplane is therefore a linear space of dimension n-1). Wealso suggest the term hyperline for a linear space of codimension2 (and, lastly, hyperpoint to designate a space of codimension3). To denote the p-dimensional polytopes within a polytope of dimension n, the following terms may be used: vertex (p=0; plural "vertices"), edge (p=1), face (p=2), cell or triface (p=3), tetraface (p=4), pentaface (p=5), hexaface (p=6), ... hypervertex (p=n-3), hyperedge (p=n-2), hyperface (p=n-1), hypercell (p=n). To establish and or memorize the n-dimensional equivalent of the Euler-Descartes formula for "ordinary" polytopes in n dimensions, it's probably best to characterize each such polytope by the open region it encloses (boundary excluded), except in dimension zero (the 0-polytope is a single point). For the formula to apply, each such region should be homeomorphic [i.e., topologically equivalent] to the entire Euclidean space of the same dimension, or equivalently to an open ball of that dimension. An edge is an open segment, a face is an open disk, a cell is an open ball, etc. [for example, the ring between two concentric circles is not allowed as a face, and the inside of a torus is disallowed as a cell]. Then we notice that a number can be assigned to any polytope (and a number of other things) called its Euler characteristic (c), which is additive for disjoint sets, equal to 1 for a point and invariant in a topological homeomorphism (so that topologically equivalent things have the same c). For our purposes, this may be considered an axiomatic definition of c. It may be used to establish (by induction) that the c of n-dimensional Euclidean space is (-1)n , which is therefore equal to the c of our "ordinary" open polytopes (HINT:Ahyperplane separates hyperspace into three parts; itself and 2 parts homeomorphic to the whole hyperspace). The c of all "ordinary" closed polytopes in n dimensions is the c of a closed n-dimensional ball and it may be obtained by inspecting the components of the boundary of any easy n-dimensional polytope like the hypercube or the simplex polytope discussed below. It turns out to be equal to 1 in any dimensionn. If the hypercell itself (the polytope's interior) is excluded from the count, as it is in the traditional 3-dimensional Euler-Descartes formula, the RHS of the formula will therefore be 2 in an odd number of dimensions and zero in an even number of dimensions. For example, in 7 dimensions, if we denote by T the number of tetrafaces, by P the number of pentafaces (hyperedges), and by H the number of hexafaces (hyperfaces), we have: V - E + F - C + T - P + H = 2 We may focus on the n-dimensional equivalent of the Platonic solids, namely the regular convex polytopes, whose hyperfaces are regular convex polytopes of a lower dimension, given the fact that the concept reduces to that of a regular polygon [equiangular and equilateral] in dimension2. In dimension 3, this gives the 5regular polyhedra. In dimension 4, we have 6regular polychora. In dimension 5 or more, only 3 regular polytopes exists which belong to one of the following three universal families (also present in spaces of lower dimensions): Family n-Polytope dim. V 0 E 1 F 2 C 3 T 4 p-Faces p Simplex n C(n+1,p+1) Point 0 1 Segment 1 2 1 Triangle 2 3 3 1 Tetrahedron 3 4 6 4 1 Pentachoron 4 5 10 10 5 1 Cross Polytope n C(n,p+1) 2p+1 Point 0 1 Segment 1 2 1 Square 2 4 4 1 Octahedron 3 6 12 8 1 Hexadecachoron 4 8 24 32 16 1 Hypercube n C(n,p) 2n-p Point 0 1 Segment 1 2 1 Square 2 4 4 1 Cube 3 8 12 6 1 Tesseract 4 16 32 24 8 1 The regular simplex polytope is obtained by considering n+1 vertices in dimension n, so that each one is at the same distance from any other (its hyperfaces are simplex polytopes of a lower dimension). Choosing as vertices all points whose Cartesian coordinates are from the set {-1,+1} , we obtain an n-dimensional hypercube (of side 2). The hyperfaces of an hypercube are hypercubes of a lower dimension. The dual of the above hypercube is the regular cross polytope whose vertices have a single nonzero coordinate, taken from the set {-1,+1} . The 5D interactive hypercube at right is from Kurt Brauchli (details here ). Click and drag with the mouse to turn the cube aound the chosen axes (H and V) indicated in the menu bar. This 5D cube projects like a 3D cube if you rotate only around axes 0, 1 or 2. The fourth and fifth dimensions appear with axes 3 and 4. With the bottom cursor, you may choose a distant (left) or close-up perspective (right). Your browser is unable to run a Java "applet" and cannot display this interactive picture... Sorry. EnolaStraight ( 2002-05-07 ) What's the radius of the circle touching 3 touching unit circles? What's the radius of the sphere touching 4 touching unit spheres? [In this (edited) question, "touching" means "externally tangent (to)".] The generalization of this question to any number of dimensions is a classic demonstration that whatever geometrical intuition we may have developed in two or three dimensions may not be trusted in a space of more dimensions. The two-dimensional case [pictured at right] shows 3 congruent circles, centered on the vertices of an equilateral triangle, touching each other and a much smaller circle [pictured as a red disk] whose radius has to be determined. Based on this 2-D case [and, to a lesser extent, on the 3-D case] it would seem that such an inner ball [disk, sphere, or n-dimensional hypersphere] would always be small enough to fit inside the simplex [equilateral triangle, regular tetrahedron, n-dimensional regular simplex] formed by the centers of the congruent balls. This happens to be true for a dimension equal to 4 or less, but fails for a dimension of 5 or more. In a very large number of dimension, the (linear) size of the inner ball is about 41% the size of the outer ones. More precisely, 2-1 = 0.41421356... is the limit of that ratio when the number of dimensions tends to infinity. Read on... Consider the center O of the n-dimensional simplex formed by the n+1 centers of the congruent balls [each of radius 1]. The critical quantity is the distance D(n) from the center O to any vertex; the radius of the inner ball is simply D(n)-1 . Well, because O is the center of gravity of n+1 vertices, it is on the line that goes from a vertex to the center of gravity of the n others. It divides that line in a 1 to n ratio. The length of that line is therefore (1+1 n)D(n) and it is also one side of a right triangle whose other side is of length D(n-1) and whose hypotenuse is of length 2 (it's the side of the simplex, the distance between the centers of two balls). In other words, D(n) is given recursively by the relations: D(1) = 1 and D(n) = [4-D(n-1)2] (1+1 n) This recursion can be used to prove by induction the following simple formula: D(n) = [ 2n (n+1) ] The simplicity of this result is a hint that there might be a more direct way to obtain it. D(2)=2 3 says that the radius of the inner circle in the above figure is 2 3-1 [about 15.47%] of the radius of any outer circle. Similarly, the corresponding ratio for spheres is 6-1 [about 22.4745%]. The limit of D(n) is 2: In a space with a very large number of dimensions, the ratio of the radius of the inner hypersphere to the radius of any outer hyperspheres is thus slightly less than 2-1 [about 41.42%]. In dimension n, the distance from the center O to any of the hyperplanes (ofdimension n-1 ) which carry the "faces" of the simplex is D(n) n. Therefore, if the radius D(n)-1 is greater than is D(n) n. , the inner hypersphere bulges outside of the n-dimensional regular simplex formed by the centers of the outer hyperspheres. This happens as soon as n2-5n+2 0 , which is the case when n is at least equal to5. Thishigher-dimensional configuration is contrary to the intuition we would form by looking only at the two-dimensional and or three-dimensional cases... Dr. Murali V.R. (2004-02-25; e-mail) What is the volume of a regular antiprism? A regular antiprism is a polyhedron whose faces are two parallel n-gonal bases [regular polygons with n sides] and 2n equilateral triangles called lateral faces. Look at the outline of such a solid from above, and what you see is a regular polygon with 2n sides (every other vertex is on the top base, and every other one is on the bottom). The angle at each vertex of this outline is thus q = p-p n . Now, each lateral face is seen as an isosceles triangle having an angle q at the top and featuring a base observed at its real size a (as the direction of observation isperpendicular to it). The height of such an isoceles triangle is thus: a cotan (q 2) = a tan (p 2n) This quantity is also equal to the length of a side of a right triangle whose hypotenuse is the true height ofa lateral face (namely a3) and whose other side is the height h of the antiprism (namely, the distance between its bases). This gives the height h of the antiprism in terms of the length a of its edges: h = a 3 - tan2(p 2n) Consider the circumscribed prism of height h whose base is the 2n-gonal outline. Each side of this outline is equal to a cos(p 2n). Its surface area is therefore : (n a2 4) sin(p n) and the volume of the prism is h times that. The antiprism is obtained from this prism by removing 2n triangular pyramids ofheight h whose bases are all congruent to the above isosceles triangle, for a combined base area of (n a2 2) tan(p 2n) and a total volume h 3 times that. The volume V of the antiprism is the difference between these two volumes: V = (n a3 24) [ 3 sin(p n) - 2 tan (p 2n) ] 3 - tan2(p 2n) This can be rewritten in a much more palatable form, using t = tan (p 2n) : V = n a3 ( 3 - t 2 ) 3 2 48t h = a 3 - tan2(p 2n) V = n h 3 6 tan(p 2n) Two noteworthy special cases (for a = 1): V = 1 72 when n=2. A regular tetrahedron! [Adegenerate but valid case.] V = 3 when n=3. A regular octahedron...
The Charged Particles Model
Polytopes and optimal packing of p points in n dimensional spheres. Contains a java applet based on a model which allows for generation of multidimensional regular and semi-regular polytopes.
Polytopes and optimal packing of p points in n dimensional spheres. Polytopes and optimal packing of p points in n dimensional spheres Introduction Definitions Assumptions The computational model in action Findings Further improvements of the computational model Regular and semi-regular convex polytopes a short historical overview References The hemipenteract of configuration [5][16] 2004, Symen H. Hovinga , the Netherlands. Sponsors: mtcint.com , Zwols.nl , Alpicasa.nl , Ledro.com , Lagodigarda.nl , 2on.Com
Polyhedra, Platonic Solids, Polytopes
Definitions, pictures, templates, and coordinates of the regular 3d and 4d polytopes. Also includes waterman polyhedra, polar and star spheres, and the time star.
Geometry G e o m e t r y Index Papers Geometry Curves Surfaces Fractals, Chaos Projection Stereographics Raytracing OpenGL Modelling Colour Textures Data Formats Fun,puzzles Other Old stuff VodCast PodCast The contents of this web site are Copyright Paul Bourke or a third party contributer where indicated. You may print or save an electronic copy of parts of this web site for your own personal use. Permission must be sought for any other use. Philosophy is written in this grand book - I mean universe - which stands continuously open to our gaze, but which cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth. Galileo (1623) The Ribbed Sphere Do plans and elevations describe unique geometry? Quadric equations Surfaces in x and y of degree 2. Fitting puzzle Slicing a torus How many ways can a torus be cut (with a single plane) so that the resulting cross sections are perfect circles? Interpretation illusion How simple geometry can be interpreted in many ways. There are holes in the sky. Where the rain gets in. But they're ever so small. That's why the rain is thin. Spike Milligan Distance between a point and a line Distance between a point and a plane The intersection of a line with another line (2D) The closest line between two lines (3D) The intersection of a line with a plane Mathematics describing a plane The intersection of two planes The intersection of three planes Polygon area and centroid calculation Inside outside polygon test Reflection of a ray Direction Cosines Determining whether a line segment intersects a facet Clipping a line with a polygon Clockwise test for polygons in 2D Test for concave convex polygon in 2D Spheres, equations and terminology The intersection of a line and a sphere (or a circle) Equation of the circle through 3 points Equation of the sphere through 4 points Intersecting area of circles on a plane Rotation of a point about an arbitrary axis Creating a plane disk perpendicular to a line segment Intersection of two circles on the plane Circumference of an ellipse Intersection of two spheres Distributing Points on a Sphere Kissing number in N dimenions A triangle was an improvement to the square wheel. It eliminated one bump. BC comics Eulers number and closed surfaces HyperSpace A Macintosh 4 dimensional geometry viewer and manual. Minus One poem Polyhedra Waterman Polyhedra Cylinder intersections Plexagons (Pleated hexagon) 4D platonic solids "We should make things as simple as possible, but not simpler." Albert Einstein Rhombic Triacontahedron Star Sphere Polar Sphere Polar + Star If triangles had a God, He'd have three sides. Old Yiddish proverb Coordinates for 4D polyhedra Parallelohedron Platonic solids Time Star To see a World in a Grain of Sand, And Heaven in a Wild Flower Hold Infinity in the palm of your hand, And Eternity in an hour William Blake Photos by Gayla Chandler Build your own Polyhedra data files Cube personalities Index Papers Geometry Curves Surfaces Fractals, Chaos Projection Stereographics Raytracing OpenGL Modelling Colour Textures Data Formats Fun,puzzles Other Old stuff VodCast PodCast
Polygons, Polyhedra, Polytopes
Regular, rectified, and truncated polytopes with normal and hidden-detail-removed projections. Images, animations, and links.
Polytopes Polygons, Polyhedra, Polytopes Polytope is the general term of the sequence, point, segment, polygon, polyhedron, ... . So we learn in H.S.M. Coxeter 's wonderful Regular Polytopes (Dover, 1973). When time permits, I may try to provide a systematic approach to higher space. Dimensional analogy is an important tool, when grappling the mysteries of hypercubes and their ilk. But let's start at the beginning, and to simplify matters, and also bring the focus to bear upon the most interesting ramifications of the subject, let us concern ourselves mostly with regular polytopes. You may wish to explore my links to some rather interesting and wonderful polyhedra and polytopes sites, at the bottom of this page. Check out an animated GIF (108K) of an unusual rhombic spirallohedron. The most symmetrical plane projection of the four-dimensional star polytope {5 2,3,3}. This polytope is bounded by 120 Great Stellated Dodecahedra, and has 600 vertices, 1200 edges, and 720 pentagrams. Yes, we shall be speaking of the fourth dimension, and, well, the 17th dimension, or for that matter, the millionth dimension. We refer to Euclidean spaces, which are flat, not curved, although such a space may contain curved objects (like circles, spheres, or hyperspheres, which are not polytopes). We are free to adopt various schemes to coordinatize such a space, so that we can specify any point within the space; but let us rely upon Cartesian coordinates, in which a point in an n-space is defined by an n-tuplet of real numbers. These real numbers specify distances from the origins along n mutually perpendicular axes, such as the familiar x-axis and y-axis of the Cartesian plane. In space of zero dimensions, the only figure possible is a point. It has no length, no breadth, no height. In space of one dimension, the only polytopes possible are line segments and points. A segment is said to be bounded by two points (its end-points). We may obtain a tessellation or close-packing of polytopes in this 1-space, by joining line segments point-to-point to fill the line of the 1-space. In space of two dimensions (a plane), we may have points, segments, and polygons. Setting aside degenerate cases like the digon (a polygon with two coincident sides) and the apeirogon (a polygon with an infinite number of sides), the simplest possible polygon is the triangle. A triangle is bounded by three segments (its sides) and three vertices (the endpoints of its three sides, which fall into coincidence in pairs). Note that if the triangle be equilateral, it is also regular: all the sides have equal length, and all the interior angles are equal. However, if a polygon has more than three sides, to be equiangular is not to be regular: for instance, all rhombs are equilateral, but only that rhomb known as the square (or, two-dimensional measure polytope, or two-dimensional hypercube, or 2-cube) is equiangular, and regular. A curious fact about regular polygons is that some have Euclidean constructions, but most do not. That is, some may be constructed with compass and unmarked straight-edge alone. Among these are the equilateral triangle, the square, the pentagon, the 17-gon, the 257-gon, and the 65,537-gon. It must be noted at the outset that a regular polygon may be regular, yet not convex: the sides may be equilateral and equiangular, yet cross one another, forming star polygons. Some examples are given below, labeled with their Schlafli symbols (Ludwig Schlafli was a pioneer in the study of regular polytopes, and devised these symbols). A regular convex n-gon has Schlafli symbol {n}, while a star n-gon in which every k-th vertex is joined (of the n vertices of the "parent" n-gon, taken cyclically) has Schlafli symbol {n k}. Three regular heptagons (7-gons), and a regular star 24-gon. Now, one thing which makes the study of regular polytopes interesting is their symmetry; but another is their departures from simple patterns and orderings. There are infinitely many different regular polygons, but only nine regular polyhedra: the famous five Platonic Solids, and the four less-known Kepler-Poinsot polyhedra, which are stellations of the Platonic pentagonal dodecahedron and icosahedron. In four dimensions, there are sixteen regular polytopes: six convex, and ten starry; but in each and every higher space, there are but three: the regular simplex, the cross polytope or orthoplex, and the hypercube. These are analogues of the Platonic Tetrahedron, Octahedron, and Cube. The five Platonic Solids, so named because Plato described them in the Timaeus. These are hidden-detail-removed projections: look carefully at the polygonal outline of each projection, and at the tiling of foreshortened polygons which compose that polygon. For images of the Kepler-Poinsot star polyhedra, and of sections and projections of the regular star-polytopes in four dimensions, visit my star polytopes page . For a QuickTime movie of the solid sections of the regular star polytope {5 2,3,3}, go here . Now: please observe that a polyhedron is bounded by polygons, which in turn are bounded by segments, which in turn are bounded by points. We say that a polyhedron is bounded by faces (polygons), edges (segments), and vertices (points). Note that every edge of a polyhedron is the join of two faces, and that every vertex of a polygon is the join if two sides. We have stumbled upon a kind of binary rule which applies to all simply-connected polytopes, I think: every (n-2)-dimensional bounding polytope is the join between exactly two (n-1)-dimensional bounding polytopes. Thus in four dimensions, where 4-polytopes are bounded by 3-polytopes (polyhedra), every polygon is the join between exactly two polyhedra. Now, the geometry of four dimensions has always and rightly been regarded as fiendishly, demonically difficult. It is. However, one may gain a foothold in this exalted space, so often the very source and fount of fantastical magical powers and events, in the eyes of this or that supposed mystic, by carefully applying dimensional analogy. Ludwig Schlafli, a Swiss, made his great advances in the study of higher space in the middle of the 19th century. His work went largely unnoticed, and in the 1880s others brought some of the same ideas forward, and even gained credit for what Schlafli had already achieved decades before. Oh, but there is a ponderous army of sines and cosines, of dot products and cross products, guarding the approaches to this magical realm. Stay, though! Not every pass is guarded; one can reach the unruly realm, using dimensional analogy! In fact, one of the more significant contributors to the facts and theories of higher space was a person without any training in linear algebra or trigonometry. Her name was Alicia Boole Stott. While geometers in the great universities, a century past, were laboring upon the broad outlines of things polytopical, Alicia Boole Stott had already gone far beyond. Her principal tool was dimensional analogy. Here is an application of dimensional analogy: just as a hidden-detail-removed projection of a convex polyhedron, onto a plane, induces a tiling of the projection's bounding polygon, by one or more (usually smaller) polygons, so also a hidden-detail-removed projection of a four-dimensional convex polytope, into a 3-space, induces a tiling of the projection's bounding polyhedron, by one or more (usually smaller) polyhedra. A pentagonal-dodecahedron-first, hidden-detail-removed projection of the regular 4-polytope {5,3,3} (or 120-cell), into a 3-space. The variously foreshortened dodecahedral cells have been exploded apart slightly so that some of the inner dodecahedra are visible. Note that, from the 4-space, all the dodecahedra would be visible at once, exploded or not. These hidden-detail-removed projections of 4-polytopes into a 3-space were something I devised using the software, Mathematica. See my Mathematica page , or just download the Regular Polytopes notebook . It is quite significant that such projections induce a tiling of polyhedra, composing a larger polyhedron. The subject here becomes rather vast and also dense, and relates to my zonotopal completions of convex polytopes, and to quasicrystals and zonotiles , and to the stellations of convex polytopes. I hope to describe these ideas in more detail soon. Go here for three animated GIFs which compare the processes of completion, stellation, and faceting, with a short discussion. From any convex polytope we may obtain related figures, the so-called rectified and truncated forms. A rectified polytope has for vertices the mid-edge points of its parent, while a truncated regular polytope is typically imagined to be truncated by hyperplanes perpendicular to the vectors to its vertices, to just such a depth as would create a regular 2n-gon, from any one of the bounding n-gons. For instance, the rectified cube (or octahedron) is the cuboctahedron, bounded by six squares and eight triangles, while the truncated cube is bounded by six octagons and eight triangles. In space of four dimensions, the rectified 120-cell, or r{5,3,3} is bounded by 120 Archimedean icosidodecahedra, and 600 Platonic tetrahedra. Above: how the hidden-detail removed, icosidodecahedron-first projection of the rectified 120-cell is built up, from its central icosidodecahedron, to its bounding polyhedron (from left to right, and top to bottom). Above: truncation of a polytope need not be of the type defined above. Here, a Platonic pentagonal dodecahedron has been slightly truncated edge-wise. An extract from my favorite chapter of Coxeter's Regular Polytopes, in which he begins the discussion of the projection of hypercubes (called measure polytopes, if of unit edge length, for then they define the unit of n-dimensional content) into zonotopes, and the fascinating subject of eutactic stars. Of which, more here ! Here is another example of dimensional analogy: Given a close-packing of equal cubes, filling a region of 3-space, we can take their sections by a plane. The result is a close-packing of polygons which fill a region of the sectioning plane. Now, let the cubes be exploded apart slightly, and take a section by a plane perpendicular to the line joining opposite vertices of any one cube. Depending upon the position of this sectioning plane, the polygons change shape. Here is one such section: Section of an array of 3-cubes. Among the regular space-fillings in four dimensions is that of the 24-cell, {3,4,3}, which is bounded by 24 Platonic octahedra. Fill a region in the 4-space with {3,4,3}'s, exploded apart slightly, and take their section by a 3-space perpendicular to the line joining opposite octahedra, of any one {3,4,3}. Here is one such section: By dimensional analogy, just as the plane section of close-packing polyhedra gives close-packing polygons, so also the solid section of a close-packing of 4-polytopes gives close-packing polyhedra. In this case, the close-packing is semi-regular, with Archimedean cuboctahedra and Platonic octahedra. Links to polyhedra and polytopes: George Hart's Pavilion of Polyhedrality George Olshevsky's Four-dimensional polytopes pages Vladimir Bulatov's polyhedra Wolfram Research's Polytope Pages! Mark Newbold's polyhedra and polytopes (Java applets, stereo pairs) David Eppstein's Totally Awesome Geometry Junkyard Flatland: A Romance of Many Dimensions James R. Buddenhagen's Uniform Polyhedra Jim McNeill's Great Polyhedra and Tessellation Site Back to Russell Towle's homepage. Contact me.
The Geometry Junkyard
Comprehensive list of links to sites on geometric properties of the shapes.
The Geometry Junkyard: Polyhedra and Polytopes Polyhedra and Polytopes This page includes pointers on geometric properties of polygons, polyhedra, and higher dimensional polytopes (particularly convex polytopes). Other pages of the junkyard collect related information on triangles, tetrahedra, and simplices , cubes and hypercubes , polyhedral models , and symmetry of regular polytopes . Adventures among the toroids . Reference to a book on polyhedral tori by B. M. Stewart. Bob Allanson's Polyhedra Page . Nice animated-GIF line art of the Platonic solids, Archimedean solids, and Archimedean duals. Almost research-related maths pictures . A. Kepert approximates superellipsoids by polyhedra. Archimedean polyhedra , Miroslav Vicher. Archimedean solids: John Conway describes some interesting maps among the Archimedean polytopes . Eric Weisstein lists properties and pictures of the Archimedean solids . Rolf Asmund's polyhedra page . Associahedron and Permutahedron . The associahedron represents the set of triangulations of a hexagon, with edges representing flips; the permutahedron represents the set of permutations of four objects, with edges representing swaps. This strangely asymmetric view of the associahedron (as an animated gif) shows that it has some kind of geometric relation with the permutahedron: it can be formed by cutting the permutahedron on two planes. A more symmetric view is below. See also a more detailed description of the associahedron , Frdric Chapoton's live3d demo , and Jean-Louis Loday's paper on associahedron coordinates . David Bailey's world of tesselations . Primarily consists of Escher-like drawings but also includes an interesting section about Kepler's work on polyhedra. Ned Batchelder's Stellated Dodecahedron T-shirt . The bellows conjecture , R. Connelly, I. Sabitov and A. Walz in Contributions to Algebra and Geometry , volume 38 (1997), No.1, 1-10. Connelly had previously discovered non-convex polyhedra which are flexible (can move through a continuous family of shapes without bending or otherwise deforming any faces); these authors prove that in any such example, the volume remains constant throughout the flexing motion. Books on polyhedra and polytopes . Collected by Tony Davie, St. Andrews U. Bounded degree triangulation . Pankaj Agarwal and Sandeep Sen ask for triangulations of convex polytopes in which the vertex or edge degree is bounded by a constant or polylog. Buckyballs . The truncated icosahedron recently acquired new fame and a new name when chemists discovered that Carbon forms molecules with its shape. The charged particle model: polytopes and optimal packing of p points in n dimensional spheres . Circumnavigating a cube and a tetrahedron , Henry Bottomley. Cognitive Engineering Lab , Java applets for exploring tilings, symmetry, polyhedra, and four-dimensional polytopes. Complex polytope . A diagram representing a complex polytope, from H. S. M. Coxeter's home page . A computational approach to tilings . Daniel Huson investigates the combinatorics of periodic tilings in two and three dimensions, including a classification of the tilings by shapes topologically equivalent to the five Platonic solids. Convex Archimedean polychoremata , 4-dimensional analogues of the semiregular solids, described by Coxeter-Dynkin diagrams representing their symmetry groups. A Counterexample to Borsuk's Conjecture , J. Kahn and G. Kalai, Bull. AMS 29 (1993). Partitioning certain high-dimensional polytopes into pieces with smaller diameter requires a number of pieces exponential in the dimension. Cuboctahedron , ink on paper, A. Glassner. Deltahedra , polyhedra with equilateral triangle faces. From Eric Weisstein's treasure trove of mathematics. Dodecafoam . A fractal froth of polyhedra fills space. Dodecahedron measures , Paul Kunkel. Domegalomaniahedron . Clive Tooth makes polyhedra out of his deep and inscrutable singular name. All the fair dice . Pictures of the polyhedra which can be used as dice, in that there is a symmetry taking any face to any other face. Chris Fearnley's 5 and 25 Frequency Geodesic Spheres rendered by POV-Ray. Five Platonic solids and a soccerball . Flexible polyhedra . From Dave Rusin's known math pages. The Fourth Dimension . John Savard provides a nice graphical explanation of the four-dimensional regular polytopes. Four-dimensional visualization . Doug Zare gives some pointers on high-dimensional visualization including a description of an interesting chain of successively higher dimensional polytopes beginning with a triangular prism. Geodesic math . Apparently this means links to pages about polyhedra. Geometria Java-based software for constructing and measuring polyhedra by transforming and slicing predefined starting blocks. Geometry, algebra, and the analysis of polygons . Notes by M. Brundage on a talk by B. Grnbaum on vector spaces formed by planar n-gons under componentwise addition. Geometry and the Imagination in Minneapolis . Notes from a workshop led by Conway, Doyle, Gilman, and Thurston. Includes several sections on polyhedra, knots, and symmetry groups. The Geometry of the Mayan TimeStar , G. de Jong. Complexes of interlocking Platonic solids animated in Java. Glowing green rhombic triacontahedra in space . Rendered by Rob Wieringa for the May-June 1997 Internet Ray Tracing Competition . The golden section and Euclid's construction of the dodecahedron , and more on the dodecahedron and icosahedron , H. Serras, Ghent. Polyhedra - homage to U. A. Graziotti . Great triambic icosidodecahedron quilt , made by Mark Newbold and Sarah Mylchreest with the aid of Mark's hyperspace star polytope slicer. Melinda Green's geometry page . Green makes models of regular sponges (infinite non-convex generalizations of Platonic solids) out of plastic "Polydron" pieces. Hecatohedra . John Conway discusses the possible symmetry groups of hundred-sided polyhedra. Hilbert's 3rd Problem and Dehn Invariants . How to tell whether two polyhedra can be dissected into each other. See also Walter Neumann's paper connecting these ideas with problems of classifying manifolds. Holyhedra . Jade Vinson solves a question of John Conway on the existence of finite polyhedra all of whose faces have holes in them (the Menger sponge provides an infinite example). HypArr, software for modeling and visualizing convex polyhedra and plane arrangements, now seems to be incorporated as a module in a larger Matlab library for multi-parametric analysis . Hypergami polyhedral playground . Rotatable wireframe models of platonic solids and of the penguinhedron. Hyperspace star polytope slicer , Java animation by Mark Newbold. The icosahedron, the great icosahedron, graph designs, and Hadamard matrices . Notes by M. Brundage from a talk by M. Rosenfeld. Icosamonohedra , icosahedra made from congruent but not necessarily equilateral triangles. Ideal hyperbolic polyhedra ray-traced by Matthias Weber. Guy Inchbald's polyhedra pages . Stellations, hendecahedra, duality, space-fillers, quasicrystals, and more. The International Bone-Roller's Guild ponders the isohedra : polyhedra that can act as fair dice, because all faces are symmetric to each other. Investigating Patterns: Symmetry and Tessellations . Companion site to a middle school text by Jill Britton, with links to many other web sites involving symmetry or tiling. Johnson Solids , convex polyhedra with regular faces. From Eric Weisstein's treasure trove of mathematics. Sndor Kabai's mathematical graphics , primarily polyhedra and 3d fractals. Kepler-Poinsot Solids , concave polyhedra with star-shaped faces. From Eric Weisstein's treasure trove of mathematics. See also H. Serras' page on Kepler-Poinsot solids . Links2go: Polyhedra Louis Bel's povray galleries: les polyhdres rguliers , knots , and more knots . Lunatic's guide to polyhedra . 3-Manifolds from regular solids . Brent Everitt lists the finite volume orientable hyperbolic and spherical 3-manifolds obtained by identifying the faces of regular solids. Maple polyhedron gallery . Martin's pretty polyhedra . Simulation of particles repelling each other on the sphere produces nice triangulations of its surface. Mathematica Graphics Gallery: Polyhedra Max. non-adjacent vertices on 120-cell . Sci.math discussion on the size of the maximum independent set on this regular 4-polytope. Apparently it is known to be between 220 and 224 inclusive . Minesweeper on Archimedean polyhedra , Robert Webb. Models of Platonic solids and related symmetric polyhedra. Netlib polyhedra . Coordinates for regular and Archimedean polyhedra, prisms, anti-prisms, and more. Nine . Drew Olbrich discovers the associahedron by evenly spacing nine points on a sphere and dualizing. Nonorthogonal polyhedra built from rectangles . Melody Donoso and Joe O'Rourke answer an open question of Biedl, Lubiw, and Sun. Occult correspondences of the Platonic solids . Some random thoughts from Anders Sandberg . Pappus on the Archimedean solids . Translation of an excerpt of a fourth century geometry text. Peek , software for visualizing high-dimensional polytopes. Penumbral shadows of polygons form projections of four-dimensional polytopes. From the Graphics Center's graphics archives. Pictures of 3d and 4d regular solids, R. Koch, U. Oregon. Koch also provides some 4D regular solid visualization applets . The Platonic solids . With Java viewers for interactive manipulation. Peter Alfeld, Utah. Platonic solids and quaternion groups , J. Baez. Platonic spheres . Java animation, with a discussion of platonic solid classification, Euler's formula, and sphere symmetries. Platonic Universe , Stephan Werbeck. What shapes can you form by gluing regular dodecahedra face-to-face? Poly , Windows Mac shareware for exploring various classes of polyhedra including Platonic solids, Archimedean solids, Johnson solids, etc. Includes perspective views, Shlegel diagrams, and unfolded nets. Polygonal and polyhedral geometry . Dave Rusin, Northern Illinois U. Polygons as projections of polytopes . Andrew Kepert answers a question of George Baloglou on whether every planar figure formed by a convex polygon and all its diagonals can be formed by projecting a three-dimensional convex polyhedron. Polygons, polyhedra, polytopes , R. Towle. Polyhedra . Bruce Fast is building a library of images of polyhedra. He describes some of the regular and semi-regular polyhedra, and lists names of many more including the Johnson solids (all convex polyhedra with regular faces). Polyhedra collection , V. Bulatov. Polyhedra exhibition . Many regular-polyhedron compounds, rendered in povray by Alexandre Buchmann. Polyhedra pastimes , links to teaching activities collected by J. Britton. A polyhedral analysis . Ken Gourlay looks at the Platonic solids and their stellations. Polyhedron, polyhedra, polytopes, ... - Numericana . Polyhedron challenge: cuboctahedron . Polyhedron web scavenger hunt PolyGloss . Wendy Krieger is unsatisfied with terminology for higher dimensional geometry and attempts a better replacement. Her geometry works include some other material on higher dimensional polytopes. Polytope movie page . GIF animations by Komei Fukuda. Proofs of Euler's Formula . V-E+F=2, where V, E, and F are respectively the numbers of vertices, edges, and faces of a convex polyhedron. Puzzles by Eric Harshbarger , mostly involving colors of and mazes on polyhedra and polyominoes. Puzzles with polyhedra and numbers , J. Rezende. Some questions about labeling edges of platonic solids with numbers, and their connections with group theory. The Puzzling World of Polyhedral Dissections . Stewart T. Coffin's classic book on geometric puzzles, now available in full text on the internet! Quark constructions . The sun4v.qc Team investigates polyhedra that fit together to form a modular set of building blocks. A quasi-polynomial bound for the diameter of graphs of polyhedra , G. Kalai and D. Kleitman, Bull. AMS 26 (1992). A famous open conjecture in polyhedral combinatorics (with applications to e.g. the simplex method in linear programming) states that any two vertices of an n-face polytope are linked by a chain of O(n) edges. This paper gives the weaker bound O(nlog d). Realization Spaces of 4-polytopes are Universal , G. Ziegler and J. Richter-Gebert, Bull. AMS 32 (1995). Regular polyhedra as intersecting cylinders . Jim Buddenhagen exhibits ray-traces of the shapes formed by extending half-infinite cylinders around rays from the center to each vertex of a regular polyhedron. The boundary faces of the resulting unions form combinatorially equivalent complexes to those of the dual polyhedra. Regular polytopes in higher dimensions . Russell Towle uses Mathematica to slice and dice simplices, hypercubes, and the other high-dimensional regular polytopes. See also Russell's 4D star polytope quicktime animations . Regular polytopes in Hilbert space . Dan Asimov asks what the right definition of such a thing should be. Regular solids . Information on Schlafli symbols, coordinates, and duals of the five Platonic solids. (This page's title says also Archimedean solids, but I don't see many of them here.) Resistance and conductance of polyhedra . Derek Locke computes formulae for networks of unit resistors in the patterns of the edges of the Platonic solids. See also the section on resistors in the rec.puzzles faq . Rhombic spirallohedra , concave rhombus-faced polyhedra that tile space, R. Towle. Rolling polyhedra . Dave Boll investigates Hamiltonian paths on (duals of) regular polyhedra. Ruler and Compass . Mathematical web site including special sections on the geometry of polyhedrons and geometry of polytopes . The Simplex: Minimal Higher Dimensional Structures . D. Anderson. Simplex hyperplane intersection . Doug Zare nicely summarizes the shapes that can arise on intersecting a simplex with a hyperplane: if there are p points on the hyperplane, m on one side, and n on the other side, the shape is (a projective transformation of) a p-iterated cone over the product of m-1 and n-1 dimensional simplices. Six-regular toroid . Mike Paterson asks whether it is possible to make a torus-shaped polyhedron in which exactly six equilateral triangles meet at each vertex. SMAPO library of polytopes encoding the solutions to optimization problems such as the TSP. Soap bubble 120-cell from the Geometry Center archives. Squares are not diamonds . Izzycat gives a nice explanation of why these shapes should be thought of differently, even though they're congruent: they generalize to different things in higher dimensions. Stella , Windows software for visualizing regular and semi-regular polyhedra and their stellations, morphing them into each other, drawing unfolded nets for making paper models, and exporting polyhedra to various 3d design packages. Stellations of the dodecahedron stereoscopically animated in Java by Mark Newbold. Sterescopic polyhedra rendered with POVray by Mark Newbold. The Story of the 120-cell , John Stillwell, Notices of the AMS. History, algebra, geometry, topology, and computer graphics of this regular 4-dimensional polytope. Structors . Panagiotis Karagiorgis thinks he can get people to pay large sums of money for exclusive rights to use four-dimensional regular polytopes as building floor plans. But he does have some pretty pictures... Student of Hyperspace . Pictures of 6 regular polytopes, E. Swab. Superliminal Geometry . Topics include deltahedra, infinite polyhedra, and flexible polyhedra. Symmetries of torus-shaped polyhedra Symmetry, tilings, and polyhedra , S. Dutch. Synergetic geometry , Richard Hawkins' digital archive. Animations and 3d models of polyhedra and tensegrity structures. Very bandwidth-intensive. The Szilassi Polyhedron . This polyhedral torus, discovered by L. Szilassi , has seven hexagonal faces, all adjacent to each other. It has an axis of 180-degree symmetry; three pairs of faces are congruent leaving one unpaired hexagon that is itself symmetric. Tom Ace has more images as well as a downloadable unfolded pattern for making your own copy. See also Dave Rusin's page on polyhedral tori with few vertices . 3D-Geometrie . T. E. Dorozinski provides a gallery of images of 3d polyhedra, 2d and 3d tilings, and subdivisions of curved surfaces. Three dimensional turtle talk description of a dodecahedron . The dodecahedron's description is "M40T72R5M40X63.435T288X296.565R5M40T72M40X63.435T288X296.565R4"; isn't that helpful? Three untetrahedralizable objects Triangles and squares . Slides from a talk I gave relating a simple 2d puzzle, Escher's drawings of 3d polyhedra, and the combinatorics of 4d polytopes, via angles in hyperbolic space. Warning: very large file (~8Mb). For more technical details see my paper with Kuperberg and Ziegler . Truncated Octahedra . Hop David has a nice picture of Coxeter's regular sponge {6,4|4}, formed by leaving out the square faces from a tiling of space by truncated octahedra. Truncated Trickery: Truncatering . Some truncation relations among the Platonic solids and their friends. Tuvel's Polyhedra Page and Tuvel's Hyperdimensional Page . Information and images on universal polyhedra and higher dimensional polytopes. Two-distance sets . Timothy Murphy and others discuss how many points one can have in an n-dimensional set, so that there are only two distinct interpoint distances. The correct answer turns out to be n2 2 + O(n). This talk abstract by Petr Lisonek (and paper in JCTA 77 (1997) 318-338) describe some related results. Uniform polychora . A somewhat generalized definition of 4d polytopes, investigated and classified by J. Bowers, the polyhedron dude. See also the dude's pages on 4d polytwisters and 3d uniform polyhedron nomenclature . Uniform polyhedra . Computed by Roman Maeder using a Mathematica implementation of a method of Zvi Har'El. Maeder also includes separately a picture of the 20 convex uniform polyhedra , and descriptions of the 59 stellations of the icosahedra . Uniform polyhedra in POV-ray format , by Russell Towle. Uniform polyhedra , R. Morris. Rotatable 3d java view of these polyhedra. An uninscribable 4-regular polyhedron . This shape can not be drawn with all its vertices on a single sphere. Variations of Uniform Polyhedra , Vince Matsko. Visual techniques for computing polyhedral volumes . T. V. Raman and M. S. Krishnamoorthy use Zome-based ideas to derive simple expressions for the volumes of the Platonic solids and related shapes. Visualization of the Carrillo-Lipman Polytope . Geometry arising from the simultaneous comparison of multiple DNA or protein sequences. Volumes in synergetics . Volumes of various regular and semi-regular polyhedra, scaled according to inscribed tetrahedra. Volumes of ideal hyperbolic hypercubes . Volumes of pieces of a dodecahedron . David Epstein (not me!) wonders why parallel slices through the layers of vertices of a dodecahedron produce equal-volume chunks. Waterman polyhedra , formed from the convex hulls of centers of points near the origin in an alternating lattice. See also Paul Bourke's Waterman Polyhedron page . Why doesn't Pick's theorem generalize? One can compute the volume of a two-dimensional polygon with integer coordinates by counting the number of integer points in it and on its boundary, but this doesn't work in higher dimensions. Why "snub cube"? John Conway provides a lesson on polyhedron nomenclature and etymology. From the geometry.research archives. Zonohedra and zonotopes . These centrally symmetric polyhedra provide another way of understanding the combinatorics of line arrangements. From the Geometry Junkyard , computational and recreational geometry pointers. Send email if you know of an appropriate page not listed here. David Eppstein , Theory Group , ICS , UC Irvine . Semi-automatically filtered from a common source file.
The Mathematical Atlas - Polytopes and Polyhedra
Collection of discussions regarding volume, vertices, dissection, g-holed tori, and other subjects.
52B: Polytopes and polyhedra [ Search ] [ Subject Index ] [ MathMap ] [ Tour ] [ Help! ] ABOUT: [ Introduction ] [ History ] [ Related areas ] [ Subfields ] POINTERS: [ Texts ] [ Software ] [ Web links ] [ Selected topics here ] 52B: Polytopes and polyhedra Introduction Here are a few files concerning geometric objects made from straight pieces: polygons, polyhedra, and generalizations. History Applications and related fields Questions regarding the underlying spaces (and algebraic invariants) are included on the topology pages. Classic questions regarding polygons and regular solids are included on the geometry pages. There is a separate section on constructibility of polygons (and other things) with ruler and compass . Many of these questions are related to themes arising in geometric visualization, a topic covered reasonably well on the net. In particular, the newsgroup comp.graphics.algorithms considers such themes from time to time. Some such topics are here, others on the page for computational geometry . Subfields 52B05 : Combinatorial properties (number of faces, shortest paths, etc.), See also 05Cxx 52B10 : Three-dimensional polytopes 52B11 : n-dimensional polytopes 52B12 : Special polytopes (linear programming, centrally symmetric, etc.) 52B15 : Symmetry properties of polytopes 52B20 : Lattice polytopes (including relations with commutative algebra and algebraic geometry), See also 06A08, 13F20, 13Hxx 52B22 : Shellability [new in 2000] 52B35 : Gale and other diagrams 52B40 : Matroids (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), See Also 05B35 52B45 : Dissections and valuations (Hilbert's third problem, etc.) 52B55 : Computational aspects related to convexity, For computational geometry and algorithms, See 68Q20, 68Q25, 68U05; for numerical algorithms, See 65Yxx 52B60 : Isoperimetric problems for polytopes 52B70 : Polyhedral manifolds 52B99 : None of the above but in this section Parent field: Convex and discrete geometry Browse all (old) classifications for this area at the AMS. Textbooks, reference works, and tutorials Coxeter, H. S. M.; Du Val, P.; Flather, H. T.; Petrie, J. F.: "The fifty-nine icosahedra", Springer-Verlag, New York-Berlin, 1982. 26 pp. ISBN 0-387-90770-X "Regular Polytopes", H.S.M Coxeter (Dover reprint) -- lots of formulas for polyhedra and so on. Schreiber, Peter: "What is the true number of semiregular (Archimedean) solids?", Festschrift on the occasion of the 65th birthday of Otto Krtenheerdt. Beitrge Algebra Geom. 35 (1994), no. 1, 91--94. MR95e:52020 "Polyhedron Models", by Magnus J. Wenninger (Cambridge University Press, London-New York, 1971): has models of all 52 uniform polyhedra and some stellations. "Space, Shapes, and Symmetry" by Holden -- lots of pictures of models, not limited to paper. Software and tables Other web sites with this focus virtual polyhedra Selected topics at this site A few basic questions keep arising with regard to polygons: How do you compute the area enclosed by a polygon? How do you find its center of mass of a polygon (program included) How do you find the centroid of a polygon (pointer) How do you decide if a point is interior How to decide if you're inside a polygon ? (pointer, citation) How can you check a corner for concavity . A combinatorial question: how many regions result when connecting all the vertices of a regular polygon? Number of regions formed by diagonals in a polygon Unbounded number of regions as intersection of convex polygons A couple of odds and ends of pointers to software (not tested here!): A pointer to code for Delaunay triangulation How to plot circular motion on a display? How to compute the convex hull of some points in the plane? (An interesting variant on computing areas of cyclic polygons is also included. For variety, here's a sample of a trigonometric approach to determining the area of a pentagon. For triangles, you may wish to use Heron's formula Then a few comments about polyhedral surfaces in 3-space. Using Cayley-Menger determinant to determine radius of circumscribed sphere around a tetrahedron What shape is a soccer ball ? Regular, semiregular polyhedra and the disphenoid Can one always unwrap the surface of a polyhedron to get something flat and nonoverlapping? (open) Here's a long spiel (with short punchline) on evaluating volumes of polyhedra. Another post computing volumes of polyhedra. How to compute the volume of a polyhedron? Pointers, citations, summary Volume of a tetrahedron (in terms of sides) How to compute the volume of a simplex in R^n in terms of its sides. Relating volumes of simplices to vertices, edges, or lengths. Finding volumes of an n-dimensional polyhedron Proving the analogue of the Pythagorean theorem in higher dimensions. realizability of polyhedral surfaces. Pointers to hexaflexagons Flexible polyhedra . Bellows theorem: flexible polyhedra maintain their volume Instructions for making a kaleidocycle (flexible polyhedron) A chance encounter with polyhedral tori led to the chance to try some models. There are ways to build these with few polygonal pieces. We find some information when looking up piecewise-linear embeddings of n-holed tori into R^3. (n=0 includes the tetrahedron, for example.) Read about Polyhedral versions of 1- and 2-holed tori which have a small number of vertices Pasting information for 1- and 2-holed tori with few cells. Image of a one-holed torus made only with triangles, in which all pairs of the (seven) vertices are joined by edges. This in turn led to a discussion of just how few vertices you need to create g-holed tori. A summary of what the questions are regarding polyhedral tori . Program and literature review (both long) for g-holed tori with few vertices Another program for polyhedral g-holed tori . A short summary of some basic data for polyhedral tori. A related post on polyhedral tori . Triangularizations of tori -- how nice can they be? Is there a polyhedral torus made of equilateral triangles? [Offsite]Models of minimal polyhedral models of genus-6 tori (evidently cannot be linearly embedded into R^3) General questions on polyhedra: Just what is a polytope and how does it differ from a polygon or polyhedron? (opinions vary!) Open question: can every convex polyhedron be cut along edges, then laid flat without self-overlap ? If you want to really build these things, here are a couple of construction tips The rhombic dodecahedron , use as space filler. Pointer: virtual polyhedra (pretty pictures). References on Euler's formula for polyhedra. The four regular nonconvex polyhedra (Kepler-Poinsot) Numerical data for many polyhedra -- pointer Pointer to numerical data on 4-dimensional polytopes Coordinates of a dodecahedron Some information about the vertices of the dodecahedron Coordinates of an icosahedron and "rhombicubeoctahedron" . Some information about the edges of the dodecahedron. Description of the truncated octahedron . Pointer to gallery of Archimedean solids Edge-transitive polyhedra in R^3 Decomposing polyhedra into convex or tetrahedral pieces Under what circumstances can we decompose a polyhedron into pieces which reassemble into another given polyhedron? (The Dehn invariant ) Can a 3-dimensional polyhedron be decomposed into tetrahedra ? (Not without adding interior points in general) Tiling S^3 with 600 congruent spherical tetrahedra Visualizing the 16-cell (solid in R^4) Construction of the 120-cell and 600-cell (solids in R^4) That four-dimensional polytope with no three-dimensional analogue. Unusual four-dimensional polyhedra : the 24-cell, 120-cell, and 600-cell. Generalizations of the Platonic solids to dimensions 4 and up. From the sci.math FAQ: How can you chop up a ball and reassemble the parts (the Banach Tarski paradox, and related issues). Decomposing a square and a circle into congruent (nonmeasurable!) parts Two polygons of equal area may be decomposed into congruent triangles . A dissection problem: how to dissect a square into pieces with minimal perimeter. Dissecting a rectangle into squares -- ratio of sides rational Dissecting an equilateral triangle into incongruent equilateral triangles (can't) Dissecting a cube into distinct smaller cubes Dissecting n-cubes into (minimal numbers of) simplices Using Kirchhoff's laws to solve geometric combinatorial problems Divide a square into acute triangles Decomposing a square into incongruent squares: [No link here, in an attempt to avoid the ASCII art. See Guy's "Unsolved Problems in Geometry", section C2.] The maximum surface-area tetrahedron inscribed in a sphere is the regular one. A " tetrahedral inequality ": under what circumstances can six line segments be joined into a tetrahedron? Pointer to KALEIDO (program for regular polyhedra) You can reach this page through http: www.math-atlas.org welcome.html Last modified 2001 02 14 by Dave Rusin. Mail: rusin@math.niu.edu
Occurrence of the Conics
Images of conics by Jill Britton.
Occurrence of the Conics OCCURRENCE OF THE CONICS Mathematicians have a habit of studying, just for the fun of it, things that seem utterly useless; then centuries later their studies turn out to have enormous scientific value. There is no better example of this than the work done by the ancient Greeks on the curves known as the conics: the ellipse, the parabola, and the hyperbola. They were first studied by one of Plato's pupils. No important scientific applications were found for them until the 17th century, when Kepler discovered that planets move in ellipses and Galileo proved that projectiles travel in parabolas. Appolonious of Perga, a 3rd century B.C. Greek geometer, wrote the greatest treatise on the curves. His work "Conics" was the first to show how all three curves, along with the circle, could be obtained by slicing the same right circular cone at continuously varying angles. CLICK ON SELECTED IMAGES FOR INTERNET SOURCES OR FOR ENLARGED VIEWS OF LINE DRAWINGS THE ELLIPSE Though not so simple as the circle, the ellipse is nevertheless the curve most often "seen" in everyday life. The reason is that every circle, viewed obliquely, appears elliptical. Any cylinder sliced on an angle will reveal an ellipse in cross-section (as seen in the Tycho Brahe Planetarium in Copenhagen). Tilt a glass of water and the surface of the liquid acquires an elliptical outline. Salami is often cut obliquely to obtain elliptical slices which are larger. The early Greek astronomers thought that the planets moved in circular orbits about an unmoving earth, since the circle is the simplest mathematical curve. In the 17th century, Johannes Kepler eventually discovered that each planet travels around the sun in an elliptical orbit with the sun at one of its foci. The orbits of the moon and of artificial satellites of the earth are also elliptical as are the paths of comets in permanent orbit around the sun. Halley's Comet takes about 76 years to travel abound our sun. Edmund Halley saw the comet in 1682 and correctly predicted its return in 1759. Although he did not live long enough to see his prediction come true, the comet is named in his honour. On a far smaller scale, the electrons of an atom move in an approximately elliptical orbit with the nucleus at one focus. The ellipse has an important property that is used in the reflection of light and sound waves. Any light or signal that starts at one focus will be reflected to the other focus. This principle is used in lithotripsy, a medical procedure for treating kidney stones. The patient is placed in a elliptical tank of water, with the kidney stone at one focus. High-energy shock waves generated at the other focus are concentrated on the stone, pulverizing it. The principle is also used in the construction of "whispering galleries" such as in St. Paul's Cathedral in London. If a person whispers near one focus, he can be heard at the other focus, although he cannot be heard at many places in between. Statuary Hall in the U.S. Capital building is elliptic. It was in this room that John Quincy Adams, while a member of the House of Representatives, discovered this acoustical phenomenon. He situated his desk at a focal point of the elliptical ceiling, easily eavesdropping on the private conversations of other House members located near the other focal point. The ability of the ellipse to rebound an object starting from one focus to the other focus can be demonstrated with an elliptical billiard table. When a ball is placed at one focus and is thrust with a cue stick, it will rebound to the other focus. If the billiard table is live enough, the ball will continue passing through each focus and rebound to the other. THE PARABOLA One of nature's best known approximations to parabolas is the path taken by a body projected upward and obliquely to the pull of gravity, as in the parabolic trajectory of a golf ball. The friction of air and the pull of gravity will change slightly the projectile's path from that of a true parabola, but in many cases the error is insignificant. This discovery by Galileo in the 17th century made it possible for cannoneers to work out the kind of path a cannonball would travel if it were hurtled through the air at a specific angle. When a baseball is hit into the air, it follows a parabolic path; the center of gravity of a leaping porpoise describes a parabola. The easiest way to visualize the path of a projectile is to observe a waterspout. Each molecule of water follows the same path and, therefore, reveals a picture of the curve. The fountains of the Bellagio Hotel in Las Vegas comprise a parabolic chorus line. Parabolas exhibit unusual and useful reflective properties. If a light is placed at the focus of a parabolic mirror (a curved surface formed by rotating a parabola about its axis), the light will be reflected in rays parallel to said axis. In this way a straight beam of light is formed. It is for this reason that parabolic surfaces are used for headlamp reflectors. The bulb is placed at the focus for the high beam and a little above the focus for the low beam. The opposite principle is used in the giant mirrors in reflecting telescopes and in antennas used to collect light and radio waves from outer space: the beam comes toward the parabolic surface and is brought into focus at the focal point. The instrument with the largest single-piece parabolic mirror is the Subaru telescope at the summit of Mauna Kea in Hawaii (effective diameter: 8.2 m). Heat waves, as well as light and sound waves, are reflected to the focal point of a parabolic surface. If a parabolic reflector is turned toward the sun, flammable material placed at the focus may ignite. (The word "focus" comes from the Latin and means fireplace.) A solar furnace produces heat by focusing sunlight by means of a parabolic mirror arrangement. Light is sent to it by set of moveable mirrors computerized to follow the sun during the day. Solar cooking involves a similar use of a parabolic mirror. Two types of images exist in nature: real and virtual. In a real image, the light rays actually come from the image. In a virtual image, they appear to come from the reflected image - but do not. For example, the virtual image of an object in a flat mirror is "inside" the mirror, but light rays do not emanate from there. Real images can form "outside" the system, where emerging light rays cross and are caught - as in a Mirage , an arrangement of two concave parabolic mirrors. Mirage's 3-D illusions are similar to the holograms formed by lasers. THE HYPERBOLA If a right circular cone is intersected by a plane parallel to its axis, part of a hyperbola is formed. Such an intersection can occur in physical situations as simple as sharpening a pencil that has a polygonal cross section or in the patterns formed on a wall by a lamp shade. A sonic boom shock wave has the shape of a cone, and it intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. Because the airplane is moving forward, the hyperbolic curve moves forward and eventually the boom can be heard by everyone in its path. A hyperbola revolving around its axis forms a surface called a hyperboloid . The cooling tower of a steam power plant has the shape of a hyperboloid, as does the architecture of the James S. McDonnell Planetarium of the St. Louis Science Center. All three conic sections can be characterized by moir patterns. If the center of each of two sets of concentric circles is the source of a radio signal, the synchronized signals would intersect one another in associated hyperbolas. This principle forms the basis of a hyperbolic radio navigation system known as Loran (Long Range Navigation). DOWNLOAD A HIGH-RES GRID (requires Adobe Acrobat Reader ) Jill Britton Home Page parabolic mirror animation: Lasse Bjrklund 13-August-2005 Copyright Jill Britton
Tim's Triangular Page
Explains the basic properties, formulas, theorems and different types of triangles. Includes proofs and illustrations.
Tim's Triangular Page The Sakharovs Alex Irina Tim Email Projects Resources Sport Photos Median Logic Math Foundations Badminton Clubs Trip Photos Propositional Validity Constraint Languages NH Skiing Whales Recursive Data Types XML Transformation Park City New Hampshire Finite State Machines Data Languages Art Nouveau Component Communication Downloads Air Show Assertion Propagation Puzzles Triangles London Spain Whales NH Junior Skiing Park City NH Nature Air Show M o r e F a c t s - - - P r o b l e m s Tim's Triangular Page This page may look rectangular but in fact it is a triangular page - just read the contents below and you will see. More Advanced Facts about Triangles Min Max Problems Related To Triangles Basic Facts Scalene Triangle Isosceles Triangle Equilateral Triangle Acute Triangle Obtuse Triangle Right Triangle Congruence Similar Triangles Outer (Exterior) Angle Median, Centroid Bisectrix (Bisector) Height (Altitude), Orthocenter Incircle, Circumcircle Perimeter, Area Pythagorean Theorem Side-Side-Side Congruence Theorem Side-Angle-Side Congruence Theorem Angle-Side-Angle Congruence Theorem Hypotenuse-Leg Congruence Theorem Golden Triangle A triangle is a closed plane figure bounded by three straight lines meeting at three different points. The three intersection points are triangle vertices. The line segments between the vertices are triangle sides. A triangle can also be defined as a three-sided polygon . Basic facts about triangles: The sum of the angles of a triangle is 180 degrees, and thus, every angle in a triangle is less than 180 degrees, and the biggest angle is at least 60 degrees. Every single side of a triangle is shorter than the sum of the lengths of the remaining two sides. The difference between the lengths of any two sides of a triangle is less than the length of the third side. An angle lying opposite to a bigger triangle side is also a bigger angle, and inversely. Scalene Triangle A triangle having three sides of different lengths is called a scalene triangle. All three angles of a scalene triangle are different. Inversely, if all three angles of a triangle are different, then this triangle is scalene. Isosceles Triangle A triangle having two sides of equal length is called an isosceles triangle. The two angles adjacent to the third side of an isoscales triangle are equal. If a triangle has two equal angles, it is an isoscales triangle, i.e. it has two equal sides too. In an isosceles triangle, the median, bisectrix, and height of the vertext between the equal sides all coincide. This median bisectrix height divides the isoscales triangle into two congruent right triangles. If a median coincides with a height, or if a bisectrix coincides with a height, or if a median coincides with a bisectrix (see definitions below), then the triangle is isosceles - the adjacent sides are equal. If BD is a median and a height of triangle ABC, then triangles ABD and CBD are equal because they are right triangles whose both legs are equal, respectively. Therefore, AB = BC. If BD is a bisectrix and is also a height, then triangles ABD and CBD are equal because they are right triangles with one common leg and equal acute angles. If BD is a median and a bisectrix of triangle ABC, then triangles ABD and CBD are equal because all three angles of ABD and CBD are equal, respectively, and AD = CD. The medians of the equal sides of an isosceles triangle are equal. The heights and bisectors crossing the equal sides are equal too (see definitions below). If AE and CD are medians of triangle ABC in which AB = BC, then triangles AEC and CDA are equal because they have a common side, AD = EC and angle BAC = BCA. Therefore, AE = CD. If AE and CD are heights of ABC, then right triangles AEC and CDA are equal because they have a common hypotenuse and angle BAC = BCA. If AE and CD are bisectors of triangle ABC, then triangles AEC and CDA are equal because angles BAC = BCA, ACD = CAE, and again they have a common side. Equilateral Triangle Equiangular Triangle A triangle having all three sides of equal length is called an equilateral triangle. A triangle having three equal angles is called an equiangular triangle. The angles of an equilateral triangle are all the same, they all measure 60 degrees. Thus, every equilateral triangle is also an equiangular triangle. Invesely, every equiangular triangle is also an equilateral triangle. The sum of the distances from any point in the interior of an equilateral triangle to all three sides is always equal to the height (see its definition below) of the equilateral triangle. Let us draw lines DE parallel to AC, FG parallel to AB, IJ parallel to BC. Triangles FPJ, DIP, and PGE are similar to ABC and therefore are equilateral. Denote the heights of the triangles FPJ, DIP, PGE, and ABC by e, f, g, h, respectively. e h = FJ AC, f h = DI AB, g h = GE BC. Let us add these thee equalities. Note that DP = AF and PE = JC because ADPF and JPEC are apallelograms. Using the fact that three sides of each of FPJ, DIP, PGE, and ABC are equal, we get the following equality: (e+f+g)) h = 1. Acute Triangle A triangle having three acute angles, i.e. angles that measure less than 90 degrees, is called an acute triangle. Obtuse Triangle A triangle having an obtuse angle, i.e. an angle that measures more than 90 degrees, is called an obtuse triangle. Note that the other two angles of an obtuse trianle are less than 90 degrees. Right Triangle A triangle having a right angle, i.e. an angle that measures 90 degrees, is called a right triangle. Note that the other two angles of a right trianle are less than 90 degrees. The two acute angles of an isosceles right triangle measure 45 degrees. The side opposite the right angle is called the hypotenuse. The two sides that form the right angle are called the legs. Every leg in every right triangle is shorter than the hypotenuse. The sum of the angles adjacent to the hypotenuse is 90 degrees. The distances between the midpoint of the hypotenuse and all three vertices are the same. In other words, the length of a hypotenuse median (see its definition below) is half of the hypotenuse length. Let us circumscribe a circle around right triangle ABC. If angle BAC is 90 degrees, then chord BC is a diameter and the midpoint M of BC is the center of the circumcircle. Thence, MA is also a radius of the circumcircle like MB and MC. Congruence Two triangles are congruent (equal) if they have identical size and shape so that thet can be exactly superimposed. Two congruent triangles ABC and CDA with a common side and equal angles located at two different ends of the common side always form a parallelogram . If AB=BC, then it is a rombuss . If angle ABC is 90 degrees, it is a rectangle . If AB=BC and angle ABC is 90 degrees, it is a square . Similar Triangles Two triangles are called similar if all their angles are equal, respectively. Note that it is sufficient for two triangles to have two pairs of equal angles to be similar. If triangles are congruent, then they are similar es well. In similar triangles, corresponding sides are proportional. Inversely, if all three sides of two triangles are proportional, then these triangles are similar. If two sides of two triangles are proportional and the angle between the sides are equal, then these triangles are similar. If r is the ratio of sides of two similar triangles, then the ratio of their areas (see its definition below) is r2. In similar triangles, heights, bisectors, and medains (see definitions below), respectively, are proportional as well. This follows from the fact that each of the pairs of triangles formed by a height, bisector, or medain in one similar triangle is similar to its counterpart in the other. It is apparent that the angles of the triangles formed by heights and bisectors are equal, respectively. As far as the triangles formed by medians are concerned, two of their sides are apparently proportional, and the angles between the proportional sides are equal. Consider triangle ABC. If D and E are the points of intersection of a line parallel to BC with sides AB and AC, respectively, then triangle ADE is similar to ABC. Inversely, if triangle ADE is similar to ABC, and D lies on AB, E lies on AC, then DE is parallel to BC. If triangles ABC and ADE are such that AB and AE are segments of the same straight line, AC and AD are also segments of the same line, BC and DE are parallel, then triangles ABC and ADE are similar. In general, if two triangles have parallel (or coinciding) sides, respectively, then they are similar. Consider right triangle ABC. If BD is a height going from the right-angle vertex, then triangles ABC, ABD, and BDC are all similar. In general, any line EF perpendicular to hypotenuse AC forms triangle AFE similar to ABC. The same would hold if E lied on BC. Outer (Exterior) angle Angle BCD (and other angles like this) is called an outer (exterior) angle of triangle ABC. It is equal to the sum of both non-adjacent angles ABC and BAC. Therefore, the outer angle is bigger than each of the two non-adjacent angles of the triangle. Median, Centroid The segment of the straight line joining a triangle vertex and the midpoint of the opposite site is called a median. All three medians always intersect in one point called a centroid. This is a corollary of Ceva's theorem . Any centroid divides all medians into a 1:2 ratio, with the larger portion toward the vertex and the smaller portion toward the side: GD:AG=1:2, GE:BG=1:2, GF:CG=1:2. Let F be the median point of AB, FD be parallel to AC. Then, triangles ABC and FBD are similar, BD = BC 2, FD = AC 2. Triangles AGC and GDF are similar too. Therefore, GD = AG 2 and GF = CG 2. Same considerations about line FE drawn parallel to BC lead to the conclusion that EG = BG 2. In a isosceles triangle, equal sides have equal medians and inverseley. A median separates the triangle into two triangles of equal area. A longer side always has a shorter median. Let BC AB. If CD and AE are medians, then DE is parallel to AC. Let us draw DF and EG perpendicular to AC. FDEG is a rectangular. ADF and GEC are right triangles whose legs DF and EG are equal. Since the hypotenuse of GEC is longer than the hypotenuse of ADF, leg GC is also longer than leg AF. (It follows from Pythagorean theorem.) In right triangles AEG and FDC, leg FC is longer than leg AG. The other legs are equal again. Therefore, hypotenuse CD is longer than hypotenuse AE. A triangle angle is acute, right, or obtuse if and only if half of the length of the opposite side is less, equal, or more than its median. If AC 2*BM, then AM BM and MC BM. Therefore, angle ABM BAM; MBC BCM, ABC BAC + BCA. Hence, angle ABC is less than 90 degrees. The cases AC = 2*BM and AC 2*BM are similar. The following is called Midpoint Theorem. Consider a triangle whose vertices (D,E,F) are the three midpoints of a given triangle ABC. Triangle DEF is congruent to triangles AFE, BFD, and CED. The medians of DEF coincide with the medians of ABC. The following pairs of lines are parallel : AB and ED, AC and FD, BC and FE. Besides, triangle ABC is similar to DEF, AFE, BFD, and CED. The lengths of the sides of DEF, AFE, BFD, and CED are half of the lengths of the respective sides of ABC: FD=AC 2, FE=BC 2, ED=AB 2. If F is the midpoint of AB, then draw FD parallel to AC and draw FE parallel to BC. Triangles AFE and ABC are similar and so are triangles BFD and BAC. Thus AE is half of AC and BD is half of BC, i.e. D and E are the midpoints of their respective sides. Triangles BFD and FAE are equal since AF = FB and their both adjacent angles are equal too because they are formed by parallel lines. Similar considerations can be applied to triangles BFD and DEC. Triangle FDE is equal to EAF because they share one side and its adjacent angkles are formed by parallel lines too. Finally, P is the midpoint of FE because triangles FDP and EAP are equal. Bisectrix (Bisector) The segment of the straight line bisecting a triangle angle and going from the angle vertex till the opposite side is called a bisectrix or bisector. The distance from any bisector point to one adjacent side is the same as the distance to the other. All three bisectors always intersect in one point. It follows from Ceva's theorem . Any point of a bisectrix is equidistant from the sides of the angle. In a isosceles triangle, equal sides have equal bisectors and inverseley. A longer side always has a shorter bisector. Suppose CM and AN be bisectors, and BC AB. Circumscribe a circle through points A, C, and N. This circle crosses MC at point K that lies inside the triangle because angle KAN = KCN. Angles KAN and KCN are inscribed angles with the same chord. Since angle BAC is bigger than ACB, angle KAN is smaller than BAN. Chord KC is bigger than chord AN because a bigger incribed angle has a bigger chord and KAC ACN. Therefore, MC AN. Any bisectrix divides the side opposite to the angel into segments proportional to the sides adjacent to the angle. If BD is a bisectrix, let us draw line AE parallel to the bisector BD. Since triangles AEC and DBC are similar, (AD+DC) DC = (EB+BC) BC. Due to the fact that AE and BD are parallel, the following equalities hold for angles: AEC = DBC, EAB = ABD. Therefore, AEB is an isosceles triangle: EB = AB. Substituting AB for EB in the first equality and opening brackets gives the following: AD DC = AB BC. In every triangle, a bisectrix alway lies between the height and median going from the same vertex. If BE is a bisectrix of a right angle, BD is its height, and BF is its median, than angles DBE and EBF are equal. Let BD, BE, and BF are the height, bisector, and median, respectively. The case of the height lying outside ABC is proved similarly to this one. Consider case BC AB. The opposite case is similar to this one. If BC = AB, then BD, BE, and BF coincide. First, angle BCA BAC. Angle ABD CBD because they complement angles BCA and BAC to 90 degrees. Consequently, angle ABD ABE, and E lies between D and A. Second, CE EA = BC AB 1. Hence, CE EA, and CE CF, that is, F lies between E and A. If angle ABC is 90 degrees, then BF = FA, angle ABF = CAB. Both angle CBD and CAB complement angle BCA to 90 degrees, and hence, they are equal. So are angles DBE and EBF. Height (Altitude), Orthocenter The segment of the straight line going from a triangle vertex till the opposite side (base) and crossing the opposite side or its continuation at the right angle is called a height or altitude. In contrast to medians and bisectors which alway lie within their triangles, heights may lie outside of their triangles. It happens in obtuse triangles. Two their heights lie outsisde the triangle. In right triangle, two heights coincide with the legs. All three heights always intersect in one point called an orthocenter. Again, this is a corollary of Ceva's theorem . Orthocenters are located inside acute triangles and outside of obtuse triangles. In right triangles, orthocenters coincide with the vertices connecting two legs. A longer side always has a shorter height. In a isosceles triangle, equal sides have equal heights and inverseley. If a, b, and c are triangle sides, and f, g, and h are their respective heights, then a*f = b*g = c*h because all three products amount to the doubled area of the triangle. Therefore, if a b, then g f, and so on. Incircle, Circumcircle, Euler Line A circle that touches each of the triangle's three sides is called an inscribed circle or incircle. A circle that passes through each of the triangles three vertices is called a circumscribed circle or circumcircle. Both incircle and circumcircle are unique for every triangle. The center of the incircle is located at the point of intersection of triangle bisectors. The center of the circumcircle is located at the point of intersection of three lines perpendicular to triangle sides and crossing them at the midpoints. The distances from the point of intersection of triangle bisectors to all three triangle sides are equal. Therefore, this point is the incircle center as well. Now, let F, G, and H are the midpoints of AB, BC, and AC, respectively. And let P be the point where perpendiculars going through F and G cross. Triangles AFP and FBP are equal (S-A-S), so are triangles BGP and GCP (Side-Angle-Side). Therefore, AP = BP and BP = CP. PH is perpendicular to AC because triangles APH and CPH are equal (Side-Side-Side) and angles AHP and CHP are bothe equal and complimentary. The center of the circumcircle of a right triangle is the midpoint of triangle's hypotenuse. The center of the circumcircle of acute triangles is located inside the triangles whereas it is located outside of obtuse triangles. The hypotenuse of a right triangle is a diameter of its circumcircle because its inscribed angle is 90 degrees. And the center of the circumcirle is the midpoint of the hypotenuse. Inversely, if the center of the circumcircle lies on a triangle side, then this is a right triangle. Let us prove the two following facts: - if the center of the circumcircle lies inside the triangle, then the triangle is acute - if the center of the circumcircle lies outside the triangle, then the triangle is obtuse These two facts along with the above fact about the circumcirles whose center is located on a triangle side imply that the center of the circumcircle of acute triangles is located inside the triangles and it is located outside of obtuse triangles. If D is the center of the circumcircle of triangle ABC, then angle EBC is 90 degrees. If D is inside triangle ABC, then angle ABC EBC. If D is outside ABC, then angle ABC EBC. The radius of the circumcircle of an equilateral triangle whose side is s equals s whereas the radius of its incircle is s (2* ). If BE is a bisector and D is the point where medians, bisectors and heights of ABC intersect, then DE = BD 2. D is the center of the incircle. It is also the center of the circumcircle because the medians are perpendicular to the respective sides. By Pythagorean theorem BE = *s 2. The radius of the incircle is BE 3, the radius of the circumcircle is 2*BE 3. If an equilateral triangle is erected on each side of any triangle and these equilateral triangles are exterior to the original triangle, then the segments connecting the centers of the circumcircles of the three equilateral triangles form an equilateral triangle. This is called Napoleon's theorem. Click here to see a proof. The orthocenter, centroid, and circumcenter of any triangle lie on the same line. It is called Euler line. Click here to see a proof. Another interesting circle related to triangles is the Nine-Point Circle. Click here to learn about the Nine-Point Circle. Perimeter, Area The sum of the lengthes of all three triangle sides is called a perimeter. Half of a perimeter is called a semiperimeter. If point P is in the interior of triangle ABC, then AP + BP + CP is bigger than ABC semiperimeter (AB + BC + CA) 2. Since any triangle side is less than the sum of the other two, AB AP + BP, AC AP + CP, BC BP + CP. Adding these three inequalities gives: AB + BC + AC 2*(AP + BP +CP). The area of a triangle is half of the product of its height and the base: (a.h) 2. Any of the three heights can be chosen for the calculation of the area. The result will be the same. The area of a right triangle can be calculated as half of the product of its legs. The area of a a equilateral triangle with side s equals *s2 4. If BE is a height, then it is also a median and EC = AC 2. By Pythagorean theorem BE = *s 2. The area of ABC is *s2 4. Heron's formula gives another expression for calculating the area. If p is the semiperimeter of a triangle, i.e. p = (a+b+c) 2, then the area is: Pythagorean Theorem The sum of the areas of two squares whose sides equal to the two legs, respectively, of a right triangle is the same as the area of the square whose sides equal to the hypotenuse: a2+b2=c2. Click here to see a collection of various proofs of Pythagorean Theorem. In every acute triangle, the area of the square whose sides are as the biggest triangle side is less than the sum of the areas of the two squares whose sides equal to two smaller sides, respectively. In every obtuse triangle, the area of the square whose sides are as the biggest triangle side is more than the sum of the areas of the two squares whose sides equal to two smaller sides, respectively. The Side-Side-Side Triangle Congruence Theorem If two triangles have the same side lengths, then the triangles are congruent. The Side-Angle-Side Triangle Congruence Theorem If two triangles have two sides equal, respectively, and the angles between the two sides are equal too, then the triangles are congruent. Corollary, if two right triangles have both legs equal, respectively, then they are congruent. The Angle-Side-Angle Triangle Congruence Theorem If two triangles have two angles and the sides between the two angles equal, respectively, then the triangles are congruent. Corollary, if two right triangles have one leg and the adjacent acute angle equal, respectively, then they are congruent. If two right triangles have hypotenuse and one acute angle equal, respectively, then they are congruent. No Angle-Side-Side Theorem! Note that two triangles with two sides and a non-included angle equal are not necessarily congruent. In many instances, there could be two different triangles with given two sides and a non-included angle. See triangles ABD and ACD on the right. The Hypotenuse-Leg Triangle Congruence Theorem If two right triangles have the hypotenuse and a leg equal, respectively, then the triangles are congruent. Golden Triangle An isosceles triangle such that the ratio of the length of the two equal sides to the length of the third side is the golden ratio is called a golden triangle. The angle between equal sides in a golden triangle is 36 degrees, which is 1 10 of the full angle or 1 5 of the straight angle . Note that the sides of the Great Pyramid of Giza are golden triangles. More Advanced Facts about Triangles Min Max Problems Related To Triangles When triangle study is over, it is time for a change. Why not to try puzzles now? Copyright (c) 2004 Timothy Sakharov, Alexander Sakharov Home Alex Irina Tim Email
Polygons
Discusses the differences between the different types of these straight sided figures. Also, explains the difference between regular and irregular shapes.
Polygons A polygon is a closed plane figure bounded by straight line segments. The line segments are called the sides of the polygon, and the points at which they intersect are called vertices. A polygon has the same number of sides as it has vertices. Polygons are classified according to the number of sides they have. A polygon with n sides is called an n-gon. Thus the polygon shown below is called an 9-gon. Nonagon Here are some more polygons that you may be familiar with. It should be recognisable at this stage that polygons surround us in our every day lives whether it be the face of the computer screen that you are looking at right now or the shape of the matt that your mouse rolls on. Other Polygons Here we have an 8-gon or more regularly known as an octagon. This is yet another polygon that we see every day that we take a drive in a car. Stop Sign Polygons can be broken into two distinct categories 1) Regular , 2) Irregular . Table of Contents What is a Polyhedron? Polygons Regular Irregular Platonic-Solids Tetrahedron Octahedron Hexahedron Icosahedron Dodecahedron Relationships Archimedean- Solids Truncated Tetrahedron Truncated Octahedron Truncated Hexahedron Truncated Icosahedron Truncated Dodecahedron Quasi-regular Polyhedra Rhombi Archimedeans Truncated Quasi-regulars Snub Polyhedra Polyhedra Spherical Geometry Prerequisite Knowledge Spherical Projection of the Cube Glossary of Terms HOME
Lessons on Perimeter and Area of Polygons from Math Goodies
Discusses the perimeter of polygons, area of trapezoids, and challenge exercises. Available on CD only are the area of triangles and parallegrams, as well as practice exercises with solutions.
Perimeter and Area of Polygons interactive click here fun CD Lessons Homework Puzzles Newsletter numbers Worksheets Forums Articles Home Ed Catalog The goal of this unit is to teach concepts on perimeter and area of polygons. Read the terms and conditions for using our free demo lessons. First time users also need to read this important information . Lesson Access Description Perimeter of Polygons To understand the concept of perimeter. To find the perimeter of various polygons and regular polygons. To express the answer using the proper units. Area of Rectangles To understand the concept of area, and how it differs from perimeter. To find the area of squares and rectangles using the proper formulas. To find the missing dimension given the area. To express the answer using the proper units. Available on CD only . Area of Parallelograms To find the area of a parallelogram using the proper formula. To find the missing dimension given the area and the other dimension. To express the answer using the proper units. Area of Triangles To find the area of various types of triangles using the proper formula. To express the answer using the proper units. Available on CD only . Area of Trapezoids To find the area of various trapezoids using the proper formula. To express the answer using the proper units. Practice Exercises To complete 10 additional exercises as practice. To assess students' understanding of all concepts learned so far. Available on CD only . Challenge Exercises To solve 10 additional problems that challenge students' understanding of perimeter and area of polygons. Problems are drawn from real-life situations. To hone students' problem-solving skills. Available on CD only . Solutions To review complete solutions to all exercises presented in this Volume. Includes the problem, step-by-step solutions, final answer and units for each exercise. Available on CD only . Want advanced notification of new lessons? Join our free Math Goodies Newsletter! Name: Email Address: Your position: Teacher Math Dept Chair Technology Coord Principal District Level Higher Ed Homeschooler Parent Student None of the above Your grade or level: K-2 3-5 6-8 9-12 K-12 College or University State or Province: Country: Delivery Format: HTML TEXT Manage Subscriptions Privacy Policy Other Lessons || leons en franais || Feedback Software Study Smart MathType Calculators Recommend numbers Advertise About Us Search Press Polls calculus geometry trigonometry algebra Last Modified Copyright 1998-2005 Mrs. Glosser's Math Goodies, Inc. All Rights Reserved. 23 Aug 2005 basic math pre-algebra pre-calculus adult learners . .
Xah: Visual Dictionary of Special Plane Curves
A Visual Dictionary of Special Plane Curves with graphics.
Xah: Visual Dictionary of Famous Plane Curves back to Xah's home A Visual Dictionary of Famous Plane Curves subscribe to mailing list CD-ROM available! Famous Plane Curves Ancient curves: Conic Sections Parabola Hyperbola Ellipse Cissoid of Diocles Conchoid of Nicomedes Quadratrix of Hippias Spirals: Archimedean Spiral Equiangular Spiral Lituus Clothoid Cyclodal curves: ( Epicycloid and Hypocycloid ( Astroid Deltoid Nephroid Cardioid )) Epitrochoid Hypotrochoid Rose ( Cycloid Trochoid ) Era of analytic geometry and calculus: Cassinian oval Cross Curve Folium of Descartes Piriform Semicubic Parabola Tractrix Trisectrix Trisectrix of Maclaurin Lemniscate of Bernoulli Lemniscate of Gerono Limacon of Pascal Witch of Agnesi Era of modern math: Sinusoid Catenary Bezier Curve Naming and Classification of Curves Methods of Generating New Curves Caustics Cissoid (General) Conchoid (General) Derivative and Integral Envelope Evolute Involute Inversion Isoptic and Orthoptic Parallel Curves Pedal Curve Radial Roulette Glissette Strophoid In the works line circle exponential curve Spiric Sections right strophoid Hyperbolic Sine Lissajous Polynomial the Bell Curve (Gaussian Normal curve) fractal curves: dragon curve, flowsnake, snowflake, Cantor dust, Peano curve math of curves (in the works). misc curves. Coordinate Systems . Cusp Curvature Info About This Site What's New: 2005-08 About this project, acknowledgement, copyright and usage . Software Aids: Mathematica , and other help softwares . Printed bibliography related web sites Robert Yates's Curves and Their Properties Wikipedia References See also: a Gallery of Famous Surfaces seashells exhibit spirals photo exhibit Curlicues photo exhibit Inversion Gallery Trochoid Animation Gallery Web XahLee.org Copyright 1995-2005 by Xah Lee . ( xah@xahlee.org ) http: xahlee.org SpecialPlaneCurves_dir specialPlaneCurves.html
Pythagorean Theorem
A collection of 43 proofs - some interactive - of the Pythagorean theorem.
Pythagorean Theorem and its many proofs Username: Password: Sites for teachers Sites for parents Terms of use Awards Interactive Activities CTK Exchange Games Puzzles Arithmetic Algebra Geometry Probability Eye Opener Analog Gadgets Inventor's Paradox Did you know?... Proofs Math as Language Things Impossible My Logo Math Poll Cut The Knot! MSET99 Talk Other Math sites Front Page Movie shortcuts Personal info Reciprocal links Privacy Policy Guest book News sites Recommend this site Sites for teachers Sites for parents Wholesale Shopping Health Information Online Student Loan Help Networking Software Management Training Courses Pythagorean Theorem Let's build up squares on the sides of a right triangle. Pythagoras' Theorem then claims that the sum of (the areas of) two small squares equals (the area of) the large one. In algebraic terms, a2 + b2 = c2 where c is the hypotenuse while a and b are the sides of the triangle. The theorem is of fundamental importance in the Euclidean Geometry where it serves as a basis for the definition of distance between two points. It's so basic and well known that, I believe, anyone who took geometry classes in high school couldn't fail to remember it long after other math notions got solidly forgotten. Below is a collection of various approaches to proving the theorem. Some of the proofs are accompanied by Java illustrations, but most have been written in plain HTML. Remark The statement of the Theorem was discovered on a Babylonian tablet circa 1900-1600 B.C. Whether Pythagoras (c.560-c.480 B.C.) or someone else from his School was the first to discover its proof can't be claimed with any degree of credibility. Euclid's (c 300 B.C.) Elements furnish the first and, later, the standard reference in Geometry. In fact Euclid supplied two very different proofs: the Proposition I.47 (First Book, Proposition 47) and VI.31. The Theorem is reversible which means that a triangle whose sides satisfy a2 + b2 = c2 is necessarily right angled. Euclid was the first (I.48) to mention and prove this fact. W. Dunham [ Mathematical Universe ] cites a book The Pythagorean Proposition by an early 20th century professor Elisha Scott Loomis. The book is a collection of 367 proofs of the Pythagorean Theorem and has been republished by NCTM in 1968. Pythagorean Theorem generalizes to spaces of higher dimensions. Some of the generalizations are far from obvious. Larry Hoehn came up with a plane generalization which is related to the law of cosines but is shorther and looks nicer. The Theorem whose formulation leads to the notion of Euclidean distance and Euclidean and Hilbert spaces, plays an important role in Mathematics as a whole. I began collecting math facts whose proof may be based on the Pythagorean Theorem. Wherever all three sides of a right triangle are integers, their lengths form a Pythagorean triple (or Pythagorean numbers). There is a general formula for obtaining all such numbers. My first math droodle has also related to the Pythagorean theorem. Unlike a proof without words , a droodle may suggest a statement, not just a proof. The Pythagorean configuration is known under many names, the Bride's Chair being probably the most popular. Besides the statement of the Pythagorean theorem, Bride's chair has many interesting properties, many quite elementary. Professor Edsger W. Dijkstra found an absolutely stunning generalization of the Pythagorean theorem. If, in a triangle, angles a, b, g lie opposite the sides of length a, b, c, then (EWD) sign(a + b - g) = sign(a2 + b2 - c2), where sign(t) is the signum function: sign(t) = -1, for t 0, sign(0) = 0, sign(t) = 1, for t 0. The theorem this page is devoted to is treated as "If g = p 2, then a2 + b2 = c2." Dijkstra deservedly finds (EWD) more symmetric and more informative. Absence of transcendental quantities (p) is judged to be an additional advantage. Proof 1 This is probably the most famous of all proofs of the Pythagorean proposition. It's the first of Euclid's two proofs (I.47). The underlying configuration became known under a variety of names, the Bride's Chair likely being the most popular. The proof has been illustrated by an award winning Java applet written by Jim Morey. I include it on a separate page with Jim's kind permission. First of all, ABF = AEC by SAS . This is because, AE = AB , AF = AC, and BAF = BAC + CAF = CAB + BAE = CAE. ABF has base AF and the altitude from B equal to AC. Its area therefore equals half that of square on the side AC. On the other hand, AEC has AE and the altitude from C equal to AM, where M is the point of intersection of AB with the line CL parallel to AE. Thus the area of AEC equals half that of the rectangle AELM. Which says that the area AC2 of the square on side AC equals the area of the rectangle AELM. Similarly, the are BC2 of the square on side BC equals that of rectangle BMLD. Finally, the two rectangles AELM and BMLD make up the square on the hypotenuse AB. Proof 2 We start with two squares with sides a and b, respectively, placed side by side. The total area of the two squares is a2+b2. The construction did not start with a triangle but now we draw two of them, both with sides a and b and hypotenuse c. Note that the segment common to the two squares has been removed. At this point we therefore have two triangles and a strange looking shape. As a last step, we rotate the triangles 90o, each around its top vertex. The right one is rotated clockwise whereas the left triangle is rotated counterclockwise. Obviously the resulting shape is a square with the side c and area c2. (A variant of this proof is found in an extant manuscript by Thbit ibn Qurra located in the library of Aya Sofya Musium in Turkey, registered under the number 4832. [R. Shloming, Thbit ibn Qurra and the Pythagorean Theorem, Mathematics Teacher 63 (Oct., 1970), 519-528]. ibn Qurra's diagram is similar to that in proof 27 . The proof itself starts with noting the presence of four equal right triangles surrounding a strangenly looking shape as in the current proof 2. These four triangles correspond in pairs to the starting and ending positions of the rotated triangles in the current proof. This same configuration could be observed in a proof by tesselation .) Proof 3 Now we start with four copies of the same triangle. Three of these have been rotated 90o, 180o, and 270o, respectively. Each has area ab 2. Let's put them together without additional rotations so that they form a square with side c. The square has a square hole with the side (a-b). Summing up its area (a-b)2 and 2ab, the area of the four triangles (4ab 2), we get c2 = (a-b)2+2ab = a2-2ab+b2+2ab = a2+b2 Proof 4 The fourth approach starts with the same four triangles, except that, this time, they combine to form a square with the side (a+b) and a hole with the side c. We can compute the area of the big square in two ways. Thus (a + b)2 = 4ab 2 + c2 simplifying which we get the needed identity. Proof 5 This proof, discovered by President J.A. Garfield in 1876 [ Pappas ], is a variation on the previous one. But this time we draw no squares at all. The key now is the formula for the area of a trapezoid - half sum of the bases times the altitude - (a+b) 2(a+b). Looking at the picture another way, this also can be computed as the sum of areas of the three triangles - ab 2 + ab 2 + cc 2. As before, simplifications yield a2+b2=c2. Two copies of the same trapezoid can be combined in two ways by attaching them along the slanted side of the trapezoid. One leads to the proof 4 , the other to proof 52 . Proof 6 We start with the original triangle, now denoted ABC, and need only one additional construct - the altitude AD. The triangles ABC, BDA and ADC are similar which leads to two ratios: AB BC = BD AB and AC BC = DC AC. Written another way these become ABAB = BDBC and ACAC = DCBC Summing up we get ABAB + ACAC = BDBC + DCBC = (BD+DC)BC = BCBC. In a private correspondence, Dr. France Dacar, Ljubljana, Slovenia, has suggested that the diagram on the right may serve two purposes. First, it gives an additional graphical representation to the present proof 6. In addition, it highlights the relation of the latter to proof 1 . Proof 7 The next proof is taken verbatim from Euclid VI.31 in translation by Sir Thomas L. Heath. The great G. Polya analyzes it in his Induction and Analogy in Mathematics (II.5) which is a recommended reading to students and teachers of Mathematics. In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle. Let ABC be a right-angled triangle having the angle BAC right; I say that the figure on BC is equal to the similar and similarly described figures on BA, AC. Let AD be drawn perpendicular. Then since, in the right-angled triangle ABC, AD has been drawn from the right angle at A perpendicular to the base BC, the triangles ABD, ADC adjoining the perpendicular are similar both to the whole ABC and to one another [VI.8]. And, since ABC is similar to ABD, therefore, as CB is to BA so is AB to BD [VI.Def.1]. And, since three straight lines are proportional, as the first is to the third, so is the figure on the first to the similar and similarly described figure on the second [VI.19]. Therefore, as CB is to BD, so is the figure on CB to the similar and similarly described figure on BA. For the same reason also, as BC is to CD, so is the figure on BC to that on CA; so that, in addition, as BC is to BD, DC, so is the figure on BC to the similar and similarly described figures on BA, AC. But BC is equal to BD, DC; therefore the figure on BC is also equal to the similar and similarly described figures on BA, AC. Therefore etc. Q.E.D. Confession I got a real appreciation of this proof only after reading the book by Polya I mentioned above. I hope that a Java applet will help you get to the bottom of this remarkable proof. Note that the statement actually proven is much more general than the theorem as it's generally known. Proof 8 Playing with the applet that demonstrates the Euclid's proof (7), I have discovered another one which, although ugly, serves the purpose nonetheless. Thus starting with the triangle 1 we add three more in the way suggested in proof 7: similar and similarly described triangles 2, 3, and 4. Deriving a couple of ratios as was done in proof 6 we arrive at the side lengths as depicted on the diagram. Now, it's possible to look at the final shape in two ways: as a union of the rectangle (1+3+4) and the triangle 2, or as a union of the rectangle (1+2) and two triangles 3 and 4. Equating areas leads to ab c (a2+b2) c + ab 2 = ab + (ab c a2 c + ab c b2 c) 2 Simplifying we get ab c (a2+b2) c 2 = ab 2, or (a2+b2) c2 = 1 Remark In hindsight, there is a simpler proof. Look at the rectangle (1+3+4). Its long side is, on one hand, plain c, while, on the other hand, it's a2 c+b2 c and we again have the same identity. Proof 9 Another proof stems from a rearrangement of rigid pieces, much like proof 2 . It makes the algebraic part of proof 4 completely redundant. There is nothing much one can add to the two pictures. (My sincere thanks go to Monty Phister for the kind permission to use the graphics.) There is an interactive simulation to toy with. (The proof has been published by Rufus Isaac in Mathematics Magazine, Vol. 48 (1975), p. 198.) Proof 10 This and the next 3 proofs came from [ PWW ]. The triangles in Proof 3 may be rearranged in yet another way that makes the Pythagorean identity obvious. (A more elucidating diagram on the right was kindly sent to me by Monty Phister .) Proof 11 Draw a circle with radius c and a right triangle with sides a and b as shown. In this situation, one may apply any of a few well known facts. For example, in the diagram three points F, G, H located on the circle form another right triangle with the altitude FK of length a. Its hypotenuse GH is split in the ratio (c+b) (c-b). So, as in Proof 6, we get a2=(c+b)(c-b)=c2-b2. Proof 12 This proof is a variation on 1, one of the original Euclid's proofs. In parts 1,2, and 3, the two small squares are sheared towards each other such that the total shaded area remains unchanged (and equal to a2+b2.) In part 3, the length of the vertical portion of the shaded area's border is exactly c because the two leftover triangles are copies of the original one. This means one may slide down the shaded area as in part 4. From here the Pythagorean Theorem follows easily. (This proof can be found in H. Eves, In Mathematical Circles , MAA, 2002, pp. 74-75) Proof 13 In the diagram there is several similar triangles (abc, a'b'c', a'x, and b'y.) We successively have y b = b' c, x a = a' c, cy + cx = aa' + bb'. And, finally, cc' = aa' + bb'. This is very much like Proof 6 but the result is more general. Proof 14 This proof by H.E.Dudeney (1917) starts by cutting the square on the larger side into four parts that are then combined with the smaller one to form the square built on the hypotenuse. Greg Frederickson from Purdue University, the author of a truly illuminating book, Dissections: Plane Fancy (Cambridge University Press, 1997), pointed out the historical inaccuracy: You attributed proof 14 to H.E. Dudeney (1917), but it was actually published earlier (1873) by Henry Perigal, a London stockbroker. A different dissection proof appeared much earlier, given by the Arabian mathematician astronomer Thabit in the tenth century. I have included details about these and other dissections proofs (including proofs of the Law of Cosines) in my recent book "Dissections: Plane Fancy", Cambridge University Press, 1997. You might enjoy the web page for the book: http: www.cs.purdue.edu homes gnf book.html Sincerely, Greg Frederickson Bill Casselman from the University of British Columbia seconds Greg's information. Mine came from Proofs Without Words by R.B.Nelsen (MAA, 1993). The proof has a dynamic version . Proof 15 This remarkable proof by K. O. Friedrichs is a generalization of the previous one by Dudeney. It's indeed general. It's general in the sense that an infinite variety of specific geometric proofs may be derived from it. (Roger Nelsen ascribes [ PWWII , p 3] this proof to Annairizi of Arabia (ca. 900 A.D.)) Proof 16 This proof is ascribed to Leonardo da Vinci (1452-1519) [ Eves ]. Quadrilaterals ABHI, JHBC, ADGC, and EDGF are all equal. (This follows from the observation that the angle ABH is 45o. This is so because ABC is right-angled, thus center O of the square ACJI lies on the circle circumscribing triangle ABC. Obviously, angle ABO is 45o.) Now, area(ABHI)+area(JHBC)=area(ADGC)+area(EDGF). Each sum contains two areas of triangles equal to ABC (IJH or BEF) removing which one obtains the Pythagorean Theorem. David King modifies the argument somewhat The side lengths of the hexagons are identical. The angles at P (right angle + angle between a c) are identical. The angles at Q (right angle + angle between b c) are identical. Therefore all four hexagons are identical. Proof 17 This proof appears in the Book IV of Mathematical Collection by Pappus of Alexandria (ca A.D. 300) [ Eves , Pappas ]. It generalizes the Pythagorean Theorem in two ways: the triangle ABC is not required to be right-angled and the shapes built on its sides are arbitrary parallelograms instead of squares. Thus build parallelograms CADE and CBFG on sides AC and, respectively, BC. Let DE and FG meet in H and draw AL and BM parallel and equal to HC. Then area(ABML)=area(CADE)+area(CBFG). Indeed, with the sheering transformation already used in proofs 1 and 12, area(CADE)=area(CAUH)=area(SLAR) and also area(CBFG)=area(CBVH)=area(SMBR). Now, just add up what's equal. Proof 18 This is another generalization that does not require right angles. It's due to Thbit ibn Qurra (836-901) [ Eves ]. If angles CAB, AC'B and AB'C are equal then AC2 + AB2 = BC(CB' + BC'). Indeed, triangles ABC, AC'B and AB'C are similar. Thus we have AB BC' = BC AB and AC CB' = BC AC which immediately leads to the required identity. In case the angle A is right, the theorem reduces to the Pythagorean proposition and proof 6. Proof 19 This proof is a variation on 6 . On the small side AB add a right-angled triangle ABD similar to ABC. Then, naturally, DBC is similar to the other two. From area(ABD) + area(ABC) = area(DBC), AD=AB2 AC and BD=ABBC AC we derive (ab2 AC)AB + ABAC = (ABBC AC)BC. Dividing by AB AC leads to AB2 + AC2 = BC2. Proof 20 This one is a cross between 7 and 19 . Construct triangles ABC', BCA', and ACB' similar to ABC , as in the diagram. By construction, ABC = A'BC. In addition, triangles ABB' and ABC' are also equal. Thus we conclude that area(A'BC) + area(AB'C) = area(ABC'). From the similarity of triangles we get as before B'C=AC2 BC and BC'=ACAB BC. Putting it all together yields ACBC + (AC2 BC)AC = AB(ACAB BC) which is the same as BC2 + AC2 = AB2. Proof 21 The following is an excerpt from a letter by Dr. Scott Brodie from the Mount Sinai School of Medicine, NY who sent me a couple of proofs of the theorem proper and its generalization to the Law of Cosines: The first proof I merely pass on from the excellent discussion in the Project Mathematics series, based on Ptolemy's theorem on quadrilaterals inscribed in a circle: for such quadrilaterals, the sum of the products of the lengths of the opposite sides, taken in pairs equals the product of the lengths of the two diagonals. For the case of a rectangle, this reduces immediately to a2 + b2 = c2. Proof 22 Here is the second proof from Dr. Scott Brodie's letter. We take as known a "power of the point" theorems: If a point is taken exterior to a circle, and from the point a segment is drawn tangent to the circle and another segment (a secant) is drawn which cuts the circle in two distinct points, then the square of the length of the tangent is equal to the product of the distance along the secant from the external point to the nearer point of intersection with the circle and the distance along the secant to the farther point of intersection with the circle. Let ABC be a right triangle, with the right angle at C. Draw the altitude from C to the hypotenuse; let P denote the foot of this altitude. Then since CPB is right, the point P lies on the circle with diameter BC; and since CPA is right, the point P lies on the circle with diameter AC. Therefore the intersection of the two circles on the legs BC, CA of the original right triangle coincides with P, and in particular, lies on AB. Denote by x and y the lengths of segments BP and PA, respectively, and, as usual let a, b, c denote the lengths of the sides of ABC opposite the angles A, B, C respectively. Then, x+y=c. Since angle C is right, BC is tangent to the circle with diameter CA, and the power theorem states that a2=xc; similarly, AC is tangent to the circle with diameter BC, and b2=yc. Adding, we find a2+b2=xc+yc=c2, Q.E.D. Dr. Brodie also created a Geometer's SketchPad file to illustrate this proof. Proof 23 Another proof is based on the Heron's formula which I already used in Proof 7 to display triangle areas. This is a rather convoluted way to prove the Pythagorean Theorem that, nonetheless reflects on the centrality of the Theorem in the geometry of the plane. Proof 24 [ Swetz ] ascribes this proof to abu' l'Hasan Thbit ibn Qurra Marwn al'Harrani (826-901). It's the second of the proofs given by Thbit ibn Qurra. The first one is essentially the 2 above. The proof resembles part 3 from proof 12. ABC = FLC = FMC = BED = AGH = FGE. On one hand, the area of the shape ABDFH equals AC2+BC2+area( ABC+ FMC+ FLC). On the other hand, area(ABDFH)=AB2+area( BED+ FGE+ AGH). This is an "unfolded" variant of the above proof. Two pentagonal regions - the red and the blue - are obviously equal and leave the same area upon removal of three equal triangles from each. The proof is popularized by Monty Phister , author of the inimitable Gnarly Math CD-ROM. Proof 25 B.F.Yanney (1903, [ Swetz ]) gave a proof using the "sliding argument" also employed in the Proofs 1 and 12. Successively, areas of LMOA, LKCA, and ACDE (which is AC2) are equal as are the areas of HMOB, HKCB, and HKDF (which BC2). BC = DF. Thus AC2+BC2 = area(LMOA)+area(HMOB) = area(ABHL) = AB2. Proof 26 This proof I discovered at the site maintained by Bill Casselman where it is presented by a Java applet. With all the above proofs, this one must be simple. Similar triangles like in proofs 6 or 13. Proof 27 The same pieces as in proof 26 may be rearrangened in yet another manner. This dissection is often attributed to the 17th century Dutch mathematician Frans van Schooten. [ Frederickson , p. 35] considers it as a hinged variant of one by ibn Qurra, see the note in parentheses following proof 2 . Dr. France Dacar from Slovenia has pointed out that this same diagram is easily explained with a tesselation in proof 15 . As a matter of fact, it may be better explained by a different tesselation . (I thank Douglas Rogers for setting this straight for me.) Proof 28 Melissa Running from MathForum has kindly sent me a link to A proof of the Pythagorean Theorem by Liu Hui (third century AD) . The page is maintained by Donald B. Wagner , an expert on history of science and technology in China. The diagram is a reconstruction from a written description of an algorithm by Liu Hui (third century AD). For details you are referred to the original page . Proof 29 A mechanical proof of the theorem deserves a page of its own. Pertinent to that proof is a page "Extra-geometric" proofs of the Pythagorean Theorem by Scott Brodie Proof 30 This proof I found in R. Nelsen's sequel Proofs Without Words II . (It's due to Poo-sung Park and was originally published in Mathematics Magazine, Dec 1999 ). Starting with one of the sides of a right triangle, construct 4 congruent right isosceles triangles with hypotenuses of any subsequent two perpendicular and apices away from the given triangle. The hypotenuse of the first of these triangles (in red in the diagram) should coincide with one of the sides. The apices of the isosceles triangles form a square with the side equal to the hypotenuse of the given triangle. The hypotenuses of those triangles cut the sides of the square at their midpoints. So that there appear to be 4 pairs of equal triangles (one of the pairs is in green). One of the triangles in the pair is inside the square, the other is outside. Let the sides of the original triangle be a, b, c (hypotenuse). If the first isosceles triangle was built on side b, then each has area b2 4. We obtain a2 + 4b2 4 = c2 There's a dynamic illustration and another diagram that shows how to dissect two smaller squares and rearrange them into the big one. Proof 31 Given right ABC, let, as usual, denote the lengths of sides BC, AC and that of the hypotenuse as a, b, and c, respectively. Erect squares on sides BC and AC as on the diagram. According to SAS , triangles ABC and PCQ are equal, so that QPC = A. Let M be the midpoint of the hypotenuse. Denote the intersection of MC and PQ as R. Let's show that MR PQ. The median to the hypotenuse equals half of the latter. Therefore, CMB is isosceles and MBC = MCB. But we also have PCR = MCB. From here and QPC = A it follows that angle CRP is right, or MR PQ. With these preliminaries we turn to triangles MCP and MCQ. We evaluate their areas in two different ways: One one hand, the altitude from M to PC equals AC 2 = b 2. But also PC = b. Therefore, Area( MCP) = b2 4. On the other hand, Area( MCP) = CMPR 2 = cPR 4. Similarly, Area( MCQ) = a2 4 and also Area( MCQ) = CMRQ 2 = cRQ 4. We may sum up the two identities: a2 4 + b2 4 = cPR 4 + cRQ 4, or a2 4 + b2 4 = cc 4. (My gratitude goes to Floor van Lamoen who brought this proof to my attention. It appeared in Pythagoras - a dutch math magazine for schoolkids - in the December 1998 issue, in an article by Bruno Ernst. The proof is attributed to an American High School student from 1938 by the name of Ann Condit.) Proof 32 Let ABC and DEF be two congruent right triangles such that B lies on DE and A, F, C, E are collinear. BC = EF = a , AC = DF = b , AB = DE = c . Obviously, AB DE. Compute the area of ADE in two different ways. Area( ADE) = ABDE 2 = c2 2 and also Area( ADE) = DFAE 2 = bAE 2. AE = AC + CE = b + CE. CE can be found from similar triangles BCE and DFE: CE = BCFE DF = aa b. Putting things together we obtain c2 2 = b(b + a2 b) 2 (This proof is a simplification of one of the proofs by Michelle Watkins, a student at the University of North Florida, that appeared in Math Spectrum 1997 98, v30, n3, 53-54.) Douglas Rogers observed that the same diagram can be treated differently: Proof 32 can be tidied up a bit further, along the lines of the later proofs added more recently, and so avoiding similar triangles. Of course, ADE is a triangle on base DE with height AB, so of area cc 2. But it can be dissected into the triangle FEB and the quadrilateral ADBF. The former has base FE and height BC, so area aa 2. The latter in turn consists of two triangles back to back on base DF with combined heights AC, so area bb 2. An alternative dissection sees triangle ADE as consisting of triangle ADC and triangle CDE, which, in turn, consists of two triangles back to back on base BC, with combined heights EF. The next two proofs have accompanied the following message from Shai Simonson , Professor at Stonehill College in Cambridge, MA: Greetings, I was enjoying looking through your site, and stumbled on the long list of Pyth Theorem Proofs. In my course "The History of Mathematical Ingenuity" I use two proofs that use an inscribed circle in a right triangle. Each proof uses two diagrams, and each is a different geometric view of a single algebraic proof that I discovered many years ago and published in a letter to Mathematics Teacher. The two geometric proofs require no words, but do require a little thought. Best wishes, Shai Proof 33 Proof 34 Proof 35 Cracked Domino - a proof by Mario Pacek (aka Pakoslaw Gwizdalski ) - also requires some thought. The proof sent via email was accompanied by the following message: This new, extraordinary and extremely elegant proof of quite probably the most fundamental theorem in mathematics (hands down winner with respect to the of proofs 367?) is superior to all known to science including the Chinese and James A. Garfield's (20th US president), because it is direct, does not involve any formulas and even preschoolers can get it. Quite probably it is identical to the lost original one - but who can prove that? Not in the Guinness Book of Records yet! The manner in which the pieces are combined may well be original. The dissection itself is well known (see Proofs 26 and 27 ) and is described in Frederickson's book, p. 29. It's remarked there that B. Brodie (1884) observed that the dissection like that also applies to similar rectangles. The dissection is also a particular instance of the superposition proof by K.O.Friedrichs . Proof 36 This proof is due to J. E. Bttcher and has been quoted by Nelsen (Proofs Without Words II, p. 6). I think cracking this proof without words is a good exercise for middle or high school geometry class. S. K. Stein, ( Mathematics: The Man-Made Universe , Dover, 1999, p. 74) gives a slightly different dissection. Both variants have a dynamic version . Proof 37 An applet by David King that demonstrates this proof has been placed on a separate page . Proof 38 This proof was also communicated to me by David King . Squares and 2 triangles combine to produce two hexagon of equal area, which might have been established as in Proof 9. However, both hexagons tessellate the plane. For every hexagon in the left tessellation there is a hexagon in the right tessellation. Both tessellations have the same lattice structure which is demonstrated by an applet . The Pythagorean theorem is proven after two triangles are removed from each of the hexagons. Proof 39 (By J. Barry Sutton, The Math Gazette, v 86, n 505, March 2002, p72.) Let in ABC, angle C = 90o. As usual, AB = c, AC = b, BC = a. Define points D and E on AB so that AD = AE = b. By construction, C lies on the circle with center A and radius b. Angle DCE subtends its diameter and thus is right: DCE = 90o. It follows that BCD = ACE. Since ACE is isosceles, CEA = ACE. Triangles DBC and EBC share DBC. In addition, BCD = BEC. Therefore, triangles DBC and EBC are similar. We have BC BE = BD BC, or a (c + b) = (c - b) a. And finally a2 = c2 - b2, a2 + b2 = c2. The diagram reminds one of Thbit ibn Qurra's proof . But the two are quite different. Proof 40 This one is by Michael Hardy from University of Toledo and was published in The Mathematical Intelligencer in 1988. It must be taken with a grain of salt. Let ABC be a right triangle with hypotenuse BC. Denote AC = x and BC = y. Then, as C moves along the line AC, x changes and so does y. Assume x changed by a small amount dx. Then y changed by a small amount dy. The triangle CDE may be approximately considered right. Assuming it is, it shares one angle (D) with triangle ABD, and is therefore similar to the latter. This leads to the proportion x y = dy dx, or a (separable) differential equation ydy - xdx = 0, which after integration gives y2 - x2 = const. The value of the constant is determined from the initial condition for x = 0. Since y(0) = a, y2 = x2 + a2 for all x. It is easy to take an issue with this proof. What does it mean for a triangle to be approximately right ? I can offer the following explanation. Triangles ABC and ABD are right by construction. We have, AB2 + AC2 = BC2 and also AB2 + AD2 = BD2, by the Pythagorean theorem. In terms of x and y, the theorem appears as x2 + a2 = y2 (x + dx)2 + a2 = (y + dy)2 which, after subtraction, gives ydy - xdx = (dx2 - dy2) 2. For small dx and dy, dx2 and dy2 are even smaller and might be neglected, leading to the approximate ydy - xdx = 0. The trick in Michael's vignette is in skipping the issue of approximation. But can one really justify the derivation without relying on the Pythagorean theorem in the first place? Regardless, I find it very much to my enjoyment to have the ubiquitous equation ydy - xdx = 0 placed in that geometric context. Proof 41 This one was sent to me by Geoffrey Margrave from Lucent Technologies. It looks very much as 8 , but is arrived at in a different way. Create 3 scaled copies of the triangle with sides a, b, c by multiplying it by a, b, and c in turn. Put together, the three similar triangles thus obtained to form a rectangle whose upper side is a2 + b2 , whereas the lower side is c2. (Which also shows that 8 might have been concluded in a shorter way.) Also, picking just two triangles leads to a variant of Proofs 6 and 19 : In this form the proof appears in [ Birkhoff , p. 92]. Yet another variant that could be related to 8 has been sent by James F.: The latter has a twin with a and b swapping their roles. Proof 42 The proof is based on the same diagram as 33 [ Pritchard , p. 226-227]. Area of a triangle is obviously rp, where r is the incircle and p = (a + b + c) 2 the semiperimeter of the triangle. From the diagram, the hypothenuse c = (a - r) + (b - r), or r = p - c. The area of the triangle then is computed in two ways: p(p - c) = ab 2, which is equivalent to (a + b + c)(a + b - c) = 2ab, or (a + b)2 - c2 = 2ab. And finally a2 + b2 - c2 = 0. (The proof is due to Jack Oliver, and was originally published in Mathematical Gazette 81 (March 1997), p 117-118.) Proof 43 By Larry Hoehn [ Pritchard , p. 229, and Math Gazette ]. Apply the Power of a Point theorem to the diagram above where the side a serves as a tangent to a circle of radius b: (c - b)(c + b) = a2. The result follows immediately. (The configuration here is essentially the same as in proof 39 . The invocation of the Power of a Point theorem may be regarded as a shortcut to the argument in proof 39 .) Proof 44 The following proof related to 39 , have been submitted by Adam Rose (Sept. 23, 2004.) Start with two identical right triangles: ABC and AFE, A the midpoint of BE and CF. Mark D on AB and G on extension of AF, such that BC = BD = FG (= EF). (For further notations refer to the above diagram.) BCD is isosceles. Therefore, BCD = p 2 - a 2. Since angle C is right, ACD = p 2 - (p 2 - a 2) = a 2. Since AFE is exterior to EFG, AFE = FEG + FGE. But EFG is also isosceles. Thus AGE = FGE = a 2. We now have two lines, CD and EG, crossed by CG with two alternate interior angles, ACD and AGE, equal. Therefore, CD||EG. Triangles ACD and AGE are similar, and AD AC = AE AG: b (c - a) = (c + a) b, and the Pythagorean theorem follows. Proof 45 This proof is due to Douglas Rogers who came upon it in the course of his investigation into the history of Chinese mathematics. The two have also online versions: D. G. Rogers, Pythagoras framed, cut up by Liu Hui D. G. Rogers, Beyond serendipity: how the Pythagorean proposition turns on the inscribed circle The proof is a variation on 33 , 34 , and 42 . The proof proceeds in two steps. First, as it may be observed from a Liu Hui identity (see also Mathematics in China ) a + b = c + d, where d is the diameter of the circle inscribed into a right triangle with sides a and b and hypotenuse c. Based on that and rearranging the pieces in two ways supplies another proof without words of the Pythagorean theorem: Proof 46 This proof is due to Tao Tong (Mathematics Teacher, Feb., 1994, Reader Reflections). I learned of it through the good services of Douglas Rogers who also brought to my attention Proofs 47 , 48 and 49 . In spirit, the proof resembles the proof 32 . Let ABC and BED be equal right triangles, with E on AB. We are going to evaluate the area of ABD in two ways: Area( ABD) = BDAF 2 = DEAB 2. Using the notations as indicated in the diagram we get c(c - x) 2 = bb 2. x = CF can be found by noting the similarity (BD AC) of triangles BFC and ABC: x = a2 c. The two formulas easily combine into the Pythagorean identity. Proof 47 This proof which is due to a high school student John Kawamura was report by Chris Davis, his geometry teacher at Head-Rouce School, Oakland, CA (Mathematics Teacher, Apr., 2005, p. 518.) The configuration is virtually identical to that of Proof 46 , but this time we are interested in the area of the quadrilateral ABCD. Both of its perpendicular diagonals have length c, so that its area equals c2 2. On the other hand, c2 2 = Area(ABCD) = Area(BCD) + Area(ABD) = aa 2 + bb 2 Multiplying by 2 yields the desired result. Proof 48 (W. J. Dobbs, The Mathematical Gazette, 8 (1915-1916), p. 268.) In the diagram, two right triangles - ABC and ADE - are equal and E is located on AB. As in President Garfield's proof , we evaluate the area of a trapezoid ABCD in two ways: Area(ABCD) = Area(AECD) + Area(BCE) = cc 2 + a(b - a) 2, where, as in the proof 47 , cc is the product of the two perpendicular diagonals of the quadrilateral AECD. On the other hand, Area(ABCD) = AB(BC + AD) 2 = b(a + b) 2. Combining the two we get c2 2 = a2 2 + b2 2, or, after multiplication by 2, c2 = a2 + b2. Proof 49 In the previous proof we may proceed a little differently. Complete a square on sides AB and AD of the two triangles. Its area is, on one hand, b2 and, on the other, b2 = Area(ABMD) = Area(AECD) + Area(CMD) + Area(BCE) = c2 2 + b(b - a) 2 + a(b - a) 2 = c2 2 + b2 2 - a2 2, which amounts to the same identity as before. Douglas Rogers who observed the relationship between the proofs 46-49 also remarked that a square could have been drawn on the smaller legs of the two triangles if the second triangle is drawn in the "bottom" position as in proofs 46 and 47 . In this case, we will again evaluate the area of the quadrilateral ABCD in two ways. With a reference to the second of the diagrams above, c2 2 = Area(ABCD) = Area(EBCG) + Area(CDG) + Area(AED) = a2 + a(b - a) 2 + b(b - a) 2 = a2 2 + b2 2, as was desired. He also pointed out that it is possible to think of one of the right triangles as sliding from its position in proof 46 to its position in proof 48 so that its short leg glides along the long leg of the other triangle. At any intermediate position there is present a quadrilateral with equal and perpendicular diagonals, so that for all positions it is possible to construct proofs analogous to the above. The triangle always remains inside a square of side b - the length of the long leg of the two triangles. Now, we can also imagine the triangle ABC slide inside that square. Which leads to a proof that directly generalizes 49 and includes configurations of proofs 46-48. See below. Proof 50 The area of the big square KLMN is b2. The square is split into 4 triangles and one quadrilateral: b2 = Area(KLMN) = Area(AKF) + Area(FLC) + Area(CMD) + Area(DNA) + Area(AFCD) = y(a+x) 2 + (b-a-x)(a+y) 2 + (b-a-y)(b-x) 2 + x(b-y) 2 + c2 2 = [y(a+x) + b(a+y) - y(a+x) - x(b-y) - aa + (b-a-y)b + x(b-y) + c2] 2 = [b(a+y) - aa + bb - (a+y)b + c2] 2 = b2 2 - a2 2 + c2 2. It's not an interesting derivation, but it shows that, when confronted with a task of simplifying algebraic expressions, multiplying through all terms as to remove all parentheses may not be the best strategy. In this case, however, there is even a better strategy that avoids lengthy computations altogether. On Douglas Rogers' suggestion, complete each of the four triangles to an appropriate rectangle: The four rectangles always cut off a square of size a, so that their total area is b2 - a2. Thus we can finish the proof as in the other proofs of this series: b2 = c2 2 + (b2 - a2) 2. Proof 51 (W. J. Dobbs, The Mathematical Gazette, 7 (1913-1914), p. 168.) This one comes courtesy of Douglas Rogers from his extensive collection. As in Proof 2 , the triangle is rotated 90o around one of its corners, such that the angle between the hypotenuses in two positions is right. The resulting shape of area b2 is then dissected into two right triangles with side lengths (c, c) and (b-a, a+b) and areas c2 2 and (b-a)(a+b) 2 = (b2 - a2) 2: b2 = c2 2 + (b2 - a2) 2. Proof 52 This proof, discovered by a high school student, Jamie deLemos (The Mathematics Teacher, 88 (1995), p. 79.), has been quoted by Larry Hoehn (The Mathematics Teacher, 90 (1997), pp. 438-441.) On one hand, the area of the trapezoid equals (2a + 2b) 2(a + b) and on the other, 2ab 2 + 2ba 2 + 2c2 2. Equating the two gives a2 + b2 = c2. The proof is closely related to President Garfield's proof . Proof 53 Larry Hoehn also published the following proof (The Mathematics Teacher, 88 (1995), p. 168.): Extend the leg AC of the right triangle ABC to D so that AD = AB = c, as in the diagram. At D draw a perpendicular to CD. At A draw a bisector of the angle BAD. Let the two lines meet in E. Finally, let EF be perpendicular to CF. By this construction, triangles ABE and ADE share side AE, have other two sides equal: AD = AB, as well as the angles formed by those sides: BAE = DAE. Therefore, triangles ABE and ADE are congruent by SAS . From here, angle ABE is right. It then follows that in right triangles ABC and BEF angles ABC and EBF add up to 90o. Thus ABC = BEF and BAC = EBF. The two triangles are similar, so that x a = u b = y c. But, EF = CD, or x = b + c, which in combination with the above proportion gives u = b(b + c) a and y = c(b + c) a. On the other hand, y = u + a, which leads to c(b + c) a = b(b + c) a + a, which is easily simplified to c2 = a2 + b2. Proof 54k Later (The Mathematics Teacher, 90 (1997), pp. 438-441.) Larry Hoehn took a second look at his proof and produced a generic one, or rather a whole 1-parameter family of proofs, which, for various values of the parameter, included his older proof as well as 41 . Below I offer a simplified variant inspired by Larry's work. To reproduce the essential point of proof 53 , i.e. having a right angled triangle ABE and another BEF, the latter being similar to ABC, we may simply place BEF with sides ka, kb, kc, for some k, as shown in the diagram. For the diagram to make sense we should restrict k so that ka b. (This insures that D does not go below A.) Now, the area of the rectangle CDEF can be computed directly as the product of its sides ka and (kb + a), or as the sum of areas of triangles BEF, ABE, ABC, and ADE. Thus we get ka(kb + a) = kakb 2 + kcc 2 + ab 2 + (kb + a)(ka - b) 2, which after simplification reduces to a2 = c2 2 + a2 2 - b2 2, which is just one step short of the Pythagorean proposition. The proof works for any value of k satisfying k b a. In particular, for k = b a we get proof 41 . Further, k = (b + c) a leads to proof 53 . Of course, we would get the same result by representing the area of the trapezoid AEFB in two ways. For k = 1, this would lead to President Garfield's proof . Obviously, dealing with a trapezoid is less restrictive and works for any positive value of k. References G. D. Birkhoff and R. Beatley, Basic Geometry , AMS Chelsea Pub, 2000 W. Dunham, The Mathematical Universe , John Wiley Sons, NY, 1994. W. Dunham, Journey through Genius , Penguin Books, 1991 H. Eves, Great Moments in Mathematics Before 1650 , MAA, 1983 G. N. Frederickson, Dissections: Plane Fancy , Cambridge University Press, 1997 G. N. Frederickson, Hinged Dissections: Swinging Twisting , Cambridge University Press, 2002 R. B. Nelsen, Proofs Without Words , MAA, 1993 R. B. Nelsen, Proofs Without Words II , MAA, 2000 J. A. Paulos, Beyond Numeracy , Vintage Books, 1992 T. Pappas, The Joy of Mathematics , Wide World Publishing, 1989 C. Pritchard, The Changing Shape of Geomtetry , Cambridge University Press, 2003 F. J. Swetz, From Five Fingers to Infinity , Open Court, 1996, third printing On Internet Pythagoras' Theorem , by Bill Casselman, The University of British Columbia. Pythagoras, biography Ask Dr. Math Another incarnation of 4 They try and try and try... President Garfield's Eric's Treasure Trove features more than 10 proofs A proof of the Pythagorean Theorem by Liu Hui (third century AD) An interesting page from which I borrowed Proof 28 An animated reincarnation of 9 Copyright 1996-2005 Alexander Bogomolny 15490011 Search: All Products Apparel Baby Beauty Books DVD Electronics Home Garden Gourmet Food Personal Care Jewelry Watches Housewares Magazines Musical Instruments Music Computers Camera Photo Software Sports Outdoors Tools Hardware Toys Games VHS Computer Games Cell Phones Keywords: Google Web CTK Latest on CTK Exchange probability Posted by tennyson 6 messages 03:13PM, Aug-29-05 The number 0.142857..... Posted by Zakatos 2 messages 10:54AM, Nov-05-05 SSA Postulate Posted by Allison 1 messages 02:23PM, Nov-16-05 Gift Exchange probability Posted by Owen 10 messages 11:27PM, Nov-16-05 A self-referencing sentence in us ... Posted by Paul R. 1 messages 09:42PM, Nov-16-05 Complex number solutions Posted by Owen 2 messages 10:16AM, Nov-14-05
Hyperspace Structures - Exploring the Fourth Dimension
Introduction to the hypercube, hyper-torus, and approximation to the hypersphere. Page includes Mpeg movie clips to illustration the various objects.
Hyperspace structures Hyperspace Structures Exploring the fourth dimension. Produced by Andy Burbanks and Keith Beardmore. Contents 1. The Hypercube A hypercube is a 4 dimensional analogue of the cube. Details of the construction of this object are given, along with movies of a spinning hypercube. 2. An ``Inflated Hypercube'' By inserting new vertices into the edges and faces of a hypercube, it may be ``inflated'' to give an approximation to the hypersphere. 3. Hyper-torus Adding together three vectors, each rotating in 4-space with different frequencies and amplitudes, produces a trajectory along the surface of a hyper-torus. 4. How the models were produced Details of the mathematics needed to model the structures, and how to turn the numbers into an animation. 5. Useful references A selection of recommended books. UP: Gallery | Mathematical Sciences | Loughborough University webmaster
Hyperspace Travel
Images and animations of a couple of four-dimensional objects, along with the math and source code used to produce them. Links to software to interactively visualize four-dimensional objects.
Hyperspace visualization
Four-Space Visualization of 4D Objects
Mathematical and computer programming oriented approach. Discusses wireframe rendering and ray-tracing.
Four-Space Visualization of 4D Objects - Index Four-Space Visualization of 4D Objects by Steven Richard Hollasch A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science Arizona State University August 1991 Supervisory Committee: Dr. Thomas Foley Dr. Gregory Nielson Dr. Gerald Farin Table of Contents Abstract Acknowledgements List of Figures Chapter 1: Introduction 1.1: Background 1.2: Previous Work 1.3: Overview of This Research 1.4: Contents of This Paper Chapter 2: Four Dimensional Geometry 2.1: Vector Operations and Points in Four-Dimensional Space 2.2: Rotations in Four Dimensions Chapter 3: Overview of Visualization in Three and Four Space 3.1: Viewing in Three Space 3.2: Viewing in Four Space Chapter 4: Wireframe Display of Four Dimensional Objects 4.1: High-Level Overview of 4D to 2D Projection 4.2: Description of 3D to 2D Projection 4.3: Description of 4D to 3D Projection 4.4: Rotations in 4D Wireframes 4.5: User Interaction and Visualization Aids 4.6: Example 4D Wireframe Images Chapter 5: Raytracing in Four Dimensions 5.1: General Description of the Raytracing Algorithm 5.2: Generating the Four-Dimensional Ray Grid 5.3: The General Raytrace Algorithm 5.4: Reflection and Refraction Rays 5.5: Illumination Calculations 5.6: Intersection Algorithms 5.6.1: Ray-Hypersphere Intersection 5.6.2: Ray-Tetrahedron Intersection 5.6.3: Ray-Parallelepiped Intersection 5.7: Display of 4D Raytrace Data 5.8: Example Ray4 Images Chapter 6: Conclusion 6.1: Research Conclusions 6.2: Future Research Areas References Appendix A: Implementation Notes and Source Code A PostScript Version of This Thesis Steve Hollasch , August 1991
The 4th Dimension
Visual and mathematical descriptions of higher dimensional objects, with Java applets for displaying and rotating images such as Steiner surfaces, Klein bottles, and minimal surfaces.
The Fourth Dimension The4thDimension The very mention of the possibility of a 4th dimension is a enticing prospect indeed. Is there a 4th dimension? If there is, is it time? Could there be a 5th dimension, or perhaps even higher? If you've come here looking for definite answers to these questions then I'm sorry to disappoint you. However, if you're an open minded critic of all that is not physically observable then you're in the right place. The navigation bar to the left is filled with links to several different graphical representations of higher dimensional objects such as the Mbius strip , Klein bottle , Steiner's Roman surface , Boy's surface , etc. The main gallery has pictures of several nonorientable , orientable , pseudospherical , minimal , and other miscellaneous surfaces . The heart of the problem is that since we are stuck in a 3-dimensional world we cannot visualize or physically experience anything more than than the 3rd dimension. Some of you may not like that last statement since time is such a strong candidate for the 4th dimension. You may say, "we can reasonably visualize or physically experience time." However, I say we most certainly cannot reasonably visualize or physically experience time, for if we could we would know for certain if it were the 4th dimension, we could point in some direction and say, "look time is that way". What we can do is physically experience the effects of time. Let us assume that time is the 4th dimension, since we are confined to a lesser dimension, we cannot feel or see anything more than a projection of time. It is essentially equivalent to "flat land" falling through the 3rd dimension. Our understanding of time is limited at best and outright pathetic at worst. Warning: embed_me.html could not be embedded. Random Quote Javascript must be enabled to generate the random quote. Number of visitors: This page was updated on April 22, 2004 7:26 AM Questions or comments about this page email: Webmaster Copyright 2002-2003
Thinking 4-d
Focuses on helping the reader to conceive of the fourth spatial dimension, and in particular discusses hypercubes and hyperspheres.
Thinking 4-d main page Perhaps, when one is using a microscope, viewing single-celled organisms, he takes note of the "cell membrane" that each one has. This "outer skin" protects the cell from other cells in this flat, 2-dimensional environment. How "protected", though, would the cell be, from the person viewing, who has access to the sum total of everything inside the "outer skin" - the cell fully unaware of being observed? The book Flatland deals with little sentient beings - "Flatlanders" - who live in such a "tabletop-like" world. Such a universe would be limited entirely to a mere length and width. What would life in such a world be like? You'll find that no Flatlander possesses a full digestive tract - it would split him into two pieces! In another very similar book, The Planiverse, we find 2-dimensional creatures who cannot 'walk past' another creature they happen to encounter - one must literally "climb over" the other to get to the other side! Even so, such a creature, like the single-celled organism under the microscope, is limited totally to its 'tabletop', unaware of the space that lies above the tabletop - (the space from which we observe the creature): the 3rd dimension. Could such creatures comprehend the 3rd dimension? Would they require our help? Or could they actually do it 'on their own'? And if they could comprehend the 3rd dimension without our help, just how did they go about 'understanding' something that they can neither sense, nor prove? -- Introduction continued -- "Thinking 4-D" cannot prove the existence of a fourth spatial dimension, in that the observable world does not provide the empirical data needed to do so: yet all concepts presented in "Thinking 4-D" are spatial. "Thinking 4-D" does not claim to present a verbatim description of four-space and its processes, but is merely a system that aids one in understanding the fourth dimension by relating it to the observable world. What do you think of Thinking 4-D? Do not hesitate to e-mail me at jeffsf@hotpop.com . Your input will shape the future of Thinking 4-D! Something So Simple "Gliding" the Cube Visualizing 4-space Visualizing the Hypersphere Understanding Space-time The Principle of Relativity Near-light Speed and Length Light's Unchanging Speed Wormholes and Hyperspace On Time Travel What are UFOs? Infinity On Space Travel coming soon: The Hypertorus The Klein Bottle The Real Projective Plane The Hypercube Unfolded to Web site main screen Comments, questions, feedback: jeffsf@hotpop.com geovisit();
Miscellaneous Mathematical Musings
Introduction to dimensions, multi-dimensional geometric units, and formulas for calculating higher-dimensional phenomena.
Miscellaneous Mathematical Musings by Bill Price Miscellaneous Mathematical Musings by Bill Price I'm no mathematician, but sometimes I like to pretend that I am. Here are some ideas that have occupied my mind from time to time. A Study of Dimensions: Introduction and chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Formula for Calculating Components of Hypercubes Repeating Decimals in Fractions of Prime Numbers Obtaining Decimal Equivalents of Fractions A Visual Definition of Prime Numbers A Calculator Trick Bill Price Home Page geovisit();
Hypercube's homepage
Philosophy, math, and cosmology related to the fourth dimension.
Hypercube's Home Page The Fourth Dimension is the next step in the series: "Length, Height, Width...." But what is it? Click here for a general explanation . Fourth Dimension: Related Math Philosophy: What is the Fourth Dimension? Non-Euclidean Geometry 4D Misconceptions What are Hypercubes? A. Square and Flatland Overall Metaphysics Einstein overview Accepted Facts of Today Brain Teasers (Enigmas) Archive of Discussions Fourth Dimension Discussion Forum - Post your own thoughts and speculations on this popular message board! Sign Guestbook | View Guestbook Discussion Forum Visitor : This site was created by: Eric Saltsman . esalts@zoomnet.net ICQ: 1932103 Much Help by: Dan Keck . geovisit();
HyperCube by Harmen
A short introduction to the fourth dimension, plus a hypercube java applet game.
HyperCube by Harmen HyperCube Experience hyperspace with a stereoscopic 4d game in and round the hypercube (Java applet). HyperCube (intro) HyperCube Applet Help 4D Stereo Links Feedback About
Hyper Dimensia
Tutorial which aims to give readers an understanding of the fourth dimension by having them use a java applet to rotate a cube and a hypercube to see multiple views of the shapes at once.
Hyper Dimensia www.HyperDimensia.com The Physics of Higher Dimensions Rough Draft - Introduction This is one of a series of sites on the thorny questions that physicists did not answer last century. Perhaps it will provide an insight into those challenges that you will have the opportunity to address during your career as well as your personal life. On this site we shall explore the implications of the existence of more than ThreeDimensions. An attempt will be made to examine the mathematical, scientific, technological, philosophical and other human implications of 4 or more spatial and or physical dimensions. I will take advantage of the interactive and multimedia properties of digital technology to provide an intuitive understanding of the physics of higher dimensions. There are many mysteries of science that can be easily explained with the use of higher dimensions. Among these are EMF and Gravity , Quantum Mechanics, Gravity, Dark Matter , Intrinsic Spin , The Hierarchy Problem , Particles vs. Waves, GUT's, TOE's, Parapsychology, UFO's, Missing Socks, Angels, Ghosts - everything except God and Love (the latter is left as an exercise for the student). An explanation of why these are Mysteries is provided here . In this section I will provide the clues, you work out the details. Scientific Facts and currently accepted Theory are presented in black text. Musings, What-If's and Speculation are presented in magenta. Do not lose sight of what is Theory and what is Speculation. Table Of Contents 1.0 HyperDimensional Viewing 1.1 3D Viewing 1.1.1 Open 3D Viewer Window 1.1.2 3D Games 1.2 4D Viewing 1.2.1 Open 4D Viewer Window 1.2.2 4D Games 1.3 Other 4D Viewing Options 1.3.1 W by X perspectiveless projection along Y axis 1.3.2 3D Cross Sections of 4D Object 1.3.3 Other 4D Objects Toolbox Bibliography A Appendix A Glossary Index Chapter One - HyperDimensional Viewing Our 'reality' is a world of 3 Space Dimensions and 1 Time Dimension. We believe this because we can see 3 dimensions - up down, left right, and front back. We can also believe that there is 1 time dimension - future past. This is known as 1+3 dimensionality - 1 Time dimension and 3 Space dimensions. But could there be different dimensionality than what we perceive? Is it possible that there are 4 spatial dimensions? Or 2 time dimensions? Or more? These alternatives can be referred to as HyperDimensionality. We can simulate these alternatives with the computer in order to get an intuitive grasp of these hyperdimensional relationships. On these pages we shall explore our 1+3D 'Reality', and a few of these alternate HyperDimensional worlds. 3D Viewing First let us practice with our common 3D world. Lets ignore Time and stick with 3D rather than 1+3D. I have provided a simple CAD -like viewer so that you can examine a 3D cube. Click here to open another window with the 3D viewer JAVA Applet. NOTE: You must have JAVA 2 installed to use the viewer. If you do not have the current JAVA installed, the viewer will tell you, and take you to the SUN JAVA site to download a copy. You must have the JAVA 2 (JRE 1.3 +) to proceed with this interaction. 3D Cube - A cube is 3 pairs of 2D surfaces, one pair for each dimension, attached at their adjacent 1D edges. So there are 6 2D surfaces. Each surface has 2 pairs of 1D edges, one pair for each dimension. They are 4 shared 1D edges for a total of 6*4 2 = 12 1D edges. The 3D Viewer shows 3 views of a cube. TOP, FRONT and SIDE. Note that there are 3 labeled axes within the cube - X, Y, Z. Notice the orientations of each of the axes in relation to the view. The FRONT view orientation is as you would expect - i.e. the X axis is horizontal, while the Y axis is vertical. The Z axis extends out of the window towards us. The TOP view is indeed looking down on the cube from the +Y axis, while the SIDE view is looking along the X axis from the right. TOP FRONT SIDE 3D: Screen X disabled 3D: Screen Y disabled 3D: Screen Ctr disabled 3D: Perspective disabled SCRAMBLE CENTER SPIN SPIN Button - Click on the SPIN button in the bottom right of the viewer. Notice that the cube is spinning about the screen's Horizontal X axis (which is coincidentally the same as the cube's X axis). Look closely until your are sure that these are 3 views of the same cube from 3 different positions. In each view, the X axis is stationary, and the cube spins about it. CENTER Button - Note that the axes spin with the cube. These are Object axes, not view (or world) axes. Note also that the Y axis appears to be rotating into the Z axis' space. This will be important later. Click on SPIN again, and the spinning will stop. Click on CENTER, and the cube will be centered back in its original position. In the FRONT window this is with X oriented along the window's horizontal axis, Y along the window's vertical axis, and Z coming out of the window towards you. This is a Right-handed Coordinate System (RHCS). Screen X Slider - The first slider (the top on the left) controls the 3D object's rotation about the screen's Horizontal (or 'X') axis. If you slide the 'thumb' from the left to the right, the FRONT view will dip down. The SIDE view will twist anti-clockwise, and the TOP view will dip towards you. Click on CENTER to reorient the cube. Screen Y Slider - Click on SPIN a third time, and the cube will spin about the next slider in sequence - the Vertical Y axis - in all 3 views. Here the Y axis is stationary while the Z axis rotates into the X axis. Click SPIN once again to halt the spinning. Click on CENTER to reorient the cube. The second slider controls the 3D object's rotation about the screen's 'Y' ( or vertical) axis. If you slide the thumb of this slider from left to right, you will see the object rotate from left to right in the FRONT view. The TOP and SIDE views will rotate appropriately. Click on CENTER to reorient the cube. Screen Ctr Slider - Click on SPIN a fifth time to activate spinning about the Z axis, or the screen's center. Here Z is stationary, and the X axis rotates into the Y axis' space. Click on SPIN once more to halt. If you clicked SPIN again, it would repeat the 3 slider sequence. Click on CENTER to reorient the cube. The third slider controls the 3D object's twist about the screen's center. If you slide the thumb from the left to the right, the object will rotate counterclockwise. (The bottom edge of the FRONT view moves with the slider from left to right vice-versa.) Click on CENTER to reorient the cube. Perspective Slider - The bottom slider controls perspective - i.e. the relationship between an object's distance and its apparent size. Minimum perspective is at the left, maximum to the right. As you slide the thumb to the right, the perspective gets more extreme. It is as if you move your eye up near the object - the closest square gets huge compared to the further square. The absolute size (or scale) of the projection is automatically adjusted to prevent overflow (very large senseless numbers). As you slide the thumb to the left, the perspective is reduced. The apparent sizes of the back and front squares will look the same. It is as if you leaned way back away from the object (but increased the magnification so it stayed the same size). The perspective slider thus allows you to adjust the perceived depth of the object. A value of around 7 seems to work best for me. Visual Cues - You can manipulate the other sliders while SPIN is active. If you get lost (can no longer 'see' the perspective) remember that the longer edges are closest to you, while the shortest edges are furthest from you. You can also press the CENTER button at any time to reorient the cube. Play 3D Games Game 1 - The 3rd button is a game. Press the SCRAMBLE Button to randomly orient the cube. Use the sliders to return it to the center position. (Note that the scrambled slider values have no meaning - in order to confuse you). Try the SCRAMBLE and restore game a few times until you feel comfortable with the 3D Viewer tool. Game 2 - Place the green square inside the red square in the FRONT view. This will seem less trivial when we get to 4D. You can close the 3D viewer window when you are done. 4D Viewing In this section we shall explore a 1+4D model. To simplify the discussion, we shall ignore the time dimension, and stick to 4 space dimensions or 4D. I find it easiest to visualize a 4D object by extrapolating the behavior of a 2D to 3D projection into a 3D to 4D projection. For example, a 3D cube is built of 6 2D squares attached at their 1D edges. To extrapolate into a higher dimension, imagine a 4D hypercube built of 8 3D cubes attached at their 2D faces. You can see that there may be mathematical patterns that can be extracted from these extrapolations. But better yet, lets take a look. The 4D interactive visualization tool is an extension of the simple 3D viewer discussed above. Click here to start the 4D Viewer in another window. 4D Hypercube - A hypercube is 4 pairs of 3D cubes, one pair for each dimension, attached at their common 2D surfaces (sound familiar?). So there are 8 3D cubes of 6 faces each, each face shared between 2 cubes, this yields 8 * 6 2= 24 surfaces - or 12 pair, three pair of 2D surfaces for each dimension, or one pair for each of the 3 dimensions along each of the 4D axes. Extrapolating, it would seem that while an infinite 2D plane can bisect an infinite 3D universe, it would require an infinite 3D object to bisect an infinite 4D universe. The 4D object shown here is a unit hypercube. It is a 4D object that is 1 x 1 x 1 x 1 (while a unit 3D cube is 1 x 1 x 1). Each of the identical cubes of which the hypercube is formed is a 1x1x1 unit cube. 4D to 3D - The Interactive Visualization that follows provides a projection of this 4D hypercube into a window. This is done by projecting a 4D hypercube into a 3D space thus creating a 3D object, in exactly the same way that 3D is projected into 2D. The resultant 3D object is then projected onto your 2D window with the 3D viewer. This is mathematically equivalent to the prior 3D visualizer. If you lose it (i.e. - can no longer see the image in proper perspective), remember that the bigger edge is closest to you (and the shortest edges are further into the screen from you). 4D Viewer - There are 4 windows. The lower left window is a FRONT view of the hypercube. The lower right window is a view of the object from the right SIDE, while the upper left window is a view of the object from the TOP. These views are identical to the 3D viewer. The upper right view is a view of the object in the 4th dimension (more later). TOP 4D FRONT SIDE 3D: Screen X 4D: Object W = X 3D: Screen Y 4D: Object W = Y 3D: Screen Ctr 4D: Object W = Z 3D: Perspective 4D: Perspective SCRAMBLE CENTER SPIN The left column is the 3D viewer, with which you are now facile thanks to your success with the 3D Games, above. The right column, which is now enabled, controls the 4D rotations and projection. The controls on the right are used to 'build' a 3D object from the projection of the 4D hypercube into 3 Space. Remember to click on the CENTER button to reset the view to the nominal CENTER position. The 3D Horizontal, Vertical and Center Sliders on the left are as described above. So let us now focus on the 4D sliders on the right. W Axis - Now that you are familiar with the conventional 3D viewer, we can get on to the fun stuff - the 4D controls - the sliders on the right. It is time to introduce the 'W' axis - the fourth spatial dimension. In the following discussion, I shall use 'W' to describe the new 4th space axis. The 4D object is colored to show the new dimension. The positive direction along the W axis is indicated by green edges, while the "-W" direction is shown with red edges. So, the green cube lies on the +W axis, while the red cube is in the -W direction. Hypercube - The 4D figure shown here is a unit Hypercube. It is a 4D object that is 1 x 1 x 1 x 1 (a 3D unit cube is 1 x 1 x 1). A Hypercube consists of 8 identical cubes, each sharing its 6 faces with the 6 adjacent cubes (in the same way a 3D cube shares each of its 4 edges with the adjacent 4 squares to form a cube of 6 faces). Each of the identical cubes appear to be different sizes and shapes due to perspective. They really are identical. The 4D viewer lets you discover this truth as you explore the 4D object. If you rotate the 3D object about Y (use the Screen Y slider) to 30 or 40 degrees, you will see a red cube inside a green cube. The red cube is the same size as the green cube. Don't believe it? Perspective makes the red cube (which is further away along the W axis) appear smaller than the green cube (which is closer than the red cube) - just like the red green squares in the 3D viewer. 4D Rotations - Consider rotations. In 3D we think of rotation about an axis. However, this is equivalent to rotating one axis into another - i.e. rather than rotating about the Y axis, rotate the X axis into the Z axis. In this manner we can extrapolate rotations into 4D. The X axis is rotated into the Z axis (about the Y,W axes pair). This is convenient mathematically, since we can use traditional 3D rotational matrix transforms to describe these rotation(s). You will note, that the sliders on the right (again we shall ignore the perspective slider) are labeled 4D with an axis pair (be sure to click on CENTER for each of the following 3 paragraphs). 4D W = Z - The first slider (top right) rotates the object's W axis into the Z axis. As the slider's thumb is moved from the left to the right, the Z axis data is transformed into W axis data. Observe the Red,Green,White (XYZ) axes in the SIDE view. You can see the W axis grow, and the Z axis shrink (and vice-versa). The SIDE view appears to be rotating about the Y axis in the SIDE projection. But as implied above, the rotation is actually about the X,Y pair. Slowly drag the "W = Y" thumb from the 0 position to the 90 position, and note that the SIDE view axes show the W axis growing to replace the Z axis which is shrinking. At exactly 90 degrees, the Z axis has been replaced with the W axis. Likewise the TOP view shows an apparent rotation of W into Z, but about the X axis (really about X,Y pair). Ouroborus Effect - In the FRONT view, the WZ axes are collinear and perpendicular to the view plane, projecting into and out of the screen, respectively. The Ouroborus Effect becomes clear if you rotate the FRONT view by 30 degrees (move the 3D: Screen Y thumb to the right near 30). As you rotate the W into Z with the top right slider, the object will appear to "eat its own tail" as the 4D portion of the object moves into our 3D space and out again. You can see the red cube move out of the green cube, then expand to consume the green cube. The green cube then expands to eat the red cube again. The motion through the W dimension (along the W axis) appears as expanding and shrinking red and green cubes. You can explore each of the rotations with the 2nd (rotate W into Y) and 3rd (rotate W into X) sliders to display effects similar to those described above. Play 4D Games The SCRAMBLE button randomly orients the hypercube in 4 Space, and scrambles the sliders. The sliders work as before, just the value (at the right of the slider) and the slider position no longer correspond to the rotation of the object. Notice that every time you click on SCRAMBLE, the hypercube is reoriented. The CENTER button takes it back to the reset position. The SPIN button will automatically spin each of the 6 rotation sliders in turn. Click on SPIN the first time to automatically slide the 1st (3D X Axis) slider. Click on it again to disable the spin. Click on it a third time to select the next slider (3D Y Axis), and a fourth time to cancel. Clink a fifth time for the next (3D Z Axis) slider, and a sixth time to cancel. Click on the SPIN button a seventh time to select the bottom right (4D Object W X), and an 8th time to cancel. Click 9 selects "4D Object W Y", 10 cancels. Click 11 selects the top right "4D Object W Z" slider, and click 12 cancels. Click 13 starts over again with "3D X". 1) Notice the position of the hypercubes when the CENTER button is pressed. Now press the SCRAMBLE button. Can you get the 4D object back into the CENTER position using just the 6 sliders? 2) Click on CENTER, then rotate the hypercube -30 degrees about the 3D Screen Y axis. Notice that the red cube is inside of the green cube. Can you place the green cube inside the red cube? Hint - it is a 180 degree 4D rotation about the W and one other axis. 3) Click on SCRAMBLE. Using the 6 sliders, get the red cube inside the green cube. 4) Try each of the above with a different axis spinning via the SPIN button. Notes To Myself A work in progress ... Additional 4D Viewer Features: 4) 3D cross sections. (Get old clipping code - modify to clip in 4D) 5) WxX projection along Y axis.( use orthogonal projection)(link to UMN animations) 6) Other 4D objects.(link to polytrope) 7) Use Dithering for probability. 8) Use shading for the W axis. 9) Hook 'amplitude' to the W axis. 10) Use alternate metrics for W, e.g. - Cylindrical, Hyperbolic, Spherical, Elliptical, as well as Infinite. 11) Try 4D 5D rotations (intrinsic spin interpretation) 12) Interpret W as Time (Create animations of 3D sections along W axis) 13) Add Time axis (1+4D 1+5D) This website is brought to you by Edutech Project, an educational resource of DigitalChoreoGraphics . Copyright2002 Don V Black
Fourth Dimension: Tetraspace
A discussion of rotation, levitation, wheels, and bodies of water. Classification of "rotatopes" (cubes, spheres, and cylinders), and a Java applet to visualize projections and intersections of the shapes. Includes book reviews, a glossary, links, and a forum.
Fourth Dimension: Tetraspace Fourth Dimension: Tetraspace Speculations on the 4th dimension The universe that we live in has only three spatial dimensions. We are limited to length, width, and height, and we can only travel along three perpendicular paths. This page attempts to explain the properties of a hypothetical universe with a spatial fourth dimension. While people generally call time the 4th dimension in the universe we live in, time will be the 5th dimension in my hypothetical universe. Many fascinating possibilities exist when a spatial fourth dimension is present.Several types of wheels are possible, very complex machines can be built, and many more shapes are possible. Objects can pass by each other more easily, but they are harder to break into multiple pieces. Energy reduces much faster with distance than in the 3rd dimension, so both light and sound are weaker. Much more things can be compacted into a small space, but its much easier to get lost. In this page, I will explore these and many other interesting properties of the fourth dimension. To get started learning about the fourth dimension, go to the introduction . Site Index [ Introduction ] Introduction (12 23 2003) - Introduces the concept of the 4th dimension. [ The adventures of Fred, Bob and Emily ] From the 2nd to the 3rd dimension (1 13 2003) - Bob, who lives in the 3rd dimension, interacts with Fred, who lives in the 2nd dimension. From the 3rd to the 4th dimension (1 19 2003) - Emily, who lives in the 4th dimension, interacts with Bob. Rotation (1 19 2003) - Explains the properties of rotation the different dimensions. Flatness and Levitation (1 18 2003) - How objects from one dimension look in other dimensions. [ The world ] Wheels in the different dimensions (1 19 2003) - Introduces the properties of wheels and tires from the 2nd to the 4th dimension. War in the dimensions (6 03 2001) - How war would work in the 2nd, 3rd, and 4th dimension. Bodies of water (6 03 2001) - Lakes, rivers, and other bodies of water. [ Higher dimensional shapes ] Index of higher dimensional shapes Rotatope properties (3 16 2003) - Properties of round shapes from the fourth dimension. Rotatope classification (3 16 2003) - Several classification schemes for round shapes from the fourth dimension. Shape Intersection Applet (2 23 2003) - Shows the intersection of shapes from the fourth dimension with lower-dimensional shapes. Shape formulas (3 14 2003) - Face, volume, and rotation formulas. Polyhedra (2 22 2003) - The regular three-dimensional solids. [ Other fourth dimension topics ] The confusion of dimension terms (1 16 2004) - Discusses the confusion between terms of the 4th dimension like hypercube and tesseract. Fourth Dimension search terms (10 25 2003) - Search terms that one can use when searching for pages on the fourth dimension. Time vs spatial 4th dimension (2 11 2003) - Discusses the debate between the 4th dimension being time or being spatial. Book Reviews (2 26 2003) - Reviews on books about the fourth dimension. Fourth Dimension Glossary (12 8 2003: 108 words total) - Terms relating to the fourth dimension. Fourth Dimension Links (10 25 2003) - Links to other sites about the fourth dimension. Old News [ Discussion ] fourth dimension forum (11 3 2003) Updates Introduction page - 12 23 2003 I rewrote the introduction to the fourth dimension , and changed one of the images. The page is now quite a lot different. Next up for revision is the page "From the 2nd dimension to the 3rd dimension". Fourth Dimension Glossary, part 4 - 12 8 2003 I made a number of changes to the fourth dimension glossary to fix errors that Polyhedron Dude pointed out, plus added some new words, bringing the total number of words from 102 to 106. Here are the changes: added to the definitions of latitude, longitude, laptitude added the words vertex, edge, face; corrected cell definition, renamed tetron - teron duocylinder no longer called a hypercylinder corrected the tesseract definition added the words perimeter and surface; corrected surcell definition added to the definitions of hole, chasm, gully added polyteron Fourth Dimension Glossary, part 3 - 12 4 2003 I changed the terms tridth tride trin to trength trong tarrow. This is because the tridth series is potentially confusing - it contains the root "tri-" but it is a measure of the fourth dimension, not the third. See Old News To begin learning about the 4th dimension, go to the introduction . Fourth Dimension: Tetraspace originally started 2 18 2000. hits since June 2002. Contact email: This website, Fourth Dimension: Tetraspace, is copyright 2000-2003 by Garrett Jones You should be drinking alkaline water: Structured Alkaline Water www.alkaline.org
The 4D Web Page
List of links to sites on people, books, art, java applets, games, and general information on the fourth dimension.
The 4D Web Page - UC Irvine VR Lab - D'Zmura Arts Design Transarchitectures - Marcos Novak Virtual unreality - R.D. Brown Henderson, Linda D. (1983). The fourth dimension and non-euclidean geometry in modern art. (Princeton: Princeton University Press). 4D Design - Jeremy Myerson The 4th dimension - Richard Brown The manifeste dimensioniste revue no. 1 - Charles Sirato Four-dimensional art - frichter Enter the fourth dimension - American salons, CWU Computer Graphic Techniques Four-space visualization of 4D objects - Steve Hollasch Virtual environments with four or more spatial dimensions ( PostScript , PDF )- Mike D'Zmura , Philippe Colantoni and Greg Seyranian, Presence Visualization of events from arbitrary spacetime perspectives ( PostScript , PDF ) - Mike D'Zmura , Philippe Colantoni and Greg Seyranian, SPIE '00 Arbitrary-dimensional solid object display algorithm - Greg Ferrar Meshview - Hanson, Ishkov Ma Hypercube visualization - Alan B. Scrivener Books Abbott, Edwin Abbott (1884). Flatland: A romance in many dimensions (Dover--see below for internet versions and commentaries). Banchoff , Thomas F. (1996). Beyond the third dimension. (New York: W.H. Freeman). Kaku, Michio (1995). Hyperspace. (Anchor) Henderson, Linda D. (1983). The fourth dimension and non-euclidean geometry in modern art. (Princeton: Princeton University Press). Pickover, Clifford A. (2000). Surfing through hyperspace . Oxford: Oxford University Press). Rucker, Rudolf (1977). Geometry, relativity and the fourth dimension. (New York: Dover). Rucker, Rudolf, Rucker, Roy and Povilaitis, David (1977). The fourth dimension. Flatland (Edwin Abbott) HTML version (E. M. Francis) HTML version ( Project Gutenberg text ) History Fourth dimension - Linda Dalrymple Henderson People Thomas Banchoff Andrew Hanson Linda Dalrymple Henderson Steve Hollasch Marcos Novak Rudy Rucker Philosophy Four dimensionalism (1997; Philosophical Review 106, 197-231). Psychology Search and navigation in environments with four spatial dimensions ( MSWordDoc , PostScript , PDF )-Greg Seyranian and Mike D'Zmura General information, Models, Illustrations and Animations 4D virtual environments - UC Irvine VR Lab Solid Segment Sculptures - George W. Hart Vocabulary of dimension - Phillips Fourth dimension - Cliff Pickover Euclidean geometry in higher dimensions - Question corner Pictures - Richard Koch Hyperspace structures - Andy Burbanks and Keith Beardmore Uniform polytopes in four dimensions - George Olshevsky The fourth dimension - Eric Saltsman Fourth Dimension: Tetraspace - Garrett Jones The Simplex - David Anderson 4D rotating cube - mathematik.com Vision 4D - Unice Rotating 4D hypercube - Andrew Hamilton 4D star polytope visualizations - Russell Towle Surfaces beyond the third dimension - Thomas Banchoff Cross Sections 4 Dimensional Star Polytope - Hop David 4D Maze Game - John McIntosh Hyperspace Bogglers - Discover Scott Kim Medicine Semi-automatic 4D analysis of cardiac image sequences - Higgins, Wang and Reinhardt Prenatal echocardiography - UCL Engineering Time-lapse oil-exploration software - Lamont 4D Geotouch: software for three and four-dimensional GIS - Jonathan M. Lees Physics Networked virtual reality for real-time 4D navigation of astrophysical turbulence data - Hudson and Malagoli An eigenmode of the Dirac operator - Kilcup Visualization Software Hyper - 4D virtual environment software - UC Irvine VR Lab Meshview - Hanson, Ishkov and Ma HyperCube 98 - Rucker and Dormishian 4va-1.2.1 - X Window 4D viewer made available by David Bagley Hypercubes Hypercubes - the Geometry Center Java-based visualization Hyperspace Star Polytope Slicer - Mark Newbold Tesseract - Harry J. Smith Stereoscopic animated hypercube - Mark Newbold 4-Cube - Rick Mabry Hyperdimension - Ishihama Yoshiaki Hypercube - Harmen van der Wal n-Dimensional visualization - Nick Jackiw 4D regular solids - Richard Koch Seeing into four dimensions - Ken Perlin Visualization of 4d hypercube - Borut Zlobec 4D Maze Game - John McIntosh 4D Viewer - Don Black Link Pages The 4D Web Page Hypercube links Hyperspace sites Google's hypercube links Ms. Guidance to Xtra dimensions Fourth Dimension: Tetraspace - Garrett Jones 4D visualization resources on the WWW - Dimitros Mitsouras Many-dimensional geometry - Geometry junkyard Selected links, articles on 4D - R. Brown Screen Savers 4D Cube - Richard Hart Games and Puzzles Chesseract - Jim Aikin's 4D chess game Rubik's Tesseract - contact Charlie Dickman for source see also Daniel J. Velleman, Rubik's Tesseract, Mathematics Magazine, Vol. 65. No. 1, Feb. 1992. Magic Cube 4D - another 4D Rubik's cube, by Hatch, Green Berkenbilt Sudden death in 4D - Jutta Four dimensional maze - Hayward 4D Maze Game - John McIntosh Adventures in 4 Dimensions - James L. Dean Please contact Mike D'Zmura at mdzmura@uci.edu if you would like to contribute further links to this page. Last update: February 2003. UC Irvine VR Lab copyright 2003
(Russia) Moscow University
V.I. Arnold's Singularity seminary.
Arnold's Seminar, Moscow University Arnold's Mathematical Seminar This seminar was started by V.I.Arnold in the 1960's. It is dedicated to singularity theory and related topics (which, of course, include a big part of mathematics). For many years, the seminar meets on Tuesdays, 16:20 -- 18:00 in room no. 14-14 of the main building of Moscow State University. Since 1993, the Parisian branch of the seminar meets at the same time in the Jussieu Mathematical Institute (formerly in Ecole Normale Superieure). Current schedule (Autumn 2005) October 4. V.I.Arnold. Problems. October 11. V.I.Arnold. Problems (continued). October 18. A.A.Karatsuba. Solution of an arithmetical problem posed by Arnold. October 25. A.V.Zarelua. On the history of the Frobenius matrix theorem. November 1. V.I.Arnold. Problems (continued). November 8. Obscure. November 15. Jacques-Olivier Moussafir. Using some results from the theory of dynamical systems for industrial mixing. Abstract (English) , Talks of previous years . Seminar information resources: seminar depository of on-line papers , data and program files . research announcements . Links to on-line papers and home-pages of the seminar members (including former members): V.I.Arnold , S.V.Chmutov , S.V.Duzhin , A.Gabrielov , M.Kazarian , M.Garay , V.Goryunov , A.G.Khovanskii , Fabien Napolitano , B.Z.Shapiro , S.Tabachnikov , S.Yakovenko . Other related pages: Arnold's problems, Mar-99 (in Russian): TeX file . PostScript . Arnold's problems about semigroups of natural numbers (Russian in transliteration): Feb-99 . Arnold's problems: Sep-98 . collected papers of Mathematical College, Independent University of Moscow, March 1998 . V.I.Arnold's letter to the seminar, 19-Feb-98 . Two problems (Sep 2001) . Arnold's problems: Jan-2002 . A movie from Trieste, 1991 . Here is the mailing list of the seminar which contains e-mail addresses of the participants and other interested people. People who want their names and addresses to be included (or deleted) are welcome to send an e-mail to duzhin(at)pdmi.ras.ru. This page was started in May 1997. It is maintained by Sergei Duzhin.
(Netherlands) Radboud University Nijmegen
Subfaculty of Mathematics, Department of Geometry. Research interests: Lie Theory; Cohomology of Varieties.
Subfaculty of Mathematics 4 februari 2002 Bron: bp Department of Geometry Head Prof. dr. J.H.M. Steenbrink Staff Research Programme Programme Lie Theory Programme Cohomology of Varieties Zoeken in de openbare Research pagina's (m.b.v. de SURFnet Search Engine): Voorbeeld van zoeken met meer termen: reports AND 1998
(Australia) University of Adelaide
Institute for Geometry and its Applications. Members, meetings, publications.
Institute for Geometry and its Applications "Where there is matter, there is geometry." Johannes Kepler (1571-1630) Aims Members Contact Seminars Workshops Preprints School of Pure Mathematics University of Adelaide Geometry lies at the core of modern mathematics with deep and wide implications in all other mathematical disciplines. In the last decade there has been an extraordinary confluence of ideas in mathematics and theoretical physics brought about by pioneering discoveries in geometry and analysis. Geometry pervades modern technology (medical imaging and information security being two well known examples). Part of the School of Pure Mathematics the Institute for Geometry and its Applications (IGA) organises symposia and workshops and provides collaborative opportunities to promote research in geometry. The IGA has expertise in mathematical physics, particularly string theory, geometric analysis, differential geometry, representation theory of Lie groups and finite geometry and its applications to information security. The activities of the Geometric Analysis group for the last five years are here. 2005 Last changed 2005 04 14 Email IGA web person BACK TO THE IGA HOME PAGE
(Slovak Republic) Comenius University, Bratislava
Faculty of Mathematics, Physics and Informatics; Department of Geometry.
SCCG 2002 - Department of Geometry Department of Geometry Head: Doc. RNDr. Milo Boek, CSc. Phone: (+421 2) 654 279 60 E-mail: kg@fmph.uniba.sk Department of Geometry, established in 1960, is one of the oldest institutes of Faculty of Mathematics and Physics. There are two sections at the department: the section of geometry and topology and the section of descriptive geometry and computer graphics. The head of the department Doc. RNDr. Milo Boek, CSc, assisted by the staff, is responsible for both the teaching and the research done at this institution. Regarding the teaching, department offers a wide range of courses in various subjects for future teachers of mathematics, such as calculus, linear algebra, basic geometry, higher geometry and history of mathematics and special courses on some parts of geometry and topology to train the specialists in computer geometry and computer graphics. It plays a major role in the education and supervision of postgraduates and in supplementary forms of study. Scientific and research activities of the department are devoted mainly to algebraic and differential geometry and topology, applied geometry and computer graphics, history of mathematics and didactics of mathematics. In the framework of these activities the members of the department staff have supervised over 25 postgraduates to their Ph.D.-degree and over 100 post-graduates to their academic degree RNDr. The department is engaged in a wide cooperation with several inland and foreign university institutions (Germany: Halle, Leipzig, Potsdam etc.; Austria: Vienna, Graz; Hungary: Budapest; Italy: Bologna), and it is a member of an institutionalized system of the international scientific and education cooperation. It has held several international summer schools. Teaching and Scientific Staff: Doc. RNDr. Eduard Boa, CSc. algebraic geometry, commutative algebra Doc. RNDr. Milo Boek,CSc. geometric principles, placement of planar bodies Doc. RNDr. Jn imr,CSc. algebraic geometry, history of mathematics Doc. RNDr. tefan Solan,CSc. algebraic geometry, commutative algebra Doc. RNDr. Valent Zako,CSc. spline theory, numerical geometry RNDr. Rbert Bohdal CG application in cartography, spline theory RNDr. Radoslav Hlek computer aided geometry, descriptive geometry RNDr. Pavel Chalmoviansk spline theory, computational geometry RNDr. Soa Kudlikov,CSc. numerical geometry, educational software RNDr. Zuzana Mederlyov computer aided geometric design RNDr. Marek Mikita spline theory RNDr. Marianna Polednov,CSc. computer graphics, education of geometry RNDr. Zita Sklenrikov,CSc. education of geometry PhD. Students: RNDr. Zuzana Machalov spline theory RNDr. Jana Plnikov spline theory Mgr. Alen Imamovi algebraic geometry Mgr. Valria Orszgov non-Euclidean geometry Mgr. Marek Szras algebraic geometry
(UK) University of Warwick
Geometry Research Area. Members, courses, meetings, specialist topics, other links.
Warwick Mathematics Institute Research Areas Search University | Contact Us | A-Z Index | GeneralInformation Admissions Undergraduate Postgraduate Research Events People University Maths Home MMV webadmin@maths Geometry The main areas of research currently represented in the department are: Algebraic Geometry Differential Geometry Geometric Analysis Hyperbolic Geometry Singularity Theory Symplectic Geometry Topology But geometry also overlaps work in many other areas represented in the department including, for example, mechanics and symmetry, stochastic analysis, etc. The people working in this area are: Elworthy , Adam Epstein , David Epstein , Gray , Jones , Micallef , Mond , Rawnsley , Reid , Rourke , Sanderson , Series , Topping , Wendland + Marie Curie and other research fellows. Jeremy Gray is Director of the Centre for the History of the Mathematical Sciences at the Open University and spends one term each year at Warwick. He carries out research in the history of mathematics in the 19th and 20th centuries, especially geometry. Seminar activities are numerous and include a wide range from informal learning groups to more formal seminars. Teaching at Graduate level . There is substantial teaching activity at the advanced level The following courses are regularly given as lectured courses or as reading courses. MA402 Advanced Partial Differential Equations MA408 Algebraic Topology MA426 Elliptic Curves MA447 Homotopy Theory MA475 Riemann Surfaces MA4xx Hyperbolic geometry MA505 Algebraic Geometry MA555 Manifolds MA575 Riemannian Geometry MA586 Symplectic Geometry MA5N0 Connections, Curvature and Characteristic Classes MA5N8 Khler Geometry Other courses may be on offer, or can be put on as reading course or as the basis for MMath projects: Algebraic curves, Algebraic K-theory, Algebraic surfaces, Cohomology operations, Complex manifolds, Discrete subgroups of Lie groups, Foliations, Gauge theory, Geometric topology, Groups of Lie type, Homology and commutative algebra, Index theory, Knot theory, 3-Manifolds, Piecewise linear topology, Sheaf cohomology, Singularities of maps, Singularities of smooth functions, Symplectic geometry of group actions. The following links may be of interest COW and Calf seminars: usually 2 talk on Thu pm, meets approx 3 times per term at a number of different UK universities. Funded by LMS. UK and European algebraic geometry For the Marie Curie training site Threefolds in algebraic geometry (3-fAG) Joint seminar in differential geometry with the Universit Libre de Bruxelles For Riemannian geometry links see EDGE website . Research Areas
(UK) University of Bath
Bath Geometry Seminar. Informal weekly seminars to discuss topics of common interest.
Bath Geometry Seminar Bath Geometry Seminar The geometry group holds informal weekly seminars to discuss topics of common interest. Everyone is welcome. The program is often decided at fairly short notice, so watch this space. If you would like to be on our e-mailing list, please contact me . Autumn 2005 Tue 4 Oct at 1:15 in 1W 3.15 Dima Vassiliev A teleparallel interpretation of Weyl's equation Tue 11 Oct at 1:15 in 1W 3.15 Udo Hertrich-Jeromin On conformally flat hypersurfaces Tue 18 Oct at 1:15 in 1W 3.15 Fran Burstall Transformations of Willmore surfaces Tue 25 Oct at 12:15 in 1W 3.15 Fran Burstall Transformations of Willmore surfaces II Tue 25 Oct at 1:15 in 1W 3.15 Barrie Cooper Temperley-Lieb and Clifford algebra representations in the Ising model Tue 1 Nov at 1:15 in 1W 3.15 Marco Lo Giudice The Kodaira-Spencer map Tue 8 Nov at 1:15 in 1W 3.15 Luis Fernandez The dimension of the set of harmonic maps from the 2-sphere to the m-sphere, part I Tue 15 Nov at 1:15 in 1W 3.15 Luis Fernandez The dimension of the set of harmonic maps from the 2-sphere to the m-sphere, part II Tue 22 Nov at 1:15 in 1W 3.15 Geoff Smith Triple perspective and porisms Tue 29 Nov at 1:15 in 1W 3.15 Udo Hertrich-Jeromin Discrete special isothermic surfaces Tue 6 Dec at 1:15 in 1W 3.15 Susana Santos Special isothermic surfaces of type n Previous Geometry Seminars Other Seminars at Bath Udo Hertrich-Jeromin Fri Oct 7 19:10:27 BST 2005
(USA) Valley Geometry Seminar
VGS usually meets somewhere near Amherst, Massachusetts.
Valley Geometry Seminar Valley Geometry Seminar The calendar for the spring of 2004 has been moved to Valley Geometry Seminar
(UK) University of Cambridge
DPMMS Geometry Seminar. Meetings.
Geometry Seminar Geometry Seminar Go to the new web site for the geometry seminar .
(Spain) University of Valencia
Department of Geometry and Topology.
Department of Geometry and Topology [ Versi en valenci | Versin en castellano ] Welcome to our WWW (World-Wide-Web) server The Department of Geometry and Topology is one of the departments which compose the Faculty of Mathematics of the University of Valencia. In this server we provide information about the following topics: Members Teaching Postgraduate courses Opening magistral lecture of the academical year 1998-99 by Professor Antonio Martnez Naveira From this WWW page you can also access our anonymous FTP file server. For any information or suggestion, please contact juan.l.monterde@uv.es Our address is: Departament of Geometry and Topology Faculty of Mathematics University of Valencia E-46100-Burjasot (Valencia) Spain Phone: +34-96-3864571 Fax: +34-96-3864735 This page is under construction.
(Austria) Vienna University of Technology
Institut fr Geometrie. Courses, members, meetings, resources.
www.geometrie.tuwien.ac.at Institute of DiscreteMathematicsandGeometry www.geometrie.tuwien.ac.at hosts the web pages of the research units Differential Geometry and Geometric Structures Geometric Modeling and Industrial Geometry Head: HellmuthStachel Head: HelmutPottmann Students are invited to have a look at information about studying geometry , curricula in general and at a list of courses and exams offered in this and previous years. You can find us at this address , e.g. to take a book out of our library . Visit the pages of the research units to have a look at people and their research activities. You can search this website via Google, and you can also search the Institute's website dmg.tuwien.ac.at . Prior to 2004, these research units belonged to the Institute of Geometry, which was established as a scientific institution, but under a different name, in 1867. Research in geometry carried out at the Institute of Geometry is presented in the pages of the two research units mentioned above. School-related activities include educational geometry software and active participation of university members in the teachers' association Fachverband fr Geometrie . Disclaimer Impressum
(Hungary) Etvs Lornd University, Budapest
History, members, events.
Department of Geometry, Etvs Lornd University Welcome to the Department of Geometry! Etvs Lornd University , Budapest , Hungary . Address of Department History of Department Staff of Department Events Conference on Differential Geometry and Physics Conference on Convexity and Discrete Geometry Back to the Home Page of Mathematics Departments maintained by Endre Dar243czy-Kiss. A DD DD last modified on November 24, 2004. DD BODY HTML
The Geometry of the Sphere
This material was the text for part of the Advanced Mathematics course in the High School Teachers Program at the IAS Park City Mathematics Institute at the Institute for Advanced Study during July of 1996.
The Geometry of the Sphere The Geometry of the Sphere John C. Polking Rice University The material on these pages was the text for part of the Advanced Mathematics course in the High School Teachers Program at the IAS Park City Mathematics Institute at the Institute for Advanced Study during July of 1996. Teachers are requested to make their own contributions to this page. These can be in the form of comments or lesson plans that they have used based on this material. Please send email to the author at polking@rice.edu to inquire. Pages can be kept at Rice or on your own server, with a link to this page. Putting mathematics onto a web page still presents a significant challenge. Much of the effort in making the following pages as nice as they are is due to Dennis Donovan . Boyd Hemphill added two nice appendices. Susan Boone helped construct the Table of Contents. All of them are teachers and members of the Rice University Site of the IAS Park City Mathematics Institute. Table of Contents Introduction Basic information about spheres Lines and spheres Planes, spheres, circles, and great circles: Exercise: Comparison with plane geometry Incidence Relations on a Sphere Spherical distance and isometries Lunes Angles on the sphere Area on the sphere The area of a lune Spherical triangles Exercise: Experimenting with spherical triangles Exercise: The area of a spherical triangle The area of a spherical triangle. Girard's Theorem Consequences of Girard's Theorem Exercise: Distortion in maps Exercise: Similarity Exercise: Congruence theorems Exercise: Small triangles on large spheres Exercise: Spherical polygons A Proof of Euler's formula Appendices Incidence relations in the plane and in space Degrees and radians , by Boyd E. Hemphill Making a "spherical straightedge" , by Boyd E. Hemphill Introduction We are interested here in the geometry of an ordinary sphere. In plane geometry we study points, lines, triangles, polygons, etc. On the sphere we have points, but there are no straight lines --- at least not in the usual sense. However, straight lines in the plane are characterized by the fact that they are the shortest paths between points. The curves on the sphere with the same property are the great circles. Therefore it is natural to use great circles as replacements for lines. Then we can talk about triangles and polygons and other geometrical objects. In these notes we will do this, and at the same time we will continuously look back to the plane to compare the spherical results with the planar results. We will study the incidence relations between great circles, the notion of angle on the sphere, and the areas of certain fundamental regions on the sphere, culminating with the area of spherical triangles. Our ultimate goal is two very nice results. First we will prove Girard's Theorem, which gives a formula for the sum of the angles in a spherical triangle. Then we will use Girard's Theorem to prove Euler's Theorem that says that in any convex, three dimensional polyhedron we have V-E+F= 2, where V is the number of vertices, E is the number of edges, and F is the number of faces. The next section contains a discussion of the basic properties of the sphere. The orthographic projection of the earth at the beginning of this page, and the earth icon used throughout these pages were produced with the program xearth , written by Kirk Johnson of the University of Colorado. John C. Polking polking@rice.edu Last modified: Fri Jan 28 15:24:14 CST 2000
Geometry Machine
3D virtual reality geometrical images of vectors, lines and planes. Needs a VRML viewer or plugin such as live3D.
Geometry Machine The Geometry Machine This machine generates 3D virtual reality geometrical images of vectors, lines and planes to aid students in understanding the geometrical interpretation of Linear Algebra. To get the most from this machine, use it to view a geometrical problem that you have already solved, eg finding the intersection point of three planes. If you dont know anything about vector equations of lines or planes, you will not be able to use the machine but you can see some samples generated by the it. You will need a VRML viewer or plugin such as live3D which comes with Netscape 3 and 4 in order to be able to see the virtual reality worlds generated by the machine. If you are not sure whether you have it, try viewing the samples before activating the machine. Programming the Machine Select the number of each of the following objects that you want displayed and then press the button to switch on the machine. When it is switched on the machine will ask you for the vector equations of each one. How many planes do you want to see? 0 1 2 3 4 How many lines do you want to see? 0 1 2 3 4 How many vectors do you want to see? 0 1 2 3 4 How many points do you want to see? 0 1 2 3 4 Back to Mike's page listing
SDCR Computional Geometry Working Group Materials
Resources and final report of the Computational Geometry Working Group, formed as part of the ACM Workshop on Strategic Directions in Computing Research, held at MIT in 1996.
SDCR Computional Geometry Working Group Materials Strategic Directions in Computing Research Computational Geometry Working Group Scope and Goals This working group was formed as part of the ACM Workshop on Strategic Directions in Computing Research , held at the Massachusetts Institute of Technology Laboratory for Computer Science, Cambridge, Massachusetts, USA, on June 14-15, 1996. The goals of the working group are to identify key research directions for the computational geometry community, to explore connections with other areas of science and engineering, and to suggest actions and initiatives to enhance the community's visibility, funding, impact and productivity. Working Group Report Final Draft (October 1996) to appear in 28(4), December 1996. Chair Roberto Tamassia Participants and Statements Pankaj K. Agarwal [ statement ] Nancy Amato [ statement ] Danny Z. Chen [ statement ] David Dobkin [ statement ] Scot Drysdale [ statement ] Steven Fortune [ statement ] Michael T. Goodrich [ statement ] John Hershberger [ statement ] Joseph O'Rourke [ statement ] Franco P. Preparata [ statement ] Jorg-Rudiger Sack [ statement ] Subhash Suri [ statement ] Ioannis G. Tollis [ statement ] Jeffrey S. Vitter [ statement ] Sue Whitesides [ statement ] Additional Contributions Chee Yap [ statement ] Reference Materials Application Challenges to Computational Geometry , a report by the Computational Geometry Impact Task Force , chaired by Bernard Chazelle Geometry in Action by David Eppstein Computational Geometry Pages by Jeff Erickson Return to the Strategic Directions in Computing Research page Last modified: Fri Apr 18 15:25:37 EDT 2003 Roberto Tamassia
Center for Geometric and Biological Computing, Duke University
Interdisciplinary research in geometric computing. Members, research areas, publications, software, resources.
Center for Geometric and Biological Computing Center for Geometric and Biological Computing The Center for Geometric and Biological Computing at Duke is a university-wide center that promotes interdisciplinary research in geometric computing. Participants are affiliated with the Departments of Computer Science, Chemistry, Biomedical Engineering, and Mathematics, the Institute for Statistics and Decision Sciences, and the Nicholas School of the Environment. The center grew out of a collaborative center with Brown University and Johns Hopkins University . The center at Duke focuses on the following research topics: Bioinformatics :. Representing, storing, searching, simulating, analyzing, and visualizing biological structures. Biomolecular Computing : Self-assembly of DNA nanostructures, storage structures. Classification : Projective clustering, approximation algorithms, mixed-dimensions clustering. External memory algorithms and data structures : I O-efficient methods for indexing and retrieving geometric objects, transparent I O protocols, efficient parallel disk access. Geographic Information Systems : Contour line extraction, range searching, point location, and map overlays on masive data sets. Graphics : Real-time vizualisation, occlusion culling and levels of detail, data structures for architectural and urban environments, maintenance of visibility in dynamic scenes. Robotics : Approximate shortest paths, weighted terrains, navigation under partial or local information, moving obstacles and changing environments, collision detection. See the list of recent publications of the center. The Center encourages participation across campus and with industrial partners. To find out more about the Center, please contact one of the co-directors. Co-Directors Pankaj Agarwal ( pankaj@cs.duke.edu ) John Harer ( harer@math.duke.edu ) Jeff Vitter ( jsv@cs.duke.edu ) Members Lars Arge ( large@cs.duke.edu ) Robert L. Bryant ( bryant@math.duke.edu ) Jeff Chase ( chase@cs.duke.edu ) Herbert Edelsbrunner ( edels@cs.duke.edu ) Mike Goodrich ( goodrich@cs.jhu.edu ) Patrick N. Halpin ( path@env.duke.edu ) Hassan Karimi ( karimi@mcnc.org ) Rao Kosaraju ( kosaraju@cs.jhu.edu ) Thom LaBean ( thl@cs.duke.edu ) Michael Prisant ( prisant@chem.duke.edu ) Franco Preparata ( franco@cs.brown.edu ) John Reif ( reif@cs.duke.edu ) George Stetten ( stetten@acpub.duke.edu ) Roberto Tamassia ( rt@cs.brown.edu ) Dean Urban ( deanu@duke.edu ) Associated Students at Duke Sathish Govindarajan ( gsat@cs.duke.edu ) Ankur Gupta ( agupta@cs.duke.edu ) Lipyeow Lim ( lipyeow@cs.duke.edu ) Nabil H. Mustafa ( nabil@cs.duke.edu ) Octavian Procopiuc ( tavi@cs.duke.edu ) Laura Toma ( laura@cs.duke.edu ) Apratim Roy ( apratim@cs.duke.edu ) Yusu Wang ( wys@cs.duke.edu ) Rajiv Wickremesinghe ( rajiv@cs.duke.edu ) Meetesh Karia ( meet@cs.duke.edu ) Recent Visitors at Duke Gerth Brodal (MPI, Saarbrucken) Tamal Dey (IIT Kharagpur) Kristoffer Dyrkorn (Norwegian University of Science Technology) Paolo Franciosa (Universita di Roma "La Sapienza") Dan Halperin (Tel Aviv University) Jan Vahrenhold (University of Muenster) Peter Varman (Rice University) Former members Rakesh Barve ( rbarve@cs.duke.edu ) Julien Basch ( jbasch@cs.duke.edu ) Pavan Desikan ( pkd@cs.duke.edu ) Jeff Erickson ( jeffe@cs.duke.edu ) Stacey Luoma ( luoma@cs.duke.edu ) T. M. Murali ( tmax@cs.duke.edu ) Paul Natsev ( natsev@cs.duke.edu ) Neill Occhiogrosso ( no@cs.duke.edu ) Cecilia Magdalena Procopiuc ( magda@cs.duke.edu ) Kasturirangan Varadarajan ( krv@cs.duke.edu ) Min Wang ( minw@cs.duke.edu ) Publications Recent Papers Software TPIE, A Transparent Parallel I O Environment Other Relevant Links 2001 Symposium on Computational Geometry Third CGC Workshop on Computational Geometry , October 11-12, 1998, Providence, RI. Second CGC Workshop On Computational Geometry , October 18-19, 1997, Durham, NC. First CGC Workshop On Computational Geometry , October 11-12, 1996, Baltimore, MD. Center for Geometric Computing at Brown Center for Algorithm Engineering at Hopkins Computational Geometry Pages (by Jeff Erickson ) ACM Workshop on Strategic Directions in Computing Research , Working Group on Computational Geometry Working Group on Storage I O Issues in Large-Scale Computation Last modified: Tue Sep 18 14:25:31 EDT 2001
Center for Geometric Computing, Brown University
A long-term project to transfer technology from Computational Geometry to applied fields. Members, publications, meetings, prototypes, resources.
Center for Geometric Computing at Brown Center for Geometric Computing at Brown University Principal Investigators Franco P. Preparata ( franco@cs.brown.edu ) Roberto Tamassia ( rt@cs.brown.edu ) Ph.D. Students Stina Bridgeman ( ssb@cs.brown.edu ) Galina Shubina ( gs@cs.brown.edu ) Sc.M. Students Benety Goh ( bg@cs.brown.edu ) Research Assistant Luca Vismara ( lv@cs.brown.edu ) Undergraduate Research Assistants Devin Borland ( dborland@cs.brown.edu ) Marco DaSilva ( mds@cs.brown.edu ) Mark Handy ( mdh@cs.brown.edu ) Andrew Schwerin ( schwerin@cs.brown.edu ) Collaborators Ryan Baker , Carnegie Mellon University HCI Institute ( rsbaker@andrew.cmu.edu ) Michael Boilen , Microsoft ( mgb@cs.brown.edu ) Jean-Daniel Boissonnat , INRIA Sophia-Antipolis ( Jean-Daniel.Boissonnat@sophia.inria.fr ) Ulrik Brandes , University of Konstanz ( Ulrik.Brandes@uni-konstanz.de ) Vasiliki Chatzi , Synopsys ( vc@cs.brown.edu ) Yi-Jen Chiang , Polytechnic University ( yjc@poly.edu ) Olivier Devillers ( Olivier.Devillers@sophia.inria.fr ) Giuseppe Di Battista , Universit degli Studi di Roma Tre ( gdb@dia.uniroma3.it ) Ashim Garg , University at Buffalo ( agarg@cs.buffalo.edu ) Natasha Gelfand , Stanford University ( ngelfand@CS.Stanford.EDU ) Benoit Hudson, NASA Ames Research Center ( bhudson@ptolemy.arc.nasa.gov ) Giuseppe Liotta , Universit di Roma "La Sapienza" ( gl@cs.brown.edu ) Maurizio Pizzonia , Universit degli Studi di Roma Tre ( map@cs.brown.edu ) Secretary Fran Palazzo ( fp@cs.brown.edu ) Sponsor Army Research Office , Mathematical and Computer Sciences Division, Discrete Mathematics and Computer Science Program (Grant DAAH04-96-1-0013) Publications Recent Papers Prototypes of WWW Geometric Computing Resources JDSL: Data Structures Library in Java Mocha: Algorithm Animation Over the World Wide Web Graph Drawing Server Links Home Page of the Center for Geometric Computing Center for Geometric Computing at Duke Center for Geometric Computing at Hopkins Graph Drawing CS 252: Computational Geometry Computational Geometry Pages (by Jeff Erickson) Computational Geometry Working Group (part of the ACM Workshop on Strategic Directions in Computing Research , June 1996) Events Short course on High-Performance Programming in Java: Collections and Beyond 3rd CGC Workshop on Computational Geometry Roberto Tamassia Last modified: Sat Oct 14 17:46:42 EDT 2000
Computational Geometry by Godfried Toussaint
Course notes and resource links.
Computational Geometry on the Web "The book of nature is written in the characters of geometry." - Galileo Go to Specific Links Related to COMP-507 (Computational Geometry course). General Links - Computational Geometry: Geometryalgorithms.com ( Fantastic Resource Page for Computational Geometry!) Jeff Erickson's Computational Geometry Pages Geometry in Action Geometry Publications by Author Godfried's Research Interests in Computational Geometry A Gallery of JAVA Sketchpad Examples in Constructive Geometry Software: JeoEdit - A Java tool for editing polygons and points on the web Directory of Computational Geometry Software CGAL Computational Geometry Library Algorithmic Solutions.Com Computational Geometry Courses and Teaching Materials David Mount's Course Notes Robert Pless's Course Computational Geometry Course at Brown University Computational Geometry 201 (Australia) Ken Clarkson's Course Algorithm Animations Basic Introduction to Computational Geometry (PostScript file) Godfried's Favourite Computational Geometry Links Computational Geometry on the Web Computational Geometry in Barcelona Computational Geometry Links Computational Geometry Applications Computational Geometry (Wolfram Research) Some Computational Geometry Algorithms Computational Geometry Algorithms Library Center for Geometric Computing Geometry Center Open Problems Qhull Home Page 3-D Models Technology in the Geometry Classroom General Links - More Geometry: Experiencing Geometry (A really nice basic geometry course by David Henderson) Geometry and the Imagination Geometry from Euclid to Today General Geometric References Geometry Formulas and Facts Interactive Geometry Interactive Geometry Miscellany Miscellaneous Topics in Geometry My Favorite Geometry Links on the Web Open Problems in Discrete and Computational Geometry Polygons Polyhedra Tessellations Geometric Probability Specific Links Related to the Course Material: Chapter Links: Classical Geometry, Basic Concepts, Theorems and Proofs Point Inclusion Problems Convexity Testing Polygon Triangulation Art-Gallery Theorems Polygonizations of Point Sets and Generating Random Polygons Distances Within and Between Sets Subdivisions Induced by Points: Triangulations, Quadrangulations... Complexity, Convexity and Unimodality Convex Hulls Visibility (Hidden-Line Problems) Updating Triangulations of Points and Line Segments Intersection Problems Proximity Graphs, Voronoi Diagrams and Polyhedral Computations Linear Programming Facility Location Mobility of Objects in Space Degeneracies in Computational Geometry Transversals of Sets Arrangements Skeletons of Polygons Visualization 1. Classical Geometry, Basic Concepts, Theorems and Proofs The Straight-Edge and Compass Computer: Francois Labelle's Tutorial on the Complexity of Ruler and Compass Constructions (with interactive Java applet) GRACE (A graphical ruler and compass editor) The straight-edge and compass Constructive geometry of the Greeks Geometric constructions Geometrography and the Lemoine simplicity of geometric constructions Wonders of Ancient Greek Geometry Models of Computation: Ceiling and Floor functions Polygons and representations Computing the area of a polygon On Proving Things: Notes on methods of proof 100% Mathematical Proof The Nuts and Bolts of Proofs Writing Down Proofs On Proofs in Mathematics Imre Lakatos on Proofs and Refutations More about Imre Lakatos Common errors in mathematics 2. Point Inclusion Problems The Jordan Curve Theorem: Octavian Cismasu's Tutorial on the Jordan Curve Theorem (with interactive Java applet) Point-in-Polygon Testing: The plumb-line algorithm More on the plumb-line algorithm Point inclusion in the query mode 3. Convexity Testing Convex Polygons: On the number of diagonals in a convex polygon (with interactive Java applet) Testing the orientation of a simple polygon Testing the convexity of a polygon 4. Polygon Triangulation History of Triangulation Algorithms Ears and Mouths of Polygons: Ian Garton's Tutorial on cutting ears and stuffing mouths (with interactive Java applets) Simple polygons have ears and mouths Meisters' Two-Ears Theorem More about Gary Meisters The one-mouth theorem: Ian Garton's Tutorial Polygons are Anthropomorphic (PostScript file) Diagonal insertion Prune-and-Search: Finding an ear in linear time (PostScript) Finding an ear in linear time (HTML) The Graham Scan triangulates simple polygons (PostScript) Fast Polygon Triangulation in Practice: Efficient polygon triangulation algorithms. Includes counter-examples to many published algorithms. (PostScript) Geometric hashing for faster Ear-Cutting in practice (PostScript) Seidel's Algorithm Triangulating monotone polygons in linear time 5. Art-Gallery Theorems The Art-Gallery problem: Chvatal's art gallery theorem Thierry Dagnino's Tutorial on Chvatal's art-gallery theorem (with interactive Java applet) Three-colorability of triangulated polygons Illuminating the Free Space Among Quadrilateral Obstacles Mobile (edge) guards Illuninating polygons with mirrored walls 6. Polygonizations of Point Sets and Generating Random Polygons Drawing Simple Polygons through Sets of Points: Tony Ruso Valeriu Sitaru's Polygonization Tutorial (with interactive Java applet) Monotonic polygonizations Star-shaped polygonizations Polygonization counter (Java applet) Generating Random Polygons Midpoint Polygonization of Points (with Java applet) 7. Distances Within and Between Sets Minkowski metrics Distance Manhattan metric: Pascal Tesson's tutorial on taxicab geometry Minkowski Operations: Minkowski sum and difference Software for computing Minkowski sums More about Hermann Minkowski The Closest Pair Problem: Closest-pair applet animation of divide-and-conquer algorithm Diameter algorithms: Diameter algorithms (Mathew Suderman's tutorial) Computing the Width of a Set: Jaqueline Chen's tutorial on width (with interactive Java applet) Width algorithms in 2 and 3 dimensions (PostScript file) The Rotating Calipers The Rotating Caliper Page of Hormoz Pirzadeh (with an awsome Java applet!) Solving five geometry problems with the rotating calipers The Rotating Caliper Graph The Reuleaux triangle (Eric's Treasure Trove) The Reuleaux triangle (The Geometry Junkyard) Shapes of Constant Width The Maximum Distance: A simple O(n log n) algorithm for computing the maximum distance between two finite sets in the plane (PostScript) A more complicated O(n log n) algorithm for the maximum distance between two point sets which generalizes to higher dimensions (PostScript) The Minimum Distance: The minimum distance between two point sets (PostScript) The minimum vertex distance between two convex polygons (PostScript) The Hausdorff Distance: The Hausdorff Distance Normand Gregoire Mikael Bouillot's Tutorial on the Hausdorff distance and its applications (with interactive Java applet) 8. Subdivisions Induced by Points: Triangulations, Quadrangulations... Triangulations: Triangles Triangulation via Polygonizations of points Divide-and-conquer The three-coins algorithm Monotone polygons Star-shaped polygons Edge-visible polygons Externally visible polygons Minimum-Weight Triangulations: Francois Labelle's interactive Java apple t Minimum-weight triangulation of a convex polygon Quadrangulations: Quadrilaterals Martin Blais' Tutorial on Quadrangulations (with interactive Java applet) Characterizing and Computing Quadrangulations of point sets (PostScript file) Converting triangulations to quadrangulations (PostScript file) Application of convex quadrangulations to font design 9. Complexity, Convexity and Unimodality Convex Set, convex function Unimodal distance functions in geometry Binary Search 10. Convex Hulls Convex hull algorithms for points in the plane (Java interactive demos) Convex hulls in 2 and 3 dimensions (interactive Java demos) Incremental (point-insertion) Randomized incremental (PostScript) The Graham scan More about Ron Graham Divide Conquer (merge hull) Quickhull (throw-away principle) The Jarvis march (gift-wrapping) Chan's Algorithm More about Timothy Chan Kirkpatrick-Seidel's ultimate algorithm (PostScript) More about David Kirkpatrick Qhull Home Page Convex hulls and duality Linear expected time algorithms Via Bucket-Sorting and Floor Functions (compressed PostScript file) The throw-away principle (PostScript) A fast convex hull algorithm - Algorithm of Akl and Toussaint (PDF file) Convex hull algorithms for polygons: 3-Coins Algorithm Tutorial (Greg Aloupis and Bohdan Kaluzny) Melkman's linear-time algorithm (Pierre Lang's tutorial with Java applet) Another tutorial on Melkman's algorithm A History of Linear-time Convex Hull Algorithms for Simple Polygons - by Greg Aloupis 11. Visibility (Hidden-Line Problems) Visibility from a point Visibility from an edge (PostScript) Visibility from a line Visibility Graphs Computing visibility graphs of line segments and polygons 12. Updating Triangulations of Points and Line Segments Updating Triangulations: Sergei Sauchenko's Tutorial on inserting and deleting vertices and edges from triangulations (with interactive Java applets) 13. Intersection Problems Intersecting convex polygons: Slab method of Shamos and Hoey More about Michael Shamos Eric Plante's Tutorial on the Rotating-Caliper method (with interactive Java applet) A simple linear algorithm for intersecting convex polygons (Rotating Caliper method) (PostScript file) Intersecting half-planes Intersecting simple polygons Applications Linear Programming Voronoi diagrams Kernel of a polygon 14. Proximity Graphs, Voronoi Diagrams and Polyhedral Computations Minimum spanning trees Nearest neighbour graphs The Relative Neighbourhood Graph (PostScript) Lune (also vescica piscis or lens) The Gabriel graph The Delaunay triangulation Delaunay Triangulations for Spatial Modelling Finding 2-D Delaunay Trinagulations from 3-D Convex Hulls (interactive Java applet) Comparison of Delaunay TrinagulationAlgorithms Computing Constrained Delaunay Trinagulations in the Plane Voronoi Diagrams: The Voronoi Web Site (Christopher Gold) VoroGlide (fantastic interactive Java applet for Voronoi diagrams and Delaunay triangulations) Another interactive applet for delaunay triangulations and Voronoi diagrams A Voronoi vertex is the circumcenter of its Delaunay triangle David Eppstein's Links for Voronoi diagrams and applications Voronoi Implementation (great applets!!!) Tutorial on Fortune's Voronoi Diagram Algorithm Higher-Order Voronoi Diagram Applet Applications of Voronoi Diagrams: Mark Grundland's Fractals from Voronoi diagrams Sphere of influence graphs (SIG's) Richard Unger's tutorial on SIG's (with interactive Java applet) Proximity Graphs (including a survey paper) Alpha Shapes and Beta Skeletons: Alpha shapes Franois Blair's Tutorial on Alpha Shapes (with interactive Java applet and a super-duper automated guided-tour demo) Gallery of alpha shapes Code for computing alpha-shapes (and convex hulls) Beta skeletons: Xiaoming Zhong's Tutorial on Beta Skeletons (with interactive Java applet) Polyhedral Computation: Frequently Asked Questions in Polyhedral Computation (by Komei Fukuda) 15. Linear Programming What is Linear Programming? Introduction to Linear Programming Interactive Linear Programming The Simplex Algorithm (Java Applet) Linear programming problems in geometry Megiddo's linear time algorithm Geometric Series Prune-and-Search Algorithm Applet 16. Facility Location The minimax facility location problem The smallest enclosing circle: An O(n log n) algorithm (with interactive Java applet) Smallest Enclosing Ball Code C++ Code for Smallest Enclosing Balls in all Dimensions The linear time algorithm of Megiddo and Dyer (Tutorial of Jacob Eliosoff and Richard Unger with applet) Generalizations to geodesic metrics Constrained facility location problems in the plane and on the sphere A survey of geometric facility location problems 17. Mobility of Objects in Space The Match-Stick Game: Fabienne Lathuliere's Tutorial on Translation Separability of Sets of Line Segments (with interactive Java applet) Translation separability of objects (compressed PostScript file) Separating objects in space (Tutorial by Kishore Anand and Anatoly Lichatchev with EXPLOSIVE applet!)interactive 4-bar linkage applet The Rhombic Dodecahedron Separating Two Monotone Polygons in Linear Time Separating Two Simple Polygons via Relative Convex Hulls Geodesic Paths: Geodesic (relative) convex hulls Steve Robbins' Tutorial on Relative Convex Hulls Shortest-paths for robotics planning (with Java applet) The algorithm of Chazelle and Lee-Preparata Linkages A historical introduction to linkages by Joseph Malkevitch Interactive 4-bar linkage applet Planar Machines' web site. An invitation to Topology. (multiple link linkages! fantastic site!!!) Paucellier's Linkage Convexifying Polygons: Linda Sun's tutorial on generating self-avoiding polygons 18. Degeneracies in Computational Geometry General position assumptions What is a degeneracy? Some examples of geometric degeneracies Robustness An arithmetic for handling intrinsic degeneracies Lower bounds for detecting affine and spherical degeneracies Avoiding Algorithm-Induced Degeneracies: (PostScript) No two points on a vertical line No three points on a vertical plane No two points with the same coordinates and many many more Computing nice viewpoints of objects in space (PostScript) 19. Transversals of Sets The Philo Line (Philo of Byzantium): Definition and computation Historical survey and characterizations Point-line duality: Interactive Java demos illustrating various definitions of duality The space of transversals Computing shortest transversals 20. Arrangements Arrangements of lines Envelopes of Arrangements of Lines (compressed PostScript file) Arrangements of line segments Arrangements of discs Arrangements of Jordan curves 21. Skeletons of Polygons The medial axis The Straight-Line skeleton 22. Visualization Nice Projections: Map projections Regular Projections and Knot Theory Aperture-Angle Optimization Problems: Viewing a statue Applet for aperture angle demo The Spindle Torus Kepler's Apples and Lemons More about Kepler Aperture-angle optimization problems in two dimensions (PostScript) Aperture-angle optimization problems in three dimensions (PostScript) Teaching Activities Homepage
The Voronoi Web Site
Christopher Gold's Computational Geometry Links.
Welcome The Voronoi Web Site "The Universal Spatial Data Structure" (Franz Aurenhammer) Welcome to all of you interested in using Voronoi diagrams for spatial analysis! My background is in the Spatial Sciences - among other things geology, geography, forestry, agriculture, cartography, surveying and, for the last few years, GIS in its many forms. Along the way I started to see many common issues in all of these, and felt that many of us were re-inventing solutions, and often doing it badly. After a while a pattern emerged: we had an extraordinarily useful tool in the Delaunay triangulation - Voronoi diagram dual representation of spatial relationships, and it should be used more often. While I have many friends in computer science, especially computational geometry, this site is not primarily about theoretical developments, although I certainly want to include their work. It is primarily for the practitioner of some spatial science who is looking for a way of expressing his particular problem. I hope you will find, as I have myself, that the Voronoi diagram often gives real insights into how to express your problem, and often how to produce an elegant solution. Best wishes, Chris Gold 2005.09.30 - We are moving the website to a new server and some pages may not work properly for some time. We are hoping to fix everything as soon as possible. "The Universal Spatial Data Structure" (Franz Aurenhammer)
Ear Cutting for Simple Polygons
Algorithms for polygonal geometry by Ian Garton.
Computational Geometry - Ear Cutting for Simple Polygons Ear Cutting for Simple Polygons by Ian Garton Contents Introduction to Ear Cutting for Simple Polygons The Two-Ears Theorem An O(kn) Time Algorithm For Finding an Ear An O(n) Time Algorithm For Finding an Ear Interactive Ear Cutting! The One-Mouth Theorem Interactive Mouth Closing! Glossary of Terms References This page was last updated on Wednesday, December 10th, 1997. 1997 Ian Inc.
Geometry Literature Database (geombib)
An ongoing project compiling a reasonably complete BibTeX bibliography of papers in computational geometry.
Geometry Literature Database -- More information about the database More information about the database The original description about history, state, and use of the database, formatted with Hyperlatex (with some later manual revisions): History and overview Bibliographers Creating entries Formatting entries Sample entries Miscellaneous comments and open problems Here is Joe O'Rourke's Computational Geometry Column on the database (in Postscript). And these are the original plain files as they are distributed with the database: The README file The authority file The announce file If you plan to use and or modify the database a lot, you should probably download your own copy . Bill Jones (jones@skdad.usask.ca) Otfried Schwarzkopf (otfried@cs.ruu.nl) Jeff Erickson (jeffe@cs.uiuc.edu)
Voronoi Diagrams
Selected references and links.
Voronoi Voronoi Diagrams Selected References Books Spatial Tessellations: Concepts and Applications of Voronoi Diagrams by Okabe, Boots and Sugihara, John Wiley Sons, 1992. This is the bible. Definitions, properties, algorithms, generalizations and applications galore! Unfortunately, it retails for $180. The King County Library System has one copy. There are several copies scattered among academic libraries in the Pacific Northwest, which you can get on interlibrary loan. Computational Geometry in C by J. O'Rourke, Cambridge University Press, 1994. Nice senior-level treatment that includes several applications and describes how Voronoi diagrams are related to minimal spanning trees and traveling salesman problems. There is a brief discussion of the "cone slicing" interpretation of Voronoi diagrams. You don't have to know C (or any programming at all) to read this book. Computational Geometry: Algorithms and Applications by de Berg, van Kreveld, Overmars and Schwarzkopf, Springer-Verlag, 1997. High-level undergraduate or low-level graduate textbook. Limited, but readable, description of the structure of algorithms for computing Voroni diagrams. You can download the fifteen-page chapter on Voroni diagrams from a web site devoted to this book: http: www.cs.ruu.nl geobook Web sites http: www.ics.uci.edu %7Eeppstein gina scot.drysdale.html Long list of applications. Definitions. Some variations of the basic Voronoi problem. Brief description of two algorithms for constructing Voronoi diagrams. http: www.beloit.edu ~biology zdravko voronoi.html The creation of Zdravko Jeremic as part of the Howard Hughes Young Scholar Program at Beloit College. A little history. Definitions. Tons of references and pointers. Jeremic's paper on modeling animal territories. Articles "Dirichlet Polygons - An Example of Geometry in Geography" by T. O'Shea, The Mathematics Teacher, March, 1986. This is the only non-technical, immediately accessible journal article we could find. It contains a couple of nice, elementary applications. | Mathographies and Snippets | Math Homepage |
Geometry Algorithms
Resources for geometry algorithm software: geometry history, monthly algorithms and archive, books and journals, videos, and website links.
Geometry Algorithm Home Geometry Algorithms [Home] [ Overview ] [ History ] [ Algorithms ] [ Books ] [ WebSites ] [ GiftShop ] Welcome to Dan Sunday's Geometry Algorithms web site. Here you will find resources for developing geometry algorithm and computer graphics software. Whether you're just interested in learning about this class of algorithms, or have a real problem to solve, we may have what you need. Look around. Visit the Geometry Gift Shop Hot Books (Click on Cover for Info) Computational Geometry in C Reviews: Computational Geometry Reviews: Geometric Tools Reviews: Overview Linear Algebra (PDF) C++ Point Vector Class Geometry History Geometers History Books History Web Sites Algorithm Archive List of Algorithm Titles Table of Contents Books Best Buys Top 10 Textbooks Computer Graphics Mathematics Journals Web Sites Bibliographies General Info Software Mathematics Gift Shop Crystals Toys Gems Paper Planes Origami Animations Knots Movies String Figures Video Lessons Featured Items The Standard Deviants: Learn Geometry by Cerebellum Corp Grow Colossal Crystals by Scientific Explorer Need Help? Contact us at: services@softsurfer.com Help SUPPORT This Site Please make purchases through this site to help support us. The cost is the same to you, but we get a small commission. Copyright 2001-2004 softSurfer. All rights reserved. Email comments and suggestions to feedback@softsurfer.com
Strategic Directions in Computational Geometry
ACM NSF Working Group Report chaired by Roberto Tamassia, intended to complement the Application Challenges to Computational Geometry by suggesting overall research directions instead of specific problem areas.
Strategic Directions in Computational Geometry Working Group Report Strategic Directions in Computational Geometry Working Group Report Roberto Tamassia (editor and working group chair), Pankaj K. Agarwal , Nancy Amato , Danny Z. Chen , David Dobkin , Robert L. Scot Drysdale , Steven Fortune , Michael T. Goodrich , John Hershberger, Joseph O'Rourke , Franco P. Preparata , Jrg-R. Sack , Subhash Suri , Ioannis G. Tollis , Jeffrey S. Vitter , and Sue Whitesides (Final Draft -- October 11, 1996 ) Contents Introduction Evolution of the Discipline Goals of the Report Organization of the Report Past Contributions Selected Major Accomplishments A Visual Depiction of the State of Advancement Methodologies Robustness Finer-Grain Complexity Analysis Implementation Experimental Evaluation Computational Paradigms Parallel and Distributed Computing External-Memory Algorithms Dynamic and Real-Time Computing Approximation and Randomized Algorithms Information Visualization Graph Drawing Algorithm Animation Interaction with Other Disciplines Education for Collaboration Fostering Collaboration Across Boundaries Dissemination of Knowledge Information Links Rewarding Experimental and Applied Research A Vision for Interaction Conclusions Contributors and Acknowledgments References Introduction Computational geometry investigates algorithms for geometric problems. For an introduction to the field, see the textbooks[ Ede87 , Mul94 , O'R94 , PS85 ] and the forthcoming handbooks[ GO97 , SU97 ]. Resources on the World Wide Web are also available(see, e.g., the Directory of Computational Geometry Software [ Ame ], the Computational Geometry Pages [ Eri ]), and Geometry in Action [ Epp ]. This report outlines the evolution of computational geometry, discusses strategic research directions with emphasis on methodological issues, and proposes a framework for interaction between computational geometry and related applied fields. Evolution of the Discipline Motivated by the need for geometric computing in science and engineering applications that deal with the physical world, about twenty years ago a community of researchers started forming around the study of algorithms for geometric problems. A new discipline, christened computational geometry, was soon chartered with the dual mission of investigating the combinatorial structure of geometric objects and providing practical tools and techniques for the analysis and solution of fundamental geometric problems. As a testimony to the original success of this mission, the initial body of computational geometry literature had a prominent presence both in the field of theory of computing and in applied areas, such as graphics, robotics, mechanical engineering, and pattern matching (see, e.g., the 1984 survey of the area[ LP84 ]). As the discipline came of age through the establishment of specialized conferences and journals, powerful new techniques of considerable mathematical sophistication were added to the existing repertory. With this formal strengthening, however, came an increased emphasis on the combinatorial aspects of computational geometry, which, in a sense, softened the original link with applications. Such an inward research orientation freed computational geometers from the unpleasantness of modeling the complexity and imperfections of the physical world and of coping with the limitations of realistic computing devices. It allowed them to focus on the analysis of a Platonic world of simple, well-behaved, geometric objects that can be manipulated by idealized computing machines with unbounded memory space and real number arithmetic. As mentioned above, the outcome of this research trend (see Section 2 ) was an impressive wealth of results on the combinatorics of objects with simple shape (such as points, lines, and polygons) in low-dimensional space (mostly two and three dimensions), and on the asymptotic complexity of fundamental geometric computations (e.g., convex hull, intersection reporting, point location, and proximity queries). The impact of computational geometry was especially strong in the field of design and analysis of algorithms. Indeed, major progress on general techniques for searching (e.g., fractional cascading), dynamic data structures (e.g., incremental rebuilding), randomized computing (e.g., random sampling), and parallel computing (e.g., cascading divide-and-conquer) was effected within the computational geometry community. However, at the same time as the accomplishments of computational geometers were celebrated within the general field of theory of computing, they also became less understood and appreciated in applied circles. The simplifying models that enabled theoretical research to flourish turned out to be major impediments to technology transfer, and hindered computational geometry from accomplishing in full its dual mission. In particular, the following apparently innocent assumptions appear to be the main culprits: the reliance on asymptotic analysis as the ultimate gauge for estimating the performance of geometric algorithms, disregarding more practical aspects of efficiency; the adoption of real arithmetic, disregarding numerical finite-precision issues; the neglect of degenerate configurations, disregarding the difficulty of taking them into account in implementations; the model of uniform access to data in memory, disregarding the huge gap between the speed of main memory and disks. As a consequence, the excitement of the computational geometry community at its theoretical accomplishments was mitigated by a sense of discomfort at the perceived loosening of ties with the very same applications that motivated the establishment of the discipline. The ``pipeline'' towards graphics, robotics, GIS, etc. continued to work successfully, but the rate of technology transfer lagged behind the growth of the ``reservoir'' of theoretical results. It became clear within the community that a correction of course was needed to achieve a more balanced evolution of the field. Indeed, the last couple of years have witnessed an increasing consensus in the computational geometry community towards a renovation of the discipline that would reconcile theory with practice and reaffirm the original dual mission. It is interesting to notice that a renewed interest in applications is also developing in the theory of computation community (see, e.g., the report on ``Strategic Directions in Theory of Computing'' [ L 96 ]). Goals of the Report A recent report entitled ``Application Challenges to Computational Geometry'' [ C 96 ], by the Computational Geometry Impact Task Force chaired by Bernard Chazelle, stresses the importance of a reorientation of the field towards providing practical solutions to the specific needs of the applications that use geometric computing. That report makes several recommendations aimed at strengthening the pipeline with geometric computing applications, and identifies ten problem areas where computational geometry can have a major impact. This report aims at complementing the Impact Task Force Report by identifying key research directions for the computational geometry community. Our focus is methodological rather than on specific problem areas. Organization of the Report In Section 2 , we highlight selected past accomplishments of computational geometry, which illustrate the richness and depth of combinatorial and algorithmic results obtained so far. Sections 3 - 4 discuss strategic directions for the field. In particular, Section 3 addresses the methodological frameworks of robustness, finer-grain complexity analysis, implementation, and experimentation, while Section 4 deals with realistic computational paradigms such as parallel and distributed computing, external-memory algorithms, real-time computing, and randomized and approximation algorithms. In Section 5 , we describe the emerging application domain (not mentioned in the Impact Task Force Report ) of information visualization. A framework for interaction between computational geometry and related applied fields, such as graphics and geographic information systems, is presented in Section 6 . Final remarks are made in Section 7 . Past Contributions In this section, we outline some of the major achievements of computational geometry, and provide a visual depiction of the state of advancement of the discipline. Selected Major Accomplishments The purpose of the the following list of accomplishments is to provide examples of fundamental results in various subfields of computational geometry. As discussed in Sections 1.1 and 2.2 , the majority of such results are combinatorial in nature and deal with asymptotic time complexity. Less explored are practical implementations and numerical robustness issues. Due to the nature of this report, all technical references have been omitted. Most of them can be found in[ O'R96 ] and by searching the Geometry Literature Database [ Jon ]. Simple polygons. For nearly every problem based on simple polygons, asymptotically optimal algorithms have been found (e.g., finding the kernel). One of the last to succumb was triangulation: a simple polygon of n vertices can be triangulated in O(n) time. Segment intersection. After many attempts, an output-size optimal algorithm was constructed for intersecting n segments in time, where k is the number of pairs of intersecting segments. Convex hulls. The convex hull of n points in d dimensions has facets, and asymptotically worst-case optimal algorithms were found both for even and odd dimensions. Voronoi diagrams and Delaunay triangulations. An optimal and practical sweepline algorithm was discovered for constructing Voronoi diagrams in the plane; implementations are now widely distributed. The deep insight that Delaunay triangulations, the duals of Voronoi diagrams, are projections of convex hulls from one higher dimension unified two principal lines of research. Linear programming. Remarkably, it was established that linear programming could be accomplished in linear time (in the number of constraints) for fixed dimension d. The doubly-exponential dependence on d was steadily improved and eventually simple randomized algorithms were found whose expected dependence on d is subexponential. These results generalize to other optimization problems, such as finding the minimum spanning ellipsoid. Point location. Point location is a classical geometric searching problem, and is used as a subroutine in a variety of geometric algorithms. Various optimal algorithms for point location in a planar map have been devised. They use O(n) space and support point location queries in time, where n is the size of the map. Simpler data structures that are not asymptotically optimal but are very efficient in practice also exist. Progress has also been made on three-dimensional point location. Range searching. A significant achievement in the last decade was the near-complete resolution of the range searching problem, with almost-matching upper and lower bounds. The introduction of -nets led to linear-size data structures with near-optimal query times for simplex range searching (reporting points inside query simplices). Trading off space with query time permits achieving faster (logarithmic) query times, using ``1 r-cuttings'': given an arrangement of n lines and a parameter r n, the plane may be quickly (and deterministically) decomposed into ``triangles'' (some unbounded), so that no triangle meets more than O(n r) of the lines. A similar result holds for arrangements of hyperplanes in d dimensions. Complexity of arrangements. Great strides were made in establishing the complexity of arrangements: The ``Zone Theorem'' established that the ``neighborhood'' of any one hyperplane in an arrangement of n hyperplanes in d-dimensions has complexity . A difficult technical achievement was showing that any m faces in an arrangement of n lines in the plane have total complexity . This bound applies, for example, to the number of incidences between n points and m lines. One long-sought result is that the complexity of the lower envelope of n surface patches in d-dimensions is (for any ). The beautiful and intricate theory of Davenport-Schinzel sequences was shown to be central to the combinatorics of arrangements, establishing, for example, that the complexity of the lower envelope of n (perhaps interpenetrating) segments in the plane is , where is the inverse Ackermann function. Visibility graphs. Efficient computation of visibility graphs has been achieved in a polygon and in a polygonal environment: for a visibility graph with n vertices and k edges, O(n + k) time suffices to find all shortest path trees (from a vertex to all others) in a polygon, and is achievable for construction of the visibility graph among obstacles in the plane. Motion planning. Many motion planning algorithms have been proved to be NP- or PSPACE-hard. Two general algorithms for solving any motion planning problem have been developed: cell decomposition and the roadmap algorithm, which run in time exponential with respect to the number of degrees of freedom of the robot, in the Turing machine model. For many special cases more efficient algorithms have been found, most notably for a polygon translating and rotating in the plane, for which a nearly-quadratic algorithm was developed. Graph drawing. A variety of techniques for constructing geometric representations of graphs have been devised. Major theoretical achievements include showing that the problems of upward planarity testing and representing a tree as a Euclidean minimum spanning tree are NP-hard. Algorithms have been discovered for upward drawings of trees with linear-area, planar straight-line drawings with integer coordinates and quadratic area, convex drawings in two and three dimensions, visibility representations and orthogonal drawings with the minimum number of bends, and upward planarity testing of embedded digraphs. In addition, algorithms with good performance in practice exist for trees, directed graphs, and undirected graphs. Randomized geometric algorithms. The introduction of random sampling techniques showed that many complex geometric problems have startlingly simple and efficient randomized solutions. Notable is the simple randomized incremental algorithm for construction of the convex hull, whose expected running time in d dimensions is asymptotically optimal. Simple optimal expected-time algorithms were also found for segment intersection (item 2 above). Derandomization led to several advances in deterministic running times, the even-d optimal hull algorithm (item 3 above) being a prominent instance. Finally, the introduction of the ``backwards analysis'' technique led to running-time analyses for randomized algorithms as simple as the algorithms themselves. Dynamic geometric algorithms. The study of dynamic algorithms and data structures has received major momentum from computational geometry. Basic dynamic geometric data structures include the segment tree, range tree, and interval tree. Based on them, efficient dynamic data structures have been devised for several fundamental geometric problems, including convex hull, point location, proximity, intersection, range searching, and path problems. Parallel geometric algorithms. Optimal or near-optimal work bounds were realized for many geometry problems under a variety of parallel computing models. For example, algorithms for the convex hull (cf.item 3 above) achieve in the EREW PRAM model an optimal runtime-processors product of for even d, and a polylog(n) factor more for odd d. A Visual Depiction of the State of Advancement The space of geometric problems so far explored can be loosely characterized by two parameters: the dimension of the geometric space and the shape complexity of the objects in that space. Most of the computational geometry accomplishments of the past deal with low-dimensional space (especially two and three dimensions), and simple objects, such as points, polygons, and subdivisions (planar maps, three-dimensional cell complexes). Higher dimensions and curved objects remain relatively unexplored. A third parameter can be used to characterize the methodology used by researchers. The majority of the computational geometry literature deals with combinatorial analysis and asymptotic computational complexity. As discussed in the next section, the pursuit of numerical robustness and the development of practical implementations appear to be strategic methodological choices for the evolution of the field. The figure below schematically illustrates the state of advancement of computational geometry research as a portion of a virtual ``cube'' whose x-, y- and z-axes are associated with methodology, dimensionality, and shape complexity. A vast portion of the cube remains to be explored. Methodologies We believe that the most important strategic direction for computational geometry is to substantially enlarge its arsenal of tools to include methods that can handle the practical aspects of geometric computing. While combinatorial and asymptotic analysis remains a cornerstone of the discipline, it is essential to start an extensive reexamination of geometric problems from the following viewpoints: robustness; finer-grain complexity analysis; implementation; and experimental evaluation. Problems considered completely solved and no longer interesting from the viewpoint of asymptotic complexity reveal unexpected challenges when studied under a new light. For example, conventional asymptotically optimal algorithms for minimum link paths in a simple polygon and proximity queries on set of planar point sites perform poorly with respect to the arithmetic precision of the numerical computations, and new data structures and approximation schemes are needed to reconcile efficiency with robustness. Robustness Geometric algorithms are usually described in the conceptual model of the real numbers, with unit-cost exact arithmetic operations. However, the original assumption of a computational model obtained by extending the traditional RAM to real-number arithmetic proved less innocent than originally thought. Implementers often substitute floating-point arithmetic for real arithmetic. This leads to the well-known problem of numerical robustness, since geometric predicates depend upon sign evaluation, which is unreliable if expression evaluation is approximate. To equate floating-point arithmetic to real-number arithmetic turned out to be indefensible in practical applications. Another convenient assumption has been the hypothesis of ``general position,'' which dispenses with the detailed consideration of special cases. Unfortunately, degenerate conditions (colinearity, cocircularity) which are likely to be generated by coarse-grid data as they occur in practice, give rise to numerically critical events. Failures originating from these assumptions have fundamentally hindered the adoption of computational geometry by practitioners. Over the years several approaches have been proposed to remedy these shortcomings. It is likely that no single approach may be capable of conferring robustness to geometric algorithms. Presumably, several tools may be included in an arsenal designed to achieve robust computations. However, an approach that has the potential to yield a useful methodology is the following. The numerical computations of a geometric algorithm are basically of two types, which we may designate as tests and constructions. These two types have clearly distinct roles. Tests are evaluations of geometric predicates associated with branching decisions in the algorithm that determine the flow of control, whereas constructions are used to produce the geometric objects that normally represent the output of the application. Approximations in the execution of the constructions give rise to approximate results, which may nevertheless be entirely acceptable as long as the maximum error does not exceed the resolution required by the application (in all cases, some more or less coarse grid). On the other hand, approximations in the execution of tests may produce incorrect branchings, which may have catastrophic consequences, since they may yield structurally (i.e., topologically) incorrect results (such as a missed intersection or an open polygon). Therefore, tests have much more stringent requirements, which leads to the conclusion that they must be carried out with complete accuracy, whereas some tolerance is permitted for constructions. (It must be observed, however, that such tolerance must be consistent with the topological structure of the result as provided by the tests.) Complete accuracy would seem to require the infinite precision implied by real-number arithmetic. Fortunately, the inherently coarse nature of the input data comes to the rescue in this connection. Each predicate is expressible as the sign of a multivariate polynomial in the input variables. If input variables are assumed of degree1, the degree of such a polynomial specifies the maximum precision required by the test in question. Of course, the maximum precision may have to be deployed only in near-degenerate cases. In typical cases, much lower precision may be sufficient to confidently evaluate the predicate. It is therefore the function of an ``arithmetic filter'' to identify the adequate precision. In this framework, the development of cost-effective filters may be one of the major challenges in the quest for robustness. There is considerable evidence that adaptive-precision arithmetic, if engineered carefully, can substantially reduce the effective cost of extended-precision evaluation. Performance comparable to floating-point arithmetic has been achieved for algorithms with relatively modest precision requirements, e.g. evaluating simple predicates on points, lines, and planes in dimensions two and three. A related issue is the problem of rounding geometric structures. The goal is to represent derived geometric structures in fixed precision, so that the key combinatorial topological properties of the structures are preserved. Finer-Grain Complexity Analysis Asymptotic performance, rather than acting as a powerful and useful analysis tool, has frequently become the ultimate focus of computational geometry research. Unfortunately, the very nature of asymptotic ``big-Oh'' worst-case analysis carries its own inadequacy: the suppression of multiplicative constants from performance functions and the overemphasis on pathological scenarios. It is quite common for algorithms that have been declared ``asymptotically optimal'' in the Random-Access Machine (RAM) computational model to be inferior to ``suboptimal'' algorithms in practice. Focusing exclusively on asymptotic analysis discounts the importance of developing practically-efficient computational tools. Remedying this shortcoming is an important and difficult task. A promising approach is to isolate significant primitives appearing in the execution of algorithms, such as pointer updates, evaluations of fixed-dimension determinants, and other data management operations (including those occurring in the handling of external and hierarchical memories), and to express performance not as a single function of application parameters (problem size, memory size, number of processors, etc.), but rather as a vector of such functions, each component of which quantifies the use of such primitives. It may even be desirable or necessary to precisely quantify some components of the performance vector (forfeiting the comforts of the big-Oh notation), in order to provide a realistic comparison between competing algorithms. Implementation Existing computational geometry algorithms are often directly relevant in industrial applications. However, the knowledge of those algorithms is not widespread, and robust, easy-to-use, well-publicized implementations are rare. The computational geometry community should strive to package its best geometric algorithms into easy-to-use software tools that can be used by non-specialists. Such tools would dramatically enlarge the set of potential users of geometric algorithms. A handful of programs--implementations of convex hull and Voronoi diagram algorithms--have been distributed successfully. In fact, these popular programs have been distributed more widely than any computational geometry publication except perhaps Preparata and Shamos's textbook[ PS85 ]. The programs have been used in a wide variety of applications, most of them unanticipated by the program authors. These successes hint at the potential influence of computational geometry in practice, and should encourage further implementation efforts. It is instructive to compare the teaching of computational geometry to that of sorting or hashing. Computational geometry is typically taught in specialized courses, or as a single short section in algorithms classes. Similarly, detailed analyses of sorting and hashing are taught in specialized algorithms courses. The crucial difference is that sorting and hashing are widely applied in other disciplines (e.g., systems programming), and are presented as black-box tools in non-algorithmic courses. Except in a few cases, computational geometry has not provided the simple, flexible tools that would enable the kind of widespread use that sorting and hashing receive. Published libraries of geometric routines (see, for example, the Directory of Computational Geometry Software [ Ame ]) often have a large granularity of adoption: potential users must adopt the whole library and its data models if they want to use any part of it. Writers of geometric software should provide lowest-common-denominator interfaces, as well as more efficient interfaces for sophisticated users. In particular, tools for geometric software development should enable non-specialists to specify their geometric problems in a straightforward manner, and to create programs for solving them by combining basic building blocks in a simple fashion. Two efforts to build libraries of geometric software are underway-- CGAL [ Ove97 ] in Europe and GeomLib [ AGKPTV95 ] in the United States. Successful examples of software tools, libraries, and repositories from the numerical, scientific, statistical, and symbolic computing communities include Netlib , LAPACK , SPSS , Mathematica , and Maple . Other models of successful publication of algorithmic results include the popular Numerical Recipes and Graphics Gems books. In the same vein, computational geometry needs books and Web sites with titles like Geometric Recipes and Geometric Tools, with accompanying software. Further ideas for dissemination of algorithms and software appear in Section 6 . Experimental Evaluation A prerequisite to deciding which algorithms to implement in geometric libraries is knowing which ones perform well in practice. The last couple of SCG (ACM Symposium on Computational Geometry) conferences have encouraged papers that do experimental work, as has SODA (ACM-SIAM Symposium on Discrete Algorithms). However, comparative studies of algorithms for solving given geometric problems are just beginning to appear. We need to encourage and reward such studies. The algorithms should be implemented to share data structures and primitives whenever possible (so that the relative speeds of the algorithms and not the cleverness of the implementers is being tested). The programs need to be ported to a range of machines, because differences in architecture seem to greatly influence the relative speeds of algorithms. A sub-area of experimental evaluation is choosing appropriate test data. We at least have some idea of what ``random'' point sets might be (uniform, normal, etc.), even if we do not know how realistically these distributions model the real world. But what is a ``random'' collection of non-intersecting line segments or rectangles or polygons? We have barely begun to consider such questions. Furthermore, how do these correspond to ``real world'' data? Using ``real world'' data is not a panacea, either, because applications differ widely in what sort of data they use. What is needed is a collection of benchmark data drawn from a wide range of applications, analogous to the SPEC benchmarks used in computer architecture. No such collections of benchmark data are currently available, and creating such collections would be a valuable service to the community. Computational Paradigms To be effective, the proposed renovation of computational geometry must be accomplished within the context of today's complex technology and computing environments. In particular, it is a key strategic issue to take into account the following realistic computational paradigms: parallel and distributed computing; external-memory algorithms; dynamic and real-time computing; and approximation and randomized algorithms. Parallel and Distributed Computing For time-critical applications, multiple processors may be needed to perform the specified computations in a short amount of time, and the data inputs for the computations may be distributed geographically. This viewpoint conforms with a general trend toward a more extensive deployment of concurrency and distributed computation. The programming ease deriving from the use of a uniform address space (shared memory) must be carefully weighed against the potentially better performance of a distributed-memory network. Particularly attractive is algorithmic research within the so-called coarse-grain parallel model, which seems particularly attuned to a variety of geometric computations. The coarse-grain model reflects very closely the most plausible parallel distributed computing technology of today or of the near future. In this model a system consists of relatively few (typically, tens hundreds) processors (typically, off-the-shelf microprocessors), each equipped with a sizable private memory. The processors are either interconnected according to one of the conventional networks, or are part of a distributed network (as it turns out, since the number of processors is very small, the interconnection does not play a central role). For p processors and problem size n, the coarse-grain algorithmic approach consists of identifying p subproblems of size n p whose solutions can be combined to solve the original problem. Thus, initially all processors act serially on their respective subproblems (solitary parallelism), and subsequently interact (cooperative parallelism) to combine the results of the first phase. For n much larger than p (a very realistic situation) the parallel time of the first phase dominates the time of the second phase, and optimal speed-ups are achievable. Problems that lend themselves to this approach are those possessing substantial data-locality. External-Memory Algorithms In the large-scale geometric databases and other data-intensive processing encountered in many applications, the main (random-access) memory of the processor is not large enough for the requirements of the application. This limitation faced by large-scale applications was recognized very early on in the computer era, and its correction (hierarchical memory) represents the first serious revision of the von Neumann model. As it happens, Input Output communication (I O) between levels of hierarchical memory is the bottleneck in many large-scale geometric applications. Algorithms designed specifically to make efficient use of two or more levels of memory are often called external-memory algorithms to emphasize the explicit use of memory beyond random-access main memory. The I O bottleneck gets accentuated as processors become faster with respect to disks (currently the typical medium of external storage) or when multiple processors are used, prompting several researchers and companies to deploy external storage systems with parallel capabilities. The issues here are closely related to those outlined in the preceding subsection in connection with the coarse-grain model, when the private memories of the processors do not satisfy the requirements of the subproblems. In many practical situations, we can restrict attention to the case of two levels; for such purposes the fairly realistic two-level multiple-disk I O model covers both uniprocessor and multiprocessor systems: there are several disks and several processors (both currently in the range 1- ); each processor is equipped with a main memory that can store a limmited amount of data (currently in the range - ). The processors and the disks are connected by a network (such as a shared-memory interconnection, hypercube, or cube-connected cycles, as also postulated in the previously described coarse-grain model) that allows for efficient execution of some basic operations like sorting. Dynamic and Real-Time Computing Dynamic (or incremental) computation considers updating the solution of a problem when the problem instance is modified. Many applications are incremental (or operation-by-operation) in nature and the typical run involves on-line processing of a mixed sequence of queries and updates. In a real-time environment, it is essential that a query be answered within a fixed time boundt. If the size of the problem is such that the query time exceedst, we should at least ensure that at timet the query algorithm has produced some useful results, i.e., an approximation of the query answer. An algorithm is called interruptible if it converges toward the exact solution by incrementally producing better and better approximations. Interruptible algorithms should be explored for a variety of geometric problems that arise in time-critical applications. With respect to real-time applications, it is also interesting to devise approximation algorithms that use substantially fewer time and space resources than exact ones, with an ensuing performance approximation trade-off. For example, consider the convex hull problem. Its exact solution needs time, but in a real-time environment we may be able to afford only O(n)-time computations. What is the best approximation of the convex hull that can be achieved with O(n) time? Most previous research has been devoted to the study of polynomial-time approximation algorithms for NP-hard problems (e.g., for the traveling salesman problem). Approximation algorithms with O(n) or time complexity should be studied for problems whose exact solution seems to require substantially more time (e.g., ). Motion is common with objects in the physical world and is a primary concern in geometric applications such as collision detection in robotics and visibility determination in computer graphics. Another facet of dynamic computation is dealing with continuously changing data. A kinetic data structure maintains attributes of mobile objects (e.g., convex hull and closest pair of a set of points in continuous motion). Previous research on this subject has focused on the case where the full motion of the objects is known in advance. Work is needed on a more realistic scenario where objects can change their motion on-line because of external impulses and interactions with each other. Approximation and Randomized Algorithms The lack of fast, simple, deterministic algorithms for many fundamental problems has motivated the study of randomized algorithms. In the last few years, a number of randomized geometric algorithms, based on elegant probability theory, have been developed that are significantly simpler and, in several cases, faster than their deterministic counterparts. Randomization is also useful for dynamic data structures, on-line algorithms, and intractable problems. A number of interesting geometric problems are intractable, including certain instances of navigation, machine learning, target acquisition and tracking, and visualization. In many applications, it suffices to obtain a good but fast solution. For example, in a typical navigation problem, it is desirable to have a real-time algorithm that computes a reasonably short path instead of one that computes an optimal path but takes longer time. These factors initiated the study of approximation algorithms, which always determine a solution that is close to optimal. Approximation algorithms are also useful for higher dimensional problems because the running time of typical geometric algorithms increases exponentially with the dimension. Information Visualization The Impact Task Force Report [ C 96 ] is an excellent resource on geometric computing problems within the following ten key application domains: computer graphics and imaging, shape reconstruction, computer vision, geographical information systems, mesh generation, robotics, manufacturing and product design, robustness, molecular biology, and astrophysics. In this section, we describe the emerging application domain of information visualization, where computational geometry can have a major impact. Two specific fields are covered within information visualization: graph drawing; and algorithm animation. Information visualization is identified as a strategic research direction also in the report on ``Strategic Directions in Human Computer Interaction'' [ MHC 96 ]. Graph Drawing The visualization of complex conceptual structures is a key component of support tools for many applications in science and engineering. Examples include software engineering (call graphs, class hierarchies), database systems (entity-relationship diagrams), digital libraries and WWW browsing (hypermedia documents), distributed computation (reachability graphs of communicating processes), VLSI (symbolic layout), electronic systems (block diagrams, circuit schematics), project management (PERT diagrams, organization charts), decision support (scheduling and logistic diagrams), medicine (concept lattices), telecommunications (ring covers of networks), and law (conceptual nets). Foremost among the visual representations used are drawings of networks, graphs, and hypergraphs. A variety of graph drawing algorithms have been developed in the last decade[ DETT94 ]. Several graphic standards such as straight-line, polyline, and orthogonal, have been used to represent graphs, depending on the application. Major geometric problems in information visualization include: optimization of measures of quality of drawings, such as number of bends, number of crossings, and area; trade-offs between quality measures of drawings, e.g., between area and angular resolution of planar straight-line drawings; conceptual ``fisheye views'' for displaying large graphs; label placement for nodes and edges of a drawing; incremental and interactive layout maintenance; three-dimensional representations. Detailed experimental studies on the practical performance of layout algorithms are also much needed. Algorithm Animation The visual nature of geometry makes it a natural area where visualization can be an effective tool for communicating ideas. This is enhanced by the observation that much research in computational geometry occurs in two and three dimensions, where visualization is highly plausible. Given these observations, it is not surprising that there has been noticeable progress during the past few years in the production of visualizations of geometric algorithms and concepts[ HD97 ]. There is every reason to believe that this will continue and even accelerate in the future. As anyone who has tried to implement a complex geometric algorithm knows, implementing geometric algorithms is a difficult task. Conventional tools are limited as aids in this process. The programmer spends time with pen and pencil drawing the geometry and data structures the program is developing. This problem could be solved by the use of visualization tools. In the ideal world, this visualization would be used for three purposes: demonstration, debugging, and isolation of degeneracies. Ideally, we would like to use the same tools for all three functions. For example, we would use the tool to help us debug the implementation of an algorithm by providing visual interaction during the debugging process. Next, we would like to use the same tool to create a visualization of the algorithm with which the user can interact. This interaction could be either passive or active. For example, a video tape provides passive interaction since the viewer's controls are limited to the VCR controls. Active interactions allow the viewer greater control over the visualization. Finally, there is the issue of isolating problems in code that is symbolically correct. Typically, such bugs come from degeneracies either in the data or in the computational model. Visualization has the potential to be a great help here as a tool allowing the user to jump into the code at (or preferably before) the point at which it breaks. The problem of creating active interactions remains largely unsolved. It is still the case that a visualization demonstrates the behavior of an algorithm on one sample input and explains the behavior of the algorithm on that input. A better scenario would allow the user not only to specify the input, but also to interact with the view (and possibly even the input data) as the algorithm is running. There are a few existing systems that allow the user to interact with a running animation. However, the interactions come at a price. The viewer must typically have the hardware that was used to develop the interaction. This limits the ability to integrate such animations into hypermedia documents. There is hope that the emergence of Java and VRML will help remove this limitation. Interaction with Other Disciplines Computational geometry has established itself both as a discipline and as a community of researchers. To realize the discipline's potential for usefulness to others and to maintain its vigor, the community now seeks closer collaboration with application domains that inspire geometry problems. At the same time, it wishes to maintain both its identity and its traditional contacts with mathematics. Previous sections have suggested computational paradigms and methodologies for computational geometers to adopt in order for their research to become usable by others. However, adopting new methods is not enough: real-world problems are often inherently interdisciplinary in nature and international in impact. Hence mechanisms are needed that facilitate the crossing of boundaries between academic disciplines, between countries and continents, and between universities, government research labs and industries. The organization of effective mechanisms for interaction is a job not only for individuals (e.g., students, researchers, university-industry liaison officers, administrators), but also for computational geometry community groups (e.g., program committees, boards) and organizations (e.g., professional societies, funding agencies). Hence the remarks below are addressed to a wide audience, and while stated in the context of geometry, many apply to other subfields of computer science as well. Education for Collaboration Certainly the horizons of computational geometers can be broadened by guest speakers at seminars and conferences who come from other, related fields. Interdisciplinary workshops and short courses can go much further. International workshops such as those run by the Dagstuhl Research Center in Germany and the summer joint research conference program run by AMS-IMS-SIAM provide models. Graduate education in computational geometry should provide an opportunity for interdisciplinary study and, where possible, industrial collaboration. This could be facilitated for example by university-industry internships, degree programs with minor options in other fields, and special topics or projects courses taught by industrial researchers. Students in disciplines that have a geometric component should be informed of the possible relevance of computational geometry courses to their program of study. Fostering Collaboration Across Boundaries Opportunities for professionals to share geometry problems across academic disciplines and across university industry boundaries should be fostered. This might take the form of short or long term visits for study or consulting, including consulting by academics within their own universities. The value of such exchanges should be recognized by agencies and institutions. Mechanisms to ease publication of interdisciplinary research, and to promote publication of research outside the boundaries of traditional, narrow subdisciplines, should be designed and considered. Researchers should be able to build and maintain a reputation within the computational geometry community, while at the same time making their applied results known to the community of the application area. Possibilities for double-publishing might be explored (e.g., a research summary for one community accompanied by a full paper for the other). Dissemination of Knowledge Unfortunately, many computational geometry algorithms are either completely unknown or otherwise inaccessible to researchers and practitioners outside the community. Computational geometers should continue to address this problem by contributing expository writing such as handbooks. Handbooks currently in preparation include[ GO97 , SU97 ]. Other possibilities include creating collections modeled after Graphics Gems, and contributing survey articles to publications such as ACM Computing Surveys, Communications of the ACM, or Scientific American. Such expository literature should provide potential users with pointers to implementation advice, performance results, and any available code. The availability of systems that contain libraries of well-documented code for geometric problems forms an important aspect of knowledge dissemination. The availability of good programming environments and usable code facilitates the development of geometric applications for both specialists and non-specialists. Hence such systems should allow easy export of code to other systems, such as a geographical information system. In return, the design and implementation of such environments give rise to a host of interesting research problems for computational geometers. Current efforts include the Workbench for Computational Geometry (at Carleton University, Ottawa), XYZ-Geobench at ETH Zrich, LEDA at Max Planck Institut fr Informatik, Saarbrcken, the CGAL project involving a consortium of seven European sites (Utrecht University, ETH Zrich, Free University Berlin, INRIA Sophia-Antipolis, Max Planck Institut fr Informatik at Saarbrcken, RISC Linz, and Tel Aviv University), and the GeomLib project of the Center for Geometric Computing, a consortium of three US sites (Brown University, Duke University, and The Johns Hopkins University). Journals can publish implementation-oriented research articles by developing standards for refereeing code and by making the code associated with accepted articles available over the Web. Code accepted by a journal would become a citable, refereed journal item. Already several conferences (e.g., ESA, SCG, SODA, WADS) have started accepting papers of the above nature. Also, journals such as the International Journal on Computational Geometry and Applications (ed.D.T. Lee), Computational Geometry: Theory and Applications (eds.J.-R. Sack and J. Urrutia), the ACM Journal of Experimental Algorithmics (ed.Bernard Moret) as well as several special issues in other journals are currently seeking such papers. Information Links Theoretical and applied areas in which computational geometry could play a role span several disciplines; indeed they span the scope of several professional societies. For example, research in fields such as computer vision, automation, manufacturing, CAD CAM, robotics, computer graphics, topology, geometry, tomography, medical imaging, polyhedral combinatorics, combinatorial optimization, cartography, and geographic information systems is reported in the conferences and journals of professional societies such as ACM , AMS , SIAM , IEEE , ASME , ORSA, and AGI (Association for Geographic Information). Technical terms are not standardized: roughly the same problem may have different names in different disciplines; conversely, different problems may receive the same name. Furthermore, applied problems typically elude precise, tidy, mathematical definition. Mechanisms are needed for the rapid cross-referencing, across discipline and professional society boundaries, of geometric concepts, problems, keywords, solutions methods, and implementations. For problems that do admit precise description, practitioners should be able to find relevant information easily, including pointers to literature and code. They should also be able to pose geometric problems, whether of a general or a specific nature, to the computational geometry community at large. The computational geometry community should consider ways to maintain, build on and cross-link resources (especially electronic ones) for geometry problems. A number of individuals and research groups have initiated efforts to create electronic sources of information. Perhaps such efforts should be imitated at the level of professional societies, to insure continuity and ease of access across disciplines. A kind of electronic encyclopedia of geometry might be designed, with mechanisms for ``looking up'' and or posing geometry problems in words and images, with cross-referencing between problems and application domains, and with pointers to code libraries and researchers. Rewarding Experimental and Applied Research The computational geometry community should continue to design strategies to encourage and evaluate experimental and applied work. However, strategies such as creating separate categories for theoretical and applied papers at conferences and such as creating bench marks and standard data sets should be constantly monitored for desired effect. Clearly, providing implementation results and comparisons requires substantial research effort as well as time investment, so suitable reward structures should be devised. These might take the form of publication in journals or established, public geometric libraries as discussed above. The commercial potential for geometric libraries provides major incentives for researchers to work on implementation issues: the possibility of financial reward, and also, the satisfaction of making something that works and gets used. Since the availability of libraries will inspire and facilitate yet more implementation and application-oriented research, the price structure for commercial code should differentiate between academic and industrial commercial users. A Vision for Interaction The vision for many in the computational geometry community is that computational geometry emerge as the discipline where for geometry, theory meets practice, where problems of an applied nature inspire and inform research problems in computational geometry and mathematics, where theoretical results are implemented, made usable, and disseminated to application domains. Conclusions Computational geometry is a lively discipline that is undergoing a crucial phase of its evolution. We have identified methodologies and computing paradigms that we consider of strategic importance for the growth of the discipline and its impact on applications. The key message is that computational geometry should reaffirm its dual mission of investigating the combinatorial structure of geometric objects and providing practical tools and techniques for the analysis and solution of fundamental geometric problems Contributors and Acknowledgments This report represents the efforts of the Computational Geometry Working Group formed as part of the ACM Workshop on Strategic Directions in Computing Research , held at the Massachusetts Institute of Technology Laboratory for Computer Science, Cambridge, Massachusetts, USA, on June 14-15, 1996. The main ideas presented in this document were discussed at the working group meetings scheduled during the workshop. The material contained in this report originates in part from the participants' position statements , from the ``Computational Geometry Column 29'' [ O'R96 ], and from the Center for Geometric Computing proposal on ``Applicable and Robust Geometric Computing''[ AGKPTV95 ]. This report benefited from a white paper on ``Exact Computation and Reliable Geometric Software'' contributed to the working group by Chee Yap, and from a previous report entitled ``Application Challenges to Computational Geometry,'' [ C 96 ] by the Computational Geometry Impact Task Force chaired by Bernard Chazelle. Comments and suggestions from Leo Guibas, Chris Hankin, and Michael Loui are gratefully acknowledged. Finally, we would like to thank Peter Wegner for encouraging the formation of this working group and for many useful discussions. References AGKPTV95 P.K. Agarwal, M.T. Goodrich, S.R. Kosaraju, F.P. Preparata, R.Tamassia and J.S. Vitter, Applicable and Robust Geometric Computing, 1995, http: www.cs.brown.edu cgc . Ame N.Amenta, Directory of Computational Geometry Software, http: www.geom.umn.edu software cglist . C 96 B.Chazelle etal., Application Challenges to Computational Geometry: CG Impact Task Force Report, Technical Report TR-521-96, Princeton Univ., Apr. 1996, http: www.cs.duke.edu ~jeffe compgeom taskforce.html. DETT94 G.Di Battista, P.Eades, R.Tamassia and I.G. Tollis, Algorithms for drawing graphs: an annotated bibliography, Comput. Geom. Theory Appl., 4, 235-282, 1994. Ede87 H.Edelsbrunner, Algorithms in Combinatorial Geometry, vol.10 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Heidelberg, West Germany, 1987. Epp D.Eppstein, Geometry in Action, http: www.ics.uci.edu ~eppstein geom.html. Eri J.Erickson, Computational Geometry Pages, http: www.cs.duke.edu ~jeffe compgeom . GO97 E.Goodman and J.O'Rourke, editors, Handbook of Discrete and Computational Geometry. CRC Press, 1997, To appear. HD97 A.Hausner and D.Dobkin, Making Geometry Visible: An introduction to the Animation of Geometric Algorithms, In J.-R. Sack and J.Urrutia, editors, Handbook on Computational Geometry, pp. ??-?? North Holland, 1997, To appear. Jon W.Jones, Geometry Literature Database, http: www.cs.duke.edu ~jeffe compgeom biblios.htmlgeombib. L 96 M.C. Loui etal., Strategic Directions in Theory of Computing, ACM Computing Surveys, 28, no.4, 1996, http: geisel.csl.uiuc.edu ~loui complete.html. LP84 D.T. Lee and F.P. Preparata, Computational geometry: a survey, IEEE Trans. Comput., C-33, 1072-1101, 1984. MHC 96 B.Myers, J.Hollan, I.Cruz etal., Strategic Directions in Human Computer Interaction, ACM Computing Surveys, 28, no.4, 1996, http: www.cs.cmu.edu ~bam nsfworkshop hcireport.html. Mul94 K.Mulmuley, Computational Geometry: An Introduction Through Randomized Algorithms. Prentice Hall, Englewood Cliffs, NJ, 1994. O'R94 J.O'Rourke, Computational Geometry in C. Cambridge University Press, 1994. O'R96 J.O'Rourke, Computational geometry column 29, Internat. J. Comput. Geom. Appl., ??, ?-?, 1996, Also in SIGACT News, 27:3 (1996), Issue 100, to appear. Ove97 M.H. Overmars, Designing the Computational Geometry Algorithms Library CGAL, In Applied Computational Geometry (Proc. WACG '96), Lecture Notes in Computer Science. Springer-Verlag, 1997. PS85 F.P. Preparata and M.I. Shamos, Computational Geometry: An Introduction. Springer-Verlag, New York, NY, 1985. SU97 J.-R. Sack and J.Urrutia, editors, Handbook on Computational Geometry. North Holland, 1997, To appear. Roberto Tamassia Fri Oct 11 02:39:30 EDT 1996
Godfried Toussaint's Research Interests
Mainly in computational geometry, e.g., mobility of objects in space, degeneracies, quadrangulations, tomography, triangulation, proximity, facility location, and polygonal approximation.
Research Interests "Everything should be made as simple as possible but not simpler." - Albert Einstein "Simplicity is embarrasing when you have to work for months to achieve it." - Kary Mullis "The search for truth is more precious than its possession." - Albert Einstein Computational Geometry: Proofs and Refutations in Geometry Mobility of Objects in Space (Robotics, Linkages, Polymer Physics and Protein Folding) Visualization Avoiding Algorithm-Induced Degeneracies Quadrangulations in Mesh Generation Geometric Tomography Practical Polygon Triangulation Proximity Graphs Hamiltonian Circuits Constrained Facility Location Geometric Problems in Manufacturing Geodesic Properties of Polygons Arrangements Intersection Problems Visibility Problems Polygonal Curve Approximation and Smoothing Computing Distances The Rotating Calipers Convexity Theory and Unimodality Computer Vision and Pattern Recognition: Document Analysis Detecting Structure in Dot Patterns Nearest Neighbor Classification (Prototype Selection in Instance-Based Learning) Shape Hulls Computational Music Theory: Rhythm and Melody Mathematics and Flamenco (matematicas y flamenco) String Similarity Measures Rhythm Generation The Complexity of Rhythm Rhythm Segmentation History of Computing: The Collapsing Compass (Euclid of Alexandria) The Philon Line (Philon of Byzantium) Homepage
Laurent Balmelli
A research staff member at the IBM T.J. Watson Center in Hawthorne, NY. His main interests and fields of research are computational geometry, digital geometry processing, data compression, data structures and optimization techniques. This site contains his recent publications, as well as demos and software.
Laurent Balmelli - computer graphics, quadtree, global error, subdivision surfaces, digital geometry processing, geometry compression, transmission WELCOME - contact , credits Ecole Polytechnique Federale (EPFL) IBM Research Homepage Java for Signal Processing 2002 Design - Laurent Balmelli - December. 17 2002 www.balmelli.net Welcome to my homesite! I am a researcher for IBM at the T.J Watson center in New York. I am working with the Visual and Geometric Computing group. Find more about my interests and background , my work and publications . My PhD thesis is available on-line . News Bulletin - last updated 12 17 2002 February I gave an overview talk at NYU on warping method for textures and volumes, hosted by Prof. Denis Zorin. ( pdf 2Mb ) March We accepted five papers for the Special Session on Multimedia Technologies for Gaming at ICME . See the annoucement here ! I spent a week at University of Tubingen in Germany as a visiting scientist, and gave a seminar on textures and volumes warping. April New paper accepted for 3DIM: we investigate the edition of 3D models using shaded 2D projections. A joint effort with Holly Rushmeier, Gabriel Taubin, Jose Gomes and Fausto Bernardini. The paper will be soon on-line! F E A T U R E D 2002 IBM SEMINARS IN COMMUNICATIONS A Series of 8 lectures in communications at the T.J.Watson Center. Learn more CALL FOR PAPERS IEEE -ICME 2003 Special session on Multimedia Technologies for Gaming Submissions are closed now ( call for papers ) LATEST WORK Space-Optimized Texture Maps - Eurographics 2002 (Best Paper Award) Volume Warping for Adaptive Isosurface extraction - IEEE Vis 2002 More downloads in DOCS Section
ArXiv: cs.CG Computational Geometry
Section of the Computing Research Repository (CoRR), moderated by Joseph O'Rourke.
Computational Geometry authors titles recent submissions Computational Geometry Authors and titles for recent submissions Tue, 1 Nov 2005 Wed, 19 Oct 2005 Tue, 18 Oct 2005 Mon, 17 Oct 2005 Wed, 12 Oct 2005 Year month listings: 2004 ( Jan , Feb , Mar , Apr , May , Jun , Jul , Aug , Sep , Oct , Nov , Dec ), 2005 ( Jan , Feb , Mar , Apr , May , Jun , Jul , Aug , Sep , Oct , Nov ) Tue, 1 Nov 2005 cs.CG 0510090 [ abs , ps , pdf , other ] : Title: A simple effective method for curvatures estimation on triangular meshes Authors: Jyh-Yang Wu , Sheng-Gwo Chen , Mei-Hsiu Chi Subj-class: Computational Geometry cs.CG 0510088 [ abs , ps , pdf , other ] : Title: Lower bounds on Locality Sensitive Hashing Authors: Rajeev Motwani , Assaf Naor , Rina Panigrahy Subj-class: Computational Geometry Wed, 19 Oct 2005 cs.CG 0510053 [ abs , src ] : Title: A pair of trees without a simultaneous geometric embedding in the plane Authors: Martin Kutz Comments: 9 pages, 9 figures Subj-class: Computational Geometry Tue, 18 Oct 2005 cs.DC 0510048 [ abs , ps , pdf , other ] : Title: Deterministic boundary recognition and topology extraction for large sensor networks Authors: Alexander Kroeller , Sandor P. Fekete , Dennis Pfisterer , Stefan Fischer Comments: 10 pages, 9 figures, Latex, to appear in Symposium on Discrete Algorithms (SODA 2006) Subj-class: Distributed, Parallel, and Cluster Computing; Computational Geometry ACM-class: C.2.1; F.2.2 Mon, 17 Oct 2005 math.CO 0510263 [ abs , ps , pdf , other ] : Title: Cubic Partial Cubes from Simplicial Arrangements Authors: David Eppstein Comments: 11 pages, 10 figures Subj-class: Combinatorics; Metric Geometry; Computational Geometry MSC-class: 05C12 (Primary) 05C78, 52C30 (Secondary) Wed, 12 Oct 2005 cs.CG 0510024 [ abs , ps , pdf , other ] : Title: Delta-confluent Drawings Authors: David Eppstein , Michael T. Goodrich , Jeremy Yu Meng Comments: 14 pages, 8 figures. A preliminary version of this work was presented at the 13th Int. Symp. Graph Drawing, Limerick, Ireland, September 2005 Subj-class: Computational Geometry ACM-class: F.2.2 Links to: arXiv , cs , find , abs , new , 0511 , ?
Computational Geometry Pages
Jeff Erickson's comprehensive directory of computational geometry resources, including bibliographies, journals, software, and related hubs.
Computational Geometry Pages Computational Geometry Pages Welcome to the Computational Geometry Pages, a (hopefully) comprehensive directory of computational geometry resources both on and off the Internet. If there is something you'd like to see here, please send me email. Contributions and suggestions from the community are always welcome! Other essential computational geometry sites include Nina Amenta 's Directory of Computational Geometry Software , Herv Brnnimann 's CG Tribune (a newsletter with events and announcements), David Eppstein 's Geometry in Action (describing applications of computational geometry in the Real World), and the Los Alamos eprint archive 's collection of computational geometry papers [ new , recent , current , search ] moderated by Joe O'Rourke . There are also several excellent Web pages devoted to theoretical computer science in general. See especially Suresh Venkatasubramanian 's Theoretical Computer Science on the Web and the ACM SIGACT home page . What's new? (07 Jan 1999) General Resources The Computational Geometry Impact Task Force Report Discussion Related resources Web Directories Really important stuff Computational geometers Other computational geometry pages Miscellaneous Discussion Forums Mailing lists Usenet newsgroups Newsletters Related Subjects Geometric application areas Geometry and discrete mathematics Theoretical computer science Computer science in general Research and Teaching Research Groups Course Materials Job Announcements Events Past Events Computational geometry Algorithms Geometric application areas Upcoming Events Upcoming events Ongoing events Upcoming Deadlines Other Calendars Literature Bibliographies The Geometry Literature Database Other bibliographies Comprehensive Specific Technical report and preprint services Journals Primary computational geometry journals Related journals Algorithms and complexity Graphics and other applications Mathematics Journal publishers Other journal lists Special Journal Issues Upcoming deadlines Published To appear Books Textbooks and primary references Surveys and collections Specialized and related topics Publishers For conference proceedings, see Past Events Software Software Libraries Nina Amenta's Directory of Computational Geometry Software Integrated libraries Other collections Code Robust lowlevel primitives Combinatorics and discrete math Geometric optimization Convex hulls and convex polytopes Voronoi diagrams and Delaunay triangulations Operations on polygons Mesh generation and manipulation Geometric modeling Visibility computation Visualization tools Other Interactive Software Delaunay triangulations and Voronoi diagrams Other There's no better way to appreciate McLuhan's adage about medium and message, I've discovered, than manually encoding fifty thousand HTML links. Typing "a-href-equals" that many times has a remarkable effect on the mind of the typist: almost Zen, if we're generous; mindnumbingly boring, if we're honest. - Jack Lynch , "Hideous Progeny, Version 0.4 Beta" (describing a much larger project than this one!) Pretty soon, anything that's not on the Web won't exist. - Jim Blinn , SIGGRAPH 1998 keynote address There's a pizza place near where I live that sells only slices. In the back you can see a guy tossing a triangle in the air. - Stephen Wright - "Zippy the Pinhead" (7 9 99) by Bill Griffith Computational Geometry Pages by Jeff Erickson Last update: 07 Jan 1999 Your feedback is always welcome. General: Taskforce Web Forums Related Research: Groups Courses Jobs Events: Past Upcoming Deadlines Calendars Literature: Biblios Journals Issues Books Software: Libraries Code Interactive
Application Challenges to Computational Geometry - Summary by Jeff Erickson
Computational Geometry Impact Task Force Report, chaired by Bernard Chazelle, about the relation between computational geometry and various application fields. This page also archives the discussion that it caused (which was intended) and related links.
Computational Geometry Impact Task Force Report The Computational Geometry Impact Task Force Report In April 1996, Bernard Chazelle 's Computational Geometry Impact Task Force published a report entitled "Application Challenges to Computational Geometry" . Anyone interested in computational geometry or geometric computing is strongly encouraged to read it! Hypertext from Seth Teller at MIT DVI (256 Kb), PostScript (467 Kb), and compressed PostScript (187 Kb) from MIT gzipped PostScript (156 Kb) from Duke Princeton technical report , also available from Bernard Chazelle's homepage . (For some reason, this version is much larger than the one from MIT.) Discussion Soon after the report's publication, comments by David Avis and Komei Fukuda sparked a lively discussion on the compgeom-discuss mailing list of some of the issues raised in the impact report. The same issues are still being discussed at computational geometry meetings - at SoCG 1998 , for example, there was a panel discussion on "The Theory Applications Interface" - but activity on the mailing lists has died off. Archives of the discussion at Bell Labs Archives of the discussion by Bernd Grtner at Freie Universitt Berlin [Many of the hypertext links in the following messages no longer work. -Jeff] Comments on the report by David Avis and Komei Fukuda (May 9, 1996) A response by Nina Amenta (May 10, 1996) Another response by Ken Clarkson (May 14, 1996) Comments on the computer vision section by Azriel Rosenfeld (May 20, 1996) Comments on the report by Paul Heckbert (June 4, 1996) A response by Trevor Coulson (June 5, 1996) Another response by Ernst Mcke (June 17, 1996) Comments on the report by Wm. Randolph Franklin (July 13, 1996) Related Events and Resources The impact report lists several relevant resources , almost all of which can be found on this web site . The ACM sponsored a Workshop on Strategic Directions in Computing Research at MIT, June 14-15, 1996. Roberto Tomassia led the Computational Geometry Working Group . Each of the fifteen members of the group (and one non-member) prepared a position statement outlining key directions for the computational geometry community. The final working group report was published in the Decmeber 1996 issue of ACM Computing Surveys. The first ACM Workshop on Applied Computational Geometry in Philadelphia, May 27-28, 1996. The workshop included a panel discussion on geometric software, led by Mark Overmars, David Dobkin, D. T. Lee, and Kurt Mehlhorn. Although everyone agreed that a geometric software library (such as CGAL or the Center for Geometic Computing's software project) is necessary, there was wide disagreement among the participants about the details. The discussion continued for a short time on the computational geometry mailing list: Position paper by David Dobkin Comments by D. T. Lee (May 30, 1996) Further suggestions by Kumar Ramaiyer (May 31, 1996) During the 1996 SoCG WACG business meeting , Mark Overmars proposed a change in the format of the ACM Symposium on Computational Geometry, which the community enthusastically endorsed (after the usual interminable discussion). Since 1997, SoCG has had both a theoretical track and an applied track, with two separate program committees but a single proceeedings. David Eppstein 's "Geometry in Action" is devoted to applications and potential applications of computational geometry, and includes pointers to over 100 individual projects and applications in areas such as astronomy, geographic information systems, CAD CAM, data mining, graph drawing, graphics, medical imaging, metrology, molecular modeling, robotics, signal processing, textile layout, typography, video games, vision, VLSI, and windowing systems. Slides from David Dobkin 's SCG WACG plenary talk "Computational Geometry: Where did it Come From, What Is It, What is it Good For?" (629Kb PostScript) Joe O'Rourke 's "Computational Geometry Column 29" , published in the 100th issue of SIGACT News , reviews past accomplishments and briefly discusses the future of the field. Joe's column has also been incorporated into the SDCR working group report . Around the same time as Bernard's group was working on the taskforce report, the broader theoretical computer science community was considering similar issues. The NSF report "Emerging Opportunities for Theoretical Computer Science" by Aho, Johnson, Karp , Kosajaru, McGeogh, Papadimitriou, and Pevzner outlines the previous successes of theoretical computer science research, and suggests a few new directions promoting stronger ties to application areas. There are important differences from the earlier version "Theory of Computation: Goals and Directions" distributed at STOC 1996.] Many people felt that the "Gang of Seven" report overemphasized application-driven research at the expense of basic theoretical research. For a vehemently opposing reponse, see "Theory of Computing: A Scientific Perspective" by Oded Goldreich and Avi Widgerson. Goldreich also prepared similar statements for the SDRC Theory of Computing working group . Many of the statements of the SDCR Theory of Computing working group try to strike a balance between the two previous extremes. The SIGACT Long Range Planning Committee compiled a short list of "Contributions of Theoretical Computer Science to Practice" . A list of further resources is maintained by Thomas Emden-Weinert . CADCOM , the Committee for the Advancement of Discrete and Combinatorial Mathematics, collects applications of discrete mathematics . You can contribute ! We can find no evidence of the methods proposed in the C[omputational ]G[eometry] literature actually being used, in spite of the fact they have been widely known for many years. - David Avis and Komei Fukuda "Comments on Application Challenges to Computational Geometry" [There was a] joke at one meeting a few years ago that first, people try to do pure theory. If there are no jobs there, they try computational geometry. Finally, only if necessary to avoid starving do they teach applications. However, this isn't a joke. This leads to bad computational geometry, and to worse applications. - Wm. Randolf Franklin "My Response to Application Challenges To Computational Geometry" Computational Geometry Pages by Jeff Erickson Last update: 11 Aug 1998 Your feedback is always welcome. General: Taskforce Web Forums Related Research: Groups Courses Jobs Events: Past Upcoming Deadlines Calendars Literature: Biblios Journals Issues Books Software: Libraries Code Interactive
London Topology and Geometry Seminar
Held on Fridays in term with interests in topology, algebraic, differential and symplectic geometry, dynamical systems and mathematical physics.
The London topology and geometry seminar The London topology and geometry seminar The seminar is held jointly by Imperial College , King's College , and Queen Mary , University of London, with visitors from Cambridge , Oxford , Warwick and other universities. It is designed for anyone visiting London on a Friday afternoon. To be put on the email list to be notified about seminars, email "subscribe" to geometry-request@ic.ac.uk . Similarly to unsubscribe. The organisers are: Martin Bridson , Shaun Bullett , Simon Donaldson, Bill Harvey , and Mark Haskins . Some older organisers of the London Geometry seminar: Du Val , Roth , and Semple . Timetable 2005 Friday 7th October. Room 140, Huxley Building, Imperial College . 1.30 Michael Anderson (Stony Brook): Dehn surgery construction of Einstein metrics in higher dimensions. Abstract Friday 14th October. Room 130, Huxley Building, Imperial College . 4.30 Marc Lackenby (Oxford): Counting covering spaces and subgroups in dimension three. Abstract Friday 21st October. No seminar - K-theory day at Oxford Friday 28th October. Room 436, King's College . 4.15 Samuel Lelievre (Warwick): Square-tiled surfaces in genus two. Abstract Tea before talk at 4pm in room 429. Friday 4th November . Room 140, Huxley Building, Imperial College . 1.30 Jason Lotay (Oxford): Deformations of coassociative submanifolds Abstract Friday 11th November. Room 140, Huxley Building, Imperial College . 1.30 Tamas Hausel (Oxford and Texas Austin): Cohomology of hyperkahler manifolds via arithmetic harmonic analysis. Abstract Friday 18th November . Room 140, Huxley Building, Imperial College . 1.30 Alexander Grigoryan (Imperial): Heat kernels of Schrodinger operators. Abstract LMS Annual General Meeting, 3-6pm at UCL has talks by Totaro and Kirwan Friday 25th November . Room 140, Huxley Building, Imperial College . 1.30 Dan Pollack (Washington): Constructing solutions to the Einstein constraint equations. Abstract Friday 2nd December. Room 140, Huxley Building, Imperial College . 1.30 Panagiota Daskalopoulos (Columbia): Type II collapsing of maximal Solutions to the Ricci flow in R2 Abstract Friday 9th December. Room 140, Huxley Building, Imperial College . 1.30 Frank Pacard (Paris XII): Blowing up Kahler manifolds of constant scalar curvature. Abstract Thursday 15th December. Room TBA, Huxley Building, Imperial College (joint seminar with Analysis). Time TBA Indira Chatterji (Ohio State): Property RD on connected Lie groups. Abstract 2006 Friday 20th January. Room 140, Huxley Building, Imperial College . 1.30 Peter Ozsvath (Columbia): TBA. Friday 27th January. Room 140, Huxley Building, Imperial College . 1.30 Barbara Fantechi (SISSA Trieste): TBA. Friday 24th February. Room 140, Huxley Building, Imperial College . 1.30 Tom Graber (Caltech): TBA. Friday 16th June. Room 140, Huxley Building, Imperial College . 1.30 Peter Topping (Warwick): TBA. Previous talks: 2005 , 2004 , 2003 , 2002 , 2001 , 2000 , and earlier . Maps London Underground map. Getting to the LMS building (De Morgan House, 57-58 Russell Square). Ring the bell and say the password that is included in the email announcing the talk. This is for LMS security reasons. Directions on joining the email list are given above. Getting to Imperial College . Do not get the tunnel to Exhibition Road; the maths department is in the Huxley Building, 180 Queen's Gate, opposite Queen's Gate Terrace. Gloucester Road is the closest tube station. Getting to King's College . Tea in room 429, seminar in 436. Getting to QMW . From Stepney Green tube go left along Mile End Road for about 200 metres. From Mile End tube go west along Mile End Road for about 300 metres. Seminar links Institute for Mathematical Sciences geometry and string theory seminars . The London Junior Geometry seminar . The London Mathematical Society . The British topology homepage . The European differential geometry homepage . The European algebraic geometry homepage and the Warwick UK algebraic geometry node. If you are interested in algebraic geometry please subscribe to EAGER-gen and add your data to the European contacts list. Relevant seminars at Imperial , King's , QMW , Cambridge , Oxford and Warwick . Imperial String theory seminar. The COW algebraic geometry seminar , and its Calf . To subscribe to the COW and Calf mailing list, go here .
Noncommutative Geometry
Research session at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK; 24 July -- 22 December 2006.
INI Programme NCG - Institute Home Page Programmes Web-Seminars Programme Home Seminars Workshops Participants Long Stay Short Stay Additional Links Contacts Mailing List Background Isaac Newton Institute for Mathematical Sciences Noncommutative Geometry 24 July - 22 December 2006 Organisers: Professor A Connes (IHES), Professor S Majid, (Queen Mary), Professor A Schwarz (UC Davis) Programme Theme Noncommutative geometry aims to carry over geometrical concepts to a a new class of spaces whose algebras of functions are no longer commutative. The central idea goes back to quantum mechanics, where classical observables such as position and momenta no longer commute. In recent years it has become appreciated that such noncommutative spaces retain a rich topology and geometry expressed first of all in K-theory and K-homology, as well as in finer aspects of the theory. The subject has also been approached from a more algebraic side with the advent of quantum groups and their quantum homogeneous spaces. The subject in its modern form has also been connected with developments in several different fields of both pure mathematics and mathematical physics. In mathematics these include fruitful interactions with analysis, number theory, category theory and representation theory. In mathematical physics, developments include the quantum Hall effect, applications to the standard model in particle physics and to renormalization in quantum field theory, models of spacetimes with noncommuting coordinates. Noncommutative geometry also appears naturally in string M-theory. The programme will be devoted to bringing together these different streams and instances of noncommutative geometry, as well as identifying new emerging directions. Three main themes of the programme will be reflected in workshops in July, September and December of 2006, covering noncommutative geometry and cyclic cohomology, noncommutative geometry and fundamental physics, and new directions in noncommutative geometry respectively.
Pacific Northwest Geometry Seminar
The Pacific Northwest Geometry Seminar (PNGS) is a regional meeting for geometers of all kinds. It is held every fall and spring, rotating among the participating institutions: University of British Columbia, Oregon State University, University of Oregon, Portland State University, University of Utah, University of Washington.
The Pacific Northwest Geometry Seminar The Pacific Northwest Geometry Seminar Basic information The next PNGS meeting: Winter 2006, Stanford University (date to be announced) Future PNGS meetings Past PNGS meetings and speakers Travel grants for participants History of the PNGS Organizers and participating institutions The Pacific Northwest Geometry Seminar (PNGS) is a regional meeting for geometers of all kinds. It is held every fall and spring, and every other winter,rotating among the following participating institutions: Oregon State University Portland State University Stanford University University of British Columbia University of Oregon University of Utah University of Washington The meetings are supported by the National Science Foundation ( NSF ), the Pacific Institute for the Mathematical Sciences ( PIMS ), and the host institutions . PNGS meetings are publicized primarily via the Geometry Conference Mailing List, an electronic mailing list used only to announce geometry or related conferences. To subscribe to the list, to unsubscribe, or to read archived messages, go to http: listserv.utk.edu archives geometry.html . Coordinator: Jim Isenberg 10C Deady and 449 Willamette Mathematics Department University of Oregon Eugene, OR 97403 Email: jim@newton.uoregon.edu Phone: (541) 346-4725 If you have comments or questions about this web page, contact the webmaster, Jack Lee lee@math.washington.edu
NonEuclid - Hyperbolic Geometry Article + Software Applet
NonEuclid is a software simulation offering straightedge and compass constructions in hyperbolic geometry.
NonEuclid - Hyperbolic Geometry Article Applet NonEuclid is Java Software for Interactively Creating Ruler and Compass Constructions in both the Poincar Disk and the Upper Half-Plane Models of Hyperbolic Geometry for use in High School and Undergraduate Education. Hyperbolic Geometry is a geometry of Einstein's General Theory of Relativity and Curved Hyperspace. Copyright: Joel Castellanos, 1994-2005 Authors: Joel Castellanos - Graduate Student, Dept. of Computer Science , University of New Mexico Joe Dan Austin - Associate Professor, Dept. of Education, Rice University Ervan Darnell - Graduate Student, Dept. of Computer Science, Rice University Italian Translation by Andrea Centomo, Scuola Media "F. Maffei", Vicenza Funding for NonEuclid has been provided by: CRPC, Rice University Institute for Advanced Study Park City Mathematics Institute Run NonEuclid Applet (click button below): If you do not see the button above, it means that your browser is not Java 1.3.0 enabled. This may be because: 1) you are running a browser that does not support Java 1.3.0, 2) there is a firewall around your Internet access, or 3) you have Java deactivated in the preferences of your browser. Both Netscape 6.2 and Microsoft Internet Explorer 6.0 include Java 1.3.0. Download NonEuclid to run on computers without an Internet connection: NonEuclid.zip Click on the link above to download a compressed archive of NonEuclid. This archive can be moved to a computer without an Internet connection, and uncompressed using WinZip . Uncompress the archive into a single directory. Then open the file named "NonEuclid.html" with Netscape, Internet Explorer or some other browser. What-to-do: Using NonEuclid - My First Triangle Activities - How to get started Exploring: - Adjacent Angles, Angles, General Triangles, Isosceles Triangles, Equilateral Triangle, Right Triangles, Congruent Triangles, Rectangles Squares, Parallelograms, Rhombus, Polygons, Circles, Tessellations of the Plane. Basic Concepts: What is Non-Euclidean Geometry: - Euclidean Geometry, Spherical Geometry, Hyperbolic Geometry, and others. The Shape of Space: - Curved Space, Flatland, Ourland, and Mercury's Orbit. The Pseudosphere: - A description of the space of which NonEuclid is a model. Parallel Lines: - In Hyperbolic Geometry, a pair of intersecting lines can both be parallel to a third line. Axioms and Theorems: - Euclid's Postulates, Hyperbolic Parallel Postulate, SAS Postulate, Hyperbolic Geometry Proofs. Area: - Exaimation of A=bh and A=s in Hyperbolic Geometry, Properties Necessary for an Area Function, Altitudes of a Hyperbolic Triangle, Defect of a Triangle, Defect of a Polygon, and an Upper Bound to Area. X-Y Coordinate System: - A description of how an x-y coordinate system can be set up in Hyperbolic Geometry. Disk and Upper Half-Plane Models: - An informal development of these two models of Hyperbolic Geometry. For The Teacher: Why is it Important for Students to Study Hyperbolic Geometry? Conceptual Mechanics of Expression in Non-Euclidean Fields by Artist Mathematician, Clifford Singer. Palm OS Application for Exploring Non-Euclidean Geometry. The package includes two files: MathLib.prc and HypGeom.prc . MathLib is a library that contains mathematical functions missing on the standard palm libraries. HypGeom is the application. This package was written by Felipe Grajales, Faculty, Universidad de los Andes, Colombia. References Further Reading. For more information, questions, bug reports, or comments send e-mail to Joel Castellanos joel@cs.unm.edu Copyright: Joel Castellanos, 1994-2005
References for Non-Euclidean Geometry
A bibliographic reference list of books and articles on non-Euclidean geometries, part of the MacTutor History of Mathematics archive.
Non-Euclidean geometry references References for: Non-Euclidean geometry Version for printing R Bonola, Non-Euclidean Geometry : A Critical and Historical Study of its Development (New York, 1955). T R Chandrasekhar, Non-Euclidean geometry from early times to Beltrami, Indian J. Hist. Sci. 24 (4) (1989), 249-256. N Daniels,Thomas Reid's discovery of a non-Euclidean geometry, Philos. Sci. 39 (1972), 219-234. F J Duarte, On the non-Euclidean geometries : Historical and bibliographical notes (Spanish), Revista Acad. Colombiana Ci. Exact. Fis. Nat. 7 (1946), 63-81. H Freudenthal, Nichteuklidische Geometrie im Altertum?, Archive for History of Exact Sciences 43 (3) (1991), 189-197. J J Gray, Euclidean and non-Euclidean geometry, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 877-886. J J Gray, Ideas of Space : Euclidean, non-Euclidean and Relativistic (Oxford, 1989). J J Gray, Non-Euclidean geometry-a re-interpretation, Historia Mathematica 6 (3) (1979), 236-258. J J Gray, The discovery of non-Euclidean geometry, in Studies in the history of mathematics (Washington, DC, 1987), 37-60. T Hawkins, Non-Euclidean geometry and Weierstrassian mathematics : the background to Killing's work on Lie algebras, Historia Mathematica 7 (3) (1980), 289-342. C Houzel, The birth of non-Euclidean geometry, in 1830-1930 : a century of geometry (Berlin, 1992), 3-21. V F Kagan, The construction of non-Euclidean geometry by Lobachevskii, Gauss and Bolyai (Russian), Akad. Nauk SSSR. Trudy Inst. Istorii Estestvoznaniya 2 (1948), 323-389. H Karzel, Development of non-Euclidean geometries since Gauss, Proceedings of the 2nd Gauss Symposium (Berlin, 1995). B Mayorga, Lobachevskii and non-Euclidean geometry (Spanish), Lect. Mat. 15 (1) (1994), 29-43. T Pati, The development of non-Euclidean geometry during the last 150 years, Bull. Allahabad Univ. Math. Assoc. 15 (1951), 1-8. B A Rosenfeld, A history of non-euclidean geometry : evolution of the concept of a geometric space (New York, 1987). B A Rozenfel'd, History of non-Euclidean geometry : Development of the concept of a geometric space (Russian) (Moscow, 1976). D M Y Sommerville, Bibliography of non-euclidean geometry (New York, 1970). B Sznssy, Remarks on Gauss's work on non-Euclidean geometry (Hungarian), Mat. Lapok 28 (1-3) (1980), 133-140. R Taton, Lobatchevski et la diffusion des gometries non-euclidiennes, The Spanish scientist before the history of science (Madrid, 1980), 39-46. I Toth, From the pre-history of non-euclidean geometry (Hungarian), Mat. Lapok 16 (1965), 300-315. R J Trudeau, The non-Euclidean revolution (Boston, Mass., 1987). A Vucinich, Nikolai Ivanovich Lobachevskii : the man behind the first non-Euclidean geometry, Isis 53 (1962), 465-481. Mainindex HistoryTopics Index BiographiesIndex Famouscurvesindex Mathematiciansoftheday Anniversariesfortheyear BirthplaceMaps Timelines SearchForm Societies,honours, etc JOC EFR February 1996 The URL of this page is: http: www-history.mcs.st-andrews.ac.uk HistTopics References Non-Euclidean_geometry.html
Minkowskian geometry and quaternion algebras
An exploration of the geometry of quaternion algebra, including a commutative variant which exhibits the properties of Minkowskian geometry.
Minkowski, Quaternions, Complex Numbers Minkowski, Quaternions, Complex Numbers Welcome to a new way of looking at the geometry of space-time. Yes, non-Euclidean 4 dimensional geometry can hurt your brain - but I've tried to pitch this at 1st-year undergrad level and explain it all step-by-step (for my benefit as much as yours, to be honest). View the maths : Commutative Quaternions and Minkowski Mail the author (NEW address): tim01.shelton-jones@virgin.net more to it than meets the i geovisit();
Smarandache Geometries
Having at least one smarandachely denied axiom: validated and invalidated; or invalidated in at least two distinct ways. Discussion groups and meetings.
Smarandache Notions Journal Linguistics Mathematics Philosophy Physics Psychology Sociology Digital Library of Science Arts Letters Geometries An axiom is said smarandachely denied if in the same space the axiom behaves differently (i.e., validated and invalided; or only invalidated but in at least two distinct ways). A Smarandache Geometry is a geometry which has at least one smarandachely denied axiom (1969). Thus, as a particular case, Euclidean, Lobachevsky-Bolyai-Gauss, and Riemannian geometries may be united altogether, in the same space, by some Smarandache geometries. These last geometries can be partially Euclidean and partially Non-Euclidean. It seems that Smarandache Geometries are connected with the Theory of Relativity (because they include the Riemannian geometry in a subspace) and with the Parallel Universes. Paper abstracts can be submitted online to the First International Conference on Smarandache Geometries , that will be held between 3-5 May, 2003, at the Griffith University, Gold Coast Campus, Queensland, Australia, organized by Dr. M. Khoshnevisan. An Introduction to the Smarandache Geometries , paper by M. Antholy, was presented to the New Zealand Mathematics Colloquium, at Palmerston North Campus, Massey University, 3-6 December 2001. You're welcome to join The Smarandache Geometries group . Smarandache Geometries ( 1 , 2 , 3 , 4 ) Books: Smarandache Manifolds, by Howard Iseri Automorphism Groups of Maps, Surfaces and Smarandache Geometries (partially post-doctoral research for the Chinese Academy of Sciences, Beijing), by Linfan Mao new Articles: A Classification of s-Lines in a Closed s-Manifold , by Howard Iseri Partially Paradoxist Smarandache Geometries, by Howard Iseri Engineering A Visual Field, by Clifford Singer An Economics Model for the Smarandache Anti-Geometry, by Roberto Torretti Download free e-books from our Digital Library of Science WebCounter
Non-Euclidean Geometry - Mathematics and the Liberal Arts
A resource for student research projects and for teachers interested in using the history of mathematics in their courses.
Non-Euclidean Geometry - Mathematics and the Liberal Arts
Spherical Trigonometry, Arc Distance Formula
Finding the shortest distance between two points on the earth given latitude and longitude. Download ARC_CALC_3, Microsoft Excel version, A Spherical Triangle Calculator by James Q. Jacobs.
Arc Distance Formula by James Q. Jacobs ARC_CALC_3 Spherical Triangle Calculator 2000 by James Q. Jacobs. Spherical Trigonometry Arc Distance Formula Finding the shortest distance between two points on the earth given latitude and longitude. Download ARC_CALC_3 . This small program will do the calculations below in an Excel 4.0 spreadsheet. You only need to input the coordinates. The program supports input of three sites and calculates the three arc distances, the area of the spherical triangle and the bearings between sites. Most spreadsheet programs should be able to import this file format. The graphic below illustrates the spreadsheet, before corrections on July 24, 2002. Previous downloads have "A to C" and "B to C" labels swapped for arcs and bearings. Let me know if there are other bugs in the applet. Epoch_2000 Temporal Epoch Calculator is a similar Excel spreadsheet. It calculates the temporal changes in astronomic constants, obliquity of the ecliptic and illumination angles at any specified latitude. You just enter the latitude and the date. Sherical Trigonometry Arc Distance Formulas Note: a and b are distinct from a (alpha) and b (beta). 1. Find distances a and b in degrees from the pole. 2. Find angle P by arithmetic comparison of longitudes. (If angle P is greater than 180 degrees subtract angle P from 360 degrees.) Subtract result from 180 degrees to find angle y. 3. Solve for 1 2 ( a - b ) and 1 2 ( a + b ) as follows: tan 1 2 ( a - b ) = - { [ sin 1 2 ( a - b ) ] [ sin 1 2 ( a + b ) ] } tan 1 2 y tan 1 2 ( a + b ) = - { [ cos 1 2 ( a - b ) ] [ cos 1 2 ( a + b ) ] } tan 1 2 y 4. Find c as follows: tan 1 2 c = { [ sin 1 2 ( a + b ) ] x [ tan 1 2 ( a - b ) ] } sin 1 2 ( a - b ) 5. Find angles A and B as follows: A = 180 - [ ( 1 2 a + b ) + ( 1 2 a - b ) ] B = 180 - [ ( 1 2 a + b ) - ( 1 2 a - b ) ] RETURN TO ARCHAEOGEODESY PAGE I | GEODESY WORLD WIDE WEB HUBS BY THE AUTHOR: Home | Anthropology | Archaeoastronomy | Photo Stock | Web Design | Art 1997 by James Q. Jacobs. All rights reserved. Your comments, etc. are appreciated: Contact . Published June 30, 2000. Cite as http: www.jqjacobs.net astro arc_form.html
Non-Euclidean Geometry with LOGO
Review of a new version of LOGO developed at Cardiff.
Non-Euclidean Geometry with LOGO Non-Euclidean Geometry with LOGO Helen Sims-Coomber and Ralph Martin, Department of Computing Mathematics, University of Wales, College of Cardiff. This article describes a version of LOGO currently under development at Cardiff that uses non-Euclidean geometry. The ultimate aim is that a final version could be given to mathematics students to help them visualise non-Euclidean geometry. The programming language LOGO with its Turtle Graphics facilities is well known in educational circles. The turtle is a small triangular pointer that appears on the display screen. Simple commands are used to move it (FORWARD or BACK) and rotate it (LEFT or RIGHT); it leaves a trail behind it as it moves around the screen. Using the turtle to draw in this way provides an easy introduction to computing for young children, but LOGO is equally suitable for older students. Many sophisticated areas of mathematics (including topology, relativity and differential geometry) can be explored through the use of turtle graphics (see [1]). The system under development at Cardiff is specifically designed for exploring non-Euclidean geometry. Euclidean geometry is the kind taught in schools. Most students will be familiar with the properties of Euclidean parallel lines; given a straight line, L, and a point, P, not on the line, we can construct exactly one line through P parallel to L. The distinguishing feature of non-Euclidean geometry is the behaviour of parallel lines. There are two main types of non-Euclidean geometry: hyperbolic geometry, in which more than one line parallel to L can be constructed, and elliptic geometry, in which parallel lines do not exist at all. Non-Euclidean geometry is difficult to visualise, and this is what makes the LOGO approach valuable. Both the hyperbolic and elliptic universes may be modelled by unit discs, with the turtle moving around inside, obeying non-Euclidean laws. The results are very striking and unfamiliar. In hyperbolic geometry, lengths (in the Euclidean sense) get smaller as the turtle moves outwards from the centre of the disc. Hence, a step of "unit length" appears different to us, depending upon the turtle's position relative to the centre of the disc. In any figure drawn with the hyperbolic turtle, boxes of the same size will appear to shrink as the turtle moves outwards. The straight lines of hyperbolic geometry are called h-lines, to avoid confusion with Euclidean straight lines. An h-line appears in the model as an arc of a circle that cuts the boundary of the unit disc at right angles. Distances tend towards zero as the turtle travels closer and closer to the edge of the disc, which it can never actually cross. From the turtle's point of view the boundary of the disc is at "infinity". A similar model exists for elliptic geometry. An elliptic "straight" line is called an e-line. Again, it is modelled by an arc of a circle, this time it cuts the unit disc diametrically (i.e. at the ends of a diameter). By convention, a pair of diametrically opposite points are considered to be one and the same point. This means that when the turtle travels to the boundary of the disc, it must "wrap around". One consequence of this property is that elliptic lines are closed. This means that if the turtle keeps on travelling along the same line without turning, it will eventually come back to its starting point after a finite distance. Further details of these models and more information about non- Euclidean geometry in general may be found in [2], [3] and [4]. Some of the standard LOGO commands are inappropriate for a non- Euclidean environment. For example, the commands FENCE, WRAP and WINDOW control the turtle's behaviour when it reaches the screen boundaries. They are not needed with non-Euclidean LOGO because the hyperbolic turtle always remains within its disc, and the elliptic turtle wraps automatically. Cartesian (x,y) co-ordinates are ambiguous in the disc models due to the "curved" nature of "straight lines." The LOGO commands used to manipulate Cartesian co-ordinates, namely SETX, SETY, XCOR and YCOR have not been implemented. Instead of Cartesian co-ordinates, we may define several non- Euclidean co-ordinate systems (see [4] for full details). The LOGO commands that change the turtle's position directly, such as POS and SETPOS, have to be aware of the co-ordinate system currently in use. We have implemented some new commands: COORDS and SETCOORDS are used for switching between the non-Euclidean co-ordinate systems; GEOM and SETGEOM are used for switching between hyperbolic and elliptic geometry. They work like the Euclidean LOGO commands PC and SETPC used to change pen colours. We mentioned earlier that the hyperbolic turtle considers the boundary of its disc as "infinity". Hence the commands FORWARD INFINITY and BACK INFINITY are meaningful in byperbolic geometry. They move the turtle forward or backwards along its current path to the boundary of its disc. Some interesting properties of hyperbolic and elliptic geometries can be illustrated using non-Euclidean LOGO. For example, many people know that the angles in a Euclidean triangle add up to 180 degrees , and that the angles in a Euclidean quadrilateral add up to 360 degrees. This not the case for non-Euclidean figures. In a triangle, the sum of the angles is 180 degrees or 180 degrees for hyperbolic or elliptic geometry respectively. It is interesting to compare a hyperbolic triangle with an elliptic triangle. The effect of drawing a series of boxes in hyperbolic geometry has already been described as appearing to shrink as the turtle moves closer to the edge of the disc. In elliptic geometry the opposite occurs, and wrap-around takes place. Many more unusual figures and patterns may be explored using non-Euclidean LOGO. Being an interpreted language, it is simple to experiment with and yields immediate results. In conclusion, non-Euclidean LOGO is an interesting tool for exploring hyperbolic and elliptic geometry. It should be exphasised, though, that the present system is at the research stage rather than a finished product to be given to end-users. We have implemented the LOGO turtle graphics commands and the facilities for defining procedures, as they are essential for creating pictures. LOGO is also notable for its list- processing facilities, but these have not been implemented as they would work in the same way no matter what geometry was in use - the emphasis of the project is on geometry and graphics rather than on general programming. Other than this, however, our LOGO implementation is complete. The next stage of the research is to develop a version of LOGO in which the user may enter the definition of a parametric surface (in the form x=x(u,v), y=y(u,v), z=z(u,v) ) and then allow the turtle to move on the given surface. This will aid students in understanding the differential geometry of curved surfaces. REFERENCES 1. H Abelson and A diSessa, Turtle Geometry, MIT Press, (1980) 2. H S M Coxeter, Non-Euclidean Geometry, 2nd ed, University of Toronto Press, (1947) 3. J Gray, Ideas of Space, 2nd ed, Oxford University Press, (1989) 4. M J Greenberg, Euclidean non-Euclidean Geometries: their development and history, 2nd ed, W H Freeman, (1980) This document was prepared by Pam Bishop. It was first published in MathsStats in November 1991 Please mail any comments to ctimath@bham.ac.uk. Copyright
Euclidean and Non-Euclidean Geometry with The Geometer's Sketchpad
Conference talk by Scott Steketee with downloadable sketches.
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Seminar on the History of Hyperbolic Geometry
Seminar notes by Greg Schreiber.
Non-Euclidean Geometry Seminar Seminar on the History of Hyperbolic Geometry Greg Schreiber In this course we traced the development of hyperbolic (non-Euclidean) geometry from ancient Greece up to the turn of the century. This was accomplished by focusing chronologically on those mathematicians who made the most significant contributions to the subject. We began with an exposition of Euclidean geometry, first from Euclid's perspective (as given in his Elements) and then from a modern perspective due to Hilbert (in his Foundations of Geometry). Almost all criticisms of Euclid up to the 19th century were centered on his fifth postulate, the so-called Parallel Postulate.The first half of the course dealt with various attempts by ancient, medieval, and (relatively) modern mathematicians to prove this postulate from Euclid's others. Some of the most noteworthy efforts were by the Roman mathematician Proclus, the Islamic mathematicians Omar Khayyam and Nasir al-Din al-Tusi, the Jesuit priest Girolamo Sacchieri, the Englishman John Wallis, and the Frenchmen Lambert and Legendre. Each one gave a flawed proof of the parallel postulate, containing some hidden assumption equivalent to that postulate. In this way properties of hyperbolic geometry were discovered, even though no one believed such a geometry to be possible. The second half of the course covered the discoveries of the 19th century. Gauss, Schweikart, and Taurinus were the first ones to consider the possibility that a non-Euclidean may be self-consistent, but it was Janos Bolyai and Lobachevskii who provided thorough descriptions of hyperbolic geometry, and it is they who are considered the founders of the subject. The analytic nature of these works contrasted sharply with the synthetic arguments of the earlier mathematicians. Riemann changed the subject even more dramatically with his introduction of differential means of describing these geometries. He expanded the class of non-Euclidean geometries to include elliptic geometry (which was then called Riemannian geometry) and also geometries whose properties may vary from point to point (which is now what is meant by Riemannian geometry). This new approach facilitated the discovery of various models of hyperbolic geometry due to Beltrami, Cayley, Poincare, and Klein. These models, the most important of which include the disk models of Beltrami and Poincare and Poincare's half-plane, were then used to prove the logical self-consistency of hyperbolic geometry, thus setting it on equal footing with Euclidean geometry, and to make the connections with complex analysis and algebra that form the points of departure for the modern treatment of the subject. The course concluded with an introduction to Klein's projective geometry, which gave the subject of hyperbolic geometry its name. References: Four general references were used throughout this course: Bonola's Non-Euclidean Geometry, Jeremy Gray's Ideas of Space, Greenberg's Euclidean and Non-Euclidean Geometries, and McCleary's Geometry from a Differential Viewpoint. In addition, original works of these mathematicians were used whenever possible, as well as biographies of them. These books included Euclid's Elements, Hilbert's Foundations of Geometry, Proclus's A Commentary on the First Book of Euclid's Elements, Saccheri's Euclid Vindicated, Bolyai's Science of Absolute Space, Lobachevskii's Geometrical Researches in the Theory of Parallels, and Riemann's "On the Hypotheses Which Lie at the Foundations of Geometry," among others.
Non-Euclidean Geometry
A historical account with links to biographies of some of the people involved.
Non-Euclidean geometry Non-Euclidean geometry Geometry and topology index HistoryTopicsIndex Version for printing In about 300 BC Euclid wrote The Elements, a book which was to become one of the most famous books ever written. Euclid stated five postulates on which he based all his theorems: To draw a straight line from any point to any other. To produce a finite straight line continuously in a straight line. To describe a circle with any centre and distance. That all right angles are equal to each other. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles. It is clear that the fifth postulate is different from the other four. It did not satisfy Euclid and he tried to avoid its use as long as possible - in fact the first 28 propositions of The Elements are proved without using it. Another comment worth making at this point is that Euclid , and many that were to follow him, assumed that straight lines were infinite. Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false 'proof'. Proclus then goes on to give a false proof of his own. However he did give the following postulate which is equivalent to the fifth postulate. Playfair 's Axiom:- Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line. Although known from the time of Proclus , this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid 's fifth postulate by this axiom. Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods of time until the mistake was found. Invariably the mistake was assuming some 'obvious' property which turned out to be equivalent to the fifth postulate. One such 'proof' was given by Wallis in 1663 when he thought he had deduced the fifth postulate, but he had actually shown it to be equivalent to:- To each triangle, there exists a similar triangle of arbitrary magnitude. One of the attempted proofs turned out to be more important than most others. It was produced in 1697 by Girolamo Saccheri . The importance of Saccheri 's work was that he assumed the fifth postulate false and attempted to derive a contradiction. Here is the Saccheri quadrilateral In this figure Saccheri proved that the summit angles at D and C were equal.The proof uses properties of congruent triangles which Euclid proved in Propositions 4 and 8 which are proved before the fifth postulate is used. Saccheri has shown: a) The summit angles are 90 (hypothesis of the obtuse angle). b) The summit angles are 90 (hypothesis of the acute angle). c) The summit angles are = 90 (hypothesis of the right angle). Euclid 's fifth postulate is c). Saccheri proved that the hypothesis of the obtuse angle implied the fifth postulate, so obtaining a contradiction. Saccheri then studied the hypothesis of the acute angle and derived many theorems of non-Euclidean geometry without realising what he was doing. However he eventually 'proved' that the hypothesis of the acute angle led to a contradiction by assuming that there is a 'point at infinity' which lies on a plane. In 1766 Lambert followed a similar line to Saccheri . However he did not fall into the trap that Saccheri fell into and investigated the hypothesis of the acute angle without obtaining a contradiction. Lambert noticed that, in this new geometry, the angle sum of a triangle increased as the area of the triangle decreased. Legendre spent 40 years of his life working on the parallel postulate and the work appears in appendices to various editions of his highly successful geometry book Elments de Gomtrie. Legendre proved that Euclid 's fifth postulate is equivalent to:- The sum of the angles of a triangle is equal to two right angles. Legendre showed, as Saccheri had over 100 years earlier, that the sum of the angles of a triangle cannot be greater than two right angles. This, again like Saccheri , rested on the fact that straight lines were infinite. In trying to show that the angle sum cannot be less than 180 Legendre assumed that through any point in the interior of an angle it is always possible to draw a line which meets both sides of the angle. This turns out to be another equivalent form of the fifth postulate, but Legendre never realised his error himself. Elementary geometry was by this time engulfed in the problems of the parallel postulate. d'Alembert , in 1767, called it the scandal of elementary geometry. The first person to really come to understand the problem of the parallels was Gauss . He began work on the fifth postulate in 1792 while only 15 years old, at first attempting to prove the parallels postulate from the other four. By 1813 he had made little progress and wrote: In the theory of parallels we are even now not further than Euclid. This is a shameful part of mathematics... However by 1817 Gauss had become convinced that the fifth postulate was independent of the other four postulates. He began to work out the consequences of a geometry in which more than one line can be drawn through a given point parallel to a given line. Perhaps most surprisingly of all Gauss never published this work but kept it a secret. At this time thinking was dominated by Kant who had stated that Euclidean geometry is the inevitable necessity of thought and Gauss disliked controversy. Gauss discussed the theory of parallels with his friend, the mathematician Farkas Bolyai who made several false proofs of the parallel postulate. Farkas Bolyai taught his son, Jnos Bolyai , mathematics but, despite advising his son not to waste one hour's time on that problem of the problem of the fifth postulate, Jnos Bolyai did work on the problem. In 1823 Bolyai wrote to his father saying I have discovered things so wonderful that I was astounded ... out of nothing I have created a strange new world. However it took Bolyai a further two years before it was all written down and he published his strange new world as a 24 page appendix to his father's book, although just to confuse future generations the appendix was published before the book itself. Gauss , after reading the 24 pages, described Jnos Bolyai in these words while writing to a friend: I regard this young geometer Bolyai as a genius of the first order . However in some sense Bolyai only assumed that the new geometry was possible. He then followed the consequences in a not too dissimilar fashion from those who had chosen to assume the fifth postulate was false and seek a contradiction. However the real breakthrough was the belief that the new geometry was possible. Gauss , however impressed he sounded in the above quote with Bolyai , rather devastated Bolyai by telling him that he ( Gauss ) had discovered all this earlier but had not published. Although this must undoubtedly be true, it detracts in no way from Bolyai 's incredible breakthrough. Nor is Bolyai 's work diminished because Lobachevsky published a work on non-Euclidean geometry in 1829. Neither Bolyai nor Gauss knew of Lobachevsky 's work, mainly because it was only published in Russian in the Kazan Messenger a local university publication. Lobachevsky 's attempt to reach a wider audience had failed when his paper was rejected by Ostrogradski . In fact Lobachevsky fared no better than Bolyai in gaining public recognition for his momentous work. He published Geometrical investigations on the theory of parallels in 1840 which, in its 61 pages, gives the clearest account of Lobachevsky 's work. The publication of an account in French in Crelle 's Journal in 1837 brought his work on non-Euclidean geometry to a wide audience but the mathematical community was not ready to accept ideas so revolutionary. In Lobachevsky 's 1840 booklet he explains clearly how his non-Euclidean geometry works. All straight lines which in a plane go out from a point can, with reference to a given straight line in the same plane, be divided into two classes - into cutting and non-cutting. The boundary lines of the one and the other class of those lines will be called parallel to the given line. Here is the Lobachevsky's diagram Hence Lobachevsky has replaced the fifth postulate of Euclid by:- Lobachevsky's Parallel Postulate. There exist two lines parallel to a given line through a given point not on the line. Lobachevsky went on to develop many trigonometric identities for triangles which held in this geometry, showing that as the triangle became small the identities tended to the usual trigonometric identities. Riemann , who wrote his doctoral dissertation under Gauss 's supervision, gave an inaugural lecture on 10 June 1854 in which he reformulated the whole concept of geometry which he saw as a space with enough extra structure to be able to measure things like length. This lecture was not published until 1868, two years after Riemann 's death but was to have a profound influence on the development of a wealth of different geometries. Riemann briefly discussed a 'spherical' geometry in which every line through a point P not on a line AB meets the line AB. In this geometry no parallels are possible. It is important to realise that neither Bolyai 's nor Lobachevsky 's description of their new geometry had been proved to be consistent. In fact it was no different from Euclidean geometry in this respect although the many centuries of work with Euclidean geometry was sufficient to convince mathematicians that no contradiction would ever appear within it. The first person to put the Bolyai - Lobachevsky non-Euclidean geometry on the same footing as Euclidean geometry was Eugenio Beltrami (1835-1900). In 1868 he wrote a paper Essay on the interpretation of non-Euclidean geometry which produced a model for 2-dimensional non-Euclidean geometry within 3-dimensional Euclidean geometry. The model was obtained on the surface of revolution of a tractrix about its asymptote. This is sometimes called a pseudo-sphere. You can see the graph of a tractrix and what the top half of a Pseudo-sphere looks like. In fact Beltrami 's model was incomplete but it certainly gave a final decision on the fifth postulate of Euclid since the model provided a setting in which Euclid 's first four postulates held but the fifth did not hold. It reduced the problem of consistency of the axioms of non-Euclidean geometry to that of the consistency of the axioms of Euclidean geometry. Beltrami 's work on a model of Bolyai - Lobachevsky 's non-Euclidean geometry was completed by Klein in 1871. Klein went further than this and gave models of other non-Euclidean geometries such as Riemann 's spherical geometry. Klein 's work was based on a notion of distance defined by Cayley in 1859 when he proposed a generalised definition for distance. Klein showed that there are three basically different types of geometry. In the Bolyai - Lobachevsky type of geometry, straight lines have two infinitely distant points. In the Riemann type of spherical geometry, lines have no (or more precisely two imaginary) infinitely distant points. Euclidean geometry is a limiting case between the two where for each line there are two coincident infinitely distant points. References (23 books articles) Other Web sites: AOL, USA Article by: J J O'Connor and E F Robertson HistoryTopicsIndex Geometry and topology index Mainindex BiographiesIndex Famouscurvesindex BirthplaceMaps Chronology Timelines Mathematiciansoftheday Anniversariesfortheyear SearchForm Societies,honours, etc JOC EFR February 1996 The URL of this page is: http: www-history.mcs.st-andrews.ac.uk HistTopics Non-Euclidean_geometry.html
The Ontology and Cosmology of Non-Euclidean Geometry
A philosophical essay.
The Ontology and Cosmology of Non-Euclidean Geometry The Ontology and Cosmology of Non-Euclidean Geometry Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence. David Hume, An Enquiry Concerning Human Understanding, Section IV, Part I, p. 20 [L.A. Shelby-Bigge, editor, Oxford University Press, 1902, 1972, p. 25] [ note ] 1. Introduction Until recently, Albert Einstein's complaints in his later years about the intelligibility of Quantum Mechanics often led philosophers and physicists to dismiss him as, essentially, an old fool in his dotage. Happily, this kind of thing is now coming to an end as a philosophers and mathematicians of the caliber of Karl Popper and Roger Penrose conspicuously point out the continuing conceptual difficulties of quantum theory [cf. Penrose's searching discussion in The Emperor's New Mind, chapter 6, "Quantum magic and quantum mystery," Oxford 1990]. The Paradox of Schrdinger's Cat is sometimes now even presented, not as a wonderful exciting implication of the theory, but for what it originally was: a reductio ad absurdum argument against the "new" quantum mechanics of Heisenberg and Bohr. Schrdinger shared the misgivings of Einstein and others. A fine statement about all this can be found in Joseph Agassi's foreword to the recent Einstein Versus Bohr, by the dissident physicist Mendel Sachs (Open Court, 1991): When I was a student of physics I was troubled by the difficulties presented and aired by Sachs in this book. Both the physicists and the philosophers of science to whom I confessed my troubles--as clearly as I could--showed me hostility rather than sympathy. I had earlier experienced the same from my religious teachers, so that I was not crushed by the hostility, but I was discouraged from pursuing my scientific interests. This book has returned me to those days and reminded me of the tremendous joys I experienced then, reading Einstein and Schrdinger and meeting Karl Popper and Alfred Land. All four expressed, one way or another, the same sentiment as Sachs: feeling difficulties about current ideas should be encouraged, not discouraged. It is amazing that such things need to be said, and it is particularly revealing that the responses Agassi got to his questions reminded him of the intolerance of religious dogmatism. Nevertheless, there is still rarely a public word spoken about the philosophical intelligibility of Einstein's own theory: the Relativistic theory of gravitation. That theory rests on the use of non-Euclidean geometry. There are still many good questions to ask about non-Euclidean geometry; but in treatment after treatment in both popular expositions and in philosophical discussion, the questions consistently seem pointedly not to get asked. A good example of this may be found in two articles published in Scientific American in 1976. J.J. Callahan's article, "The Curvature of Space in a Finite Universe" in August, makes the argument that Riemann's geometry of a positively curved, finite and unbounded space, which was used by Einstein for his theory, answers the paradox of Kant's Antinomy of Space, avoiding both finite space and infinite space as they had been traditionally understood. This is the philosophically satisfying aspect of Einstein's theory that clearly continues to exercise profound influence on contemporary physicists like Stephen Hawking, as may be seen in his recent Brief History of Time. On the other hand, in March of 1976, Scientific American also published an article by J. Richard Gott III (et al.), "Will the Universe Expand Forever?" This article detailed the evidence then available indicating that the universe was not positively curved, finite and unbounded, as Einstein, and everyone since, has wished. Instead, the universe is more likely to be infinite, either with a Lobachevskian non-Euclidean geometry, or even with a Euclidean(!) geometry after all. Now, my point is not that scientific theory is in flux. It usually is. My problem is that the philosophical implications of the likelihood that observation will continue to reveal an infinite universe (despite "missing mass," "dark matter," etc.) have not been explored. The clear implications of the observational March article for the significance of the philosophical August article have never been considered in any other cosmological article in Scientific American--or anywhere else that I have ever seen, inside or outside of philosophy. It is as though everyone is waiting around in the hope that the "missing mass" turns up. Meanwhile there is virtually a conspiracy of silence. If we just don't think--ostrich-like--about facing an infinite universe again, then it won't happen. This is not intellectually or philosophically honest. But it is of a piece with much of the way non-Euclidean geometry and its related cosmological issues have been dealt with for a long time. The closest we seem to have come to a more open consideration of these matters is when both Stephen Hawking and Karl Popper [Karl Popper, Unended Quest, Open Court, 1990; p.16] point out that Einstein, whether or not he successfully answered Kant's Antinomy of Space, did not answer the Antinomy of Time: despite decades of everyone glorifying in the philosophical revelation of a finite but unbounded universe, they simply didn't notice that the solution proposed for space didn't work with time. It is to Hawking's great philosophical credit that he faces this question squarely. In what follows I will attempt to ask questions about non-Euclidean geometry that I do not often, or ever, see asked. In the section three I will then briefly attempt to suggest how the philosophical implications Einstein's application of geometry in his theory of gravitation may be reconsidered. 2. Curved Space and Non-Euclidean Geometry Euclid's parallel postulate, in its modern reformulation, holds that, on a plane, given a line and a point not on the line, only one line can be drawn through the point parallel to the line. Gerolamo Saccheri (1667-1733) brilliantly attempted to prove this through a reductio ad absurdum argument. There were two ways to contradict the postulate: space could have 1) no parallel lines (straight lines in a plane will always meet if extended far enough), or 2) multiple straight lines through a given point parallel to a given line in the plane. These become non-Euclidean axioms. Saccheri convincingly achieved his reductio for the first possibility with the innocent assumption that straight lines are infinite [cf. Jeremy Gray, Ideas of Space Euclidean, Non-Euclidean, and Relativistic, Oxford, 1989; p. 64]. Later David Hilbert (1862-1953) would point out that the same reductio proof could be achieved by assuming that given three points on a line only one can be between the other two [David Hilbert and S. Cohn-Vossen Geometry and the Imagination (Anschauliche Geometrie--better translated Intuitive Geometry), Chelsea Publishing Company, 1952; p. 240]. For the second possibility, however, Saccheri did not achieve a good proof. And it was using just such an axiom that the first complete non-Euclidean geometries were achieved by Bolyai (1802-1860) and Lobachevskii (1792-1856). If by "flat" we mean a plane of straight lines as understood by Euclid, then true non-Euclidean manifolds (i.e. areas, volumes, spacetimes, etc.), in order to really contradict Euclid, who was talking about straight lines, would have to be flat. They could not be curved. Straight lines would be Euclidean straight, but the properties specified by non-Euclidean axioms would be satisfied. Nevertheless, since Bernhard Riemann (1826-1866), non-Euclidean manifolds are said to be "curved," and only Euclidean space itself is called "flat." Contradiction 1 above produces "positively" curved space ("spherical" or "elliptical" geometry, first described by Riemann himself), and contradiction 2 "negatively" curved space ("hyperbolic" or Lobachevskian geometry). To Euclid, this doubtlessly would seem to prove his point: the parallel postulate is about straight lines, so using curved lines hardly produces an honest non-Euclidean geometry. "Curvature" in this respect, however, is used in an unusual sense. "Intrinsic" curvature is distinguished from "extrinsic" curvature. A space can possess "intrinsic" curvature yet contain lines ("geodesics") that will be straight according to any form of measurement intrinsic to that space. A geodesic is "straight" in relation to its own manifold. Euclidean straightness thus characterizes the geodesic of a three dimensional space with no intrinsic curvature, and it is simply a matter of convention and convenience that we call Euclidean geodesics "straight" and generalized straight lines "geodesics" [Note that my references to "Euclidean" space will always mean three dimensional space as understood by Euclid himself (or Kant). "Flat" spaces of more than three dimensions may be called "Euclidean" because of their lack of curvature; but this is an extension of geometry that would have very much been news to Euclid, and I wish to retain the historical connection between "Euclidean" and Euclid]. What "curvature" would have meant to Euclid is now "extrinsic" curvature: that for a line or a plane or a space to be "curved" it must occupy a space of higher dimension, i.e. that a curved line requires a plane, a curved plane requires a volume, a curved volume requires some fourth dimension, etc. Now "intrinsic" curvature has nothing to do with any higher dimension. But how did this happen? Why did "curvature" come to have this unusual meaning? Why should we confuse ourselves by saying that "intrinsic" straight lines, geodesics, in non-Euclidean spaces have curvature? This happened because non-Euclidean planes can be modeled as extrinsically curved surfaces within Euclidean space. Thus the surface of a sphere is the classic model of a two-dimensional, positively curved Riemannian space; but while great circles are the straight lines (geodesics) according to the intrinsic properties of that surface, we see the surface as itself curved into the third dimension of Euclidean space. A sphere is such a good representation of a non-Euclidean surface, and spherical trigonometry was so well developed at the time, that it now is a little surprising that it was not the basis of the first non-Euclidean geometry developed [cf. Gray ibid. p.171]. However, as noted, such a geometry does contradict other axioms that can easily be posited for geometry. Accepting positively curved spaces means that those axioms must be rejected. Also, and more importantly, these models in Euclidean space are not always successful. The biggest problem is with Lobachevskian space. A saddle shaped surface is a Lobachevskian space at the center of the saddle, but a true Lobachevskian space does not have a center. Other Lobachevskian models distort shapes and sizes. There is no representation of a Lobachevskian surface that shares the virtues of a sphere in having no center, no singularities (i.e. points that do not belong to the space), and in allowing figures to be moved around without distortion in shape or size. Three dimensional non-Euclidean spaces of course cannot be modeled at all using Euclidean space. This raises two questions: 1) what can we spatially visualize? (a question of psychology) And 2) what can exist in reality? (a question of ontology). We cannot visualize any true Lobachevskian spaces or any non-Euclidean spaces at all with more than two dimensions--or any spaces at all with more than three dimensions. Also we can only visualize a positively curved surface if this is embedded in a Euclidean volume with an explicit extrinsic curvature. "Curvature" was thus a natural term for intrinsic properties because there always was extrinsic curvature for any model that could be visualized. Why are there these limits on what we can visualize? Why is our visual imagination confined to three Euclidean dimensions? It is now common to say that computer graphics are breaking through these limitations, but such references are always to projections of non-Euclidean or multi-dimensional spaces onto two dimensional computer screens. Such projections could be done, laboriously, long before computers; but they never produced more, and can produce no more, than flat Euclidean drawings of curves. If such graphics are expected to alter our minds so that we can see things differently, this is no more than a prediction, or a hope, not a fact. And considering that non-Euclidean geometries have been conceived for almost two centuries, the transformation of our imagination seems a bit tardy, however much help computers can now give to it. Mathematicians don't have to worry about these questions of visualization because visualization is not necessary for the analytic formulas that describe the spaces. The formulas gave meaningfulness to non-Euclidean geometry as common sense never could. The Euclidean nature of our imagination led Kant to say that although the denial of the axioms of Euclid could be conceived without contradiction, our intuition is limited by the form of space imposed by our own minds on the world. While it is not uncommon to find claims that the very existence of non-Euclidean geometry refutes Kant's theory, such a view fails to take into account the meaning of the term "synthetic," which is that a synthetic proposition can be denied without contradiction. Leonard Nelson realized that Kant's theory implies a prediction of non-Euclidean geometry, not a denial of it, and that the existence of non-Euclidean geometry vindicates Kant's claim that the axioms of geometry are synthetic [Leonard Nelson, "Philosophy and Axiomatics," Socratic Method and Critical Philosophy, Dover, 1965; p.164]. The intelligibility of non-Euclidean geometry for Kantian theory is neither a psychological nor an ontological question, but simply a logical one--using Hume's criterion of possibility as logically consistent conceivability. Something of the sort is admitted with hesitation by Jeremy Gray: As I read Kant, he does not say non-Euclidean geometry is logically impossible, but that is only because he does not claim that any geometry is logically true; geometry in his view is synthetic, not analytic. And Kant's belief that Euclidean geometry was true, because our intuitions tell us so, seems to me to be either unintelligible or wrong. [Gray, Ibid. p. 85] If we are unable to visualize non-Euclidean geometries without using extrinsically curved lines, however, the intelligibility of Kant's theory is not hard to find. The sense of the truth of Euclidean geometry for Kant is no more or less than the confidence that centuries of geometers had in the parallel postulate, a confidence based on our very real spatial imagination. If Kant's claim is "unintelligible," then Gray has not reflected on why everyone in history until the 19th century believed that the parallel postulate was true. That is the psychological question, not the logical or ontological one. The sense of ancient confidence can be recovered at any time today simply by trying to explain non-Euclidean geometry to undergraduate students who have never heard of it before. We might say that attempts to prove the postulate show that people were uneasy about it; but the universal expectation was that the postulate was really a theorem, and no one cashed in their unease by trying to construct geometry with a denial of it. Saccheri denied it, but only because he was constructing reductio ad absurdum proofs. Non-Euclidean geometry did not change our spatial imagination, it only proved what Kant had already implicitly claimed: the synthetic and axiomatically independent character of the first principles of geometry. It could well be the case that Kant is right and that we will never be able to imagine the appearance of Lobachevskian or multi-dimensional non-Euclidean spaces, or to model them without extrinsic curvature, however well we understand the analytic equations. This is purely a question of psychology and not at all one of logic, mathematics, physics, or ontology. Mathematicians are free to ignore the limitations of our imagination, although they then run the risk of wandering so far from common sense that the frontiers of mathematics will never be intelligible to even well-informed persons of general knowledge. Furthermore, since Kant believed that space was a form imposed by our minds on the world, he did not believe that space actually existed apart from our experience. This leads us to the ontological question: what can exist in reality? Non-Euclidean geometry was no more than a mathematical curiosity until Einstein applied it to physics. Now the whole issue seems much deeper and complex than it did in Kant's day, or Riemann's. If our imagination is necessarily Euclidean, hard-wired into the brain as we might now think by analogy with computers, but Einstein found a way to apply non-Euclidean geometry to the world, then we might think that space does have a reality and a genuine structure in the world however we are able to visually imagine it. In light of the distinction between intrinsic and extrinsic curvature, we must consider all the kinds of ontological axioms that will cover all the possible spaces that Euclidean and non-Euclidean geometries can describe. If the limitations imposed by our imaginations present us with features of real space, we would have to say that intrinsic curvature, despite being analytically independent of extrinsic curvature, can only exist in conjunction with extrinsic curvature and so with an embedding in higher dimensions. This could be called the axiom of ortho-curvature, according to which there would actually be no true non-Euclidean geometry, for non-Euclidean geodesics would necessarily have extrinsic curvature and so would never be the actual straight lines that we need ex hypothese to contradict Euclid. The geometry of the surface of a sphere would thus involve ortho-curvature because its intrinsic straight lines, the great circles, must be simultaneously visualized and understood to be curved lines in three dimensional Euclidean space. On the other hand, it may be that intrinsically curved spaces can exist in reality without extrinsic curvature and so without being embedded in a higher dimension. This could be called the axiom of hetero-curvature, and it would make true non-Euclidean geometry possible, since lines with non-Euclidean relations to each other would be straight in the common meaning of the term understood by Euclid or Kant. A further ontological distinction can be made. Even if the ortho-curvature axiom is true, a functionally non-Euclidean geometry would be possible if a higher dimension that allows for extrinsic curvature exists but is hidden from us. We must consider whether only the three dimensions of space exist or whether there may be additional dimensions which somehow we do not experience but which can produce an intrinsic curvature whose extrinsic properties cannot be visualized or imaginatively inspected by us. Thus we should distinguish between an axiom of closed ortho-curvature, which says that three dimensional space is all there is, and an axiom of open ortho-curvature, which says that higher dimensions can exist. This gives us three possibilities: That, with the axiom of closed ortho-curvature, there are no true non-Euclidean geometries (and no spatial dimensions beyond three), but only pseudo-geometries consisting of curves in Euclidean space; That, with the axiom of open ortho-curvature, there are no true non-Euclidean geometries but we may be faced with a functional non-Euclidean geometry in Euclidean space whose external curvature is concealed from us in dimensions (more than the three familiar spatial dimensions) not available to our inspection--this is an apparent hetero-curvature; And that, with the axiom of hetero-curvature, there are real non-Euclidean geometries whose intrinsic properties do not ontologically presuppose higher dimensions (whether or not there are more than three spatial dimensions). It is necessary to keep in mind that these axioms are answers to questions concerning reality that would be asked in physics or metaphysics and are logically entirely separate from the status of geometry in logic or mathematics or from our psychological powers of visual imagination. The second axiom leaves open the question whether "hidden" dimensions are just hidden from our perception or actually separate from our own dimensional existence. With these ontological alternatives in mind, we can now examine the philosophical implications of Einstein's use of non-Euclidean geometry. 3. Geometry in Einstein's Theory of Relativity Einstein's general theory of relativity proposes that the "force" of gravity actually results from an intrinsic curvature of spacetime, not from Newtonian action-at-a-distance or from a quantum mechanical exchange of virtual particles. If we view Einstein's philosophical project as an answer to Kant's Antinomy of Space--to explain how straight lines in space can be finite but unbounded--the introduction of time reckoned as the fourth dimension suggests that we may separate the intrinsic curvature of spacetime into curvature based on the relationship between space and time: we can think of Einstein's theory as one that satisfies the axiom of open ortho-curvature, with the peculiarity that it is indeed time, rather than a higher dimension of space, that is posited beyond our familiar three spatial dimensions. This is a metaphysically elegant theory, since is gives us the mathematical use of a higher dimension without the need to postulate a real spatial dimension beyond our experience or our existence. Time is a dimension that is present to us only one spatial slice at a time, just as the third dimension is only intersected at one (radial) point by the curved surface of a sphere in our previous model of a positively curved space. Our spherical model for non-Euclidean spacetime, however, is not quite right; for on the analogy, the intrinsic lines in space should be the geodesics and so should appear straight to us. They should appear curved only from the perspective of the higher dimension, as the great circles on the sphere appear curved from our three dimensional perspective. That is not true in terms of astronomical space, where the lines drawn by freefalling bodies in gravitational fields are most evidently curved to our three dimensional imaginations, even while they are understood to be geodesics only in terms of their form in the higher dimension of spacetime. That is exactly the opposite of the case in the model: Freefalling paths ("world lines") are geodesics in spacetime but extrinsically curved lines in space, while in the model great circles are extrinsically curved lines in solid space (corresponding to spacetime) but geodesics in plane space (corresponding to space). Intrinsic curvature, which was introduced by Riemann to explain how straight lines could have the properties associated with curvature without being curved in the ordinary sense, is now used to explain how something which is obviously curved, e.g. the orbit of a planet, is really straight. Something has gotten turned around. If the curvature of spacetime is evident to us in extrinsically curved lines in three dimensional space, then the form of the analogy forces us to posit the "higher" or extrinsic dimension, into which the straight lines are curved, as a spatial one, not the temporal one. If three dimensional space is not extrinsically curved into time according to the axiom of open ortho-curvature, then it must be time that is extrinsically curved into the dimensions of space. In the model, where before the surface of the sphere was analogous to solid space, now the surface must be analogous to two dimensions of space plus time, with the third dimension of space as that into which the geodesics of spacetime are extrinsically curved. Switching the role of time suddenly makes the model very non-intuitive, but it is compelled by the feature of the model that the geodesic is on the surface of the sphere. It does not help the philosophical issue to eject the complications of the axiom of open ortho-curvature and simply take the four dimensions of spacetime as satisfying hetero-curvature; for this loses sight of Kant's Antinomy of Space, which we hope to answer, and of the circumstance that even in Relativity the dimension of time is not exactly the same as the dimensions of space. That is the most intuitively obvious in the "separation" formula: s2 = t2 - ( x2 + y2 + z2) c2 . Here the Pythagorean formula for changes in spatial location, divided by the velocity of light squared, is subtracted from the change in time squared, to give the spacetime "separation" in units of time. Thus time is not treated as simply another spatial dimension. Thus we must consider the differences between space and time, and the axiom of open ortho-curvature alone allows for this. The result of attributing extrinsic curvature to time is also suggested by the peculiarity of using "curved space" alone to explain gravity, as is common in museums and textbooks around the world; for curved space conjures up images of hills and valleys through which moving objects describe curved paths. However, those images presuppose motion, and motion is the very thing to be explained. Gravity does not just direct motion; it causes it. An object passing by the earth is accelerated towards the earth and thereby acquires a velocity along a vector where it previously may have had no velocity at all. An object placed at rest with respect to the earth, with no initial velocity in any direction, will be accelerated with a velocity towards the earth. If there are no "forces" acting on the body, as Einstein says, then the only change that takes place is the body's movement along the temporal axis; and if the body is thereby displaced in space, it must be displaced by its movement along that axis. The temporal axis can displace the object if the axis is itself curved; so the curvature of spacetime in a gravitational field must result from the curvature of time, not of space. The extrinsic dimension of ortho-curvature, into which the straight lines curve, is a dimension of ordinary Euclidean space. This can be intuitively shown, not so much in our non-Euclidean models, but simply in a graph plotting time (t) against one dimension of space (r). An accelerating body will describe a curved line that changes its coordinate in the r axis as its coordinate in the t axis changes. If the acceleration comes from spacetime itself, then the coordinate grid will itself be curved: the t axis lines will curve, displacing themselves against the r axis (spatial location), while the r axis lines will not curve. The curvature of time itself is hidden from us because, indeed, we intersect only one point on the temporal axis. Consequently, how do we know we are being accelerated by gravity? In free fall we are being displaced with space itself, and so we move with our entire frame of reference and would not be able to detect that locally. Indeed, we cannot. It is Einstein's own "equivalence" principle of General Relativity that we cannot tell the difference between free fall in a gravitational field and free floating in the absence of a gravitational field. The motion induced in us by the curvature of time is evident only because we can observe distant objects that are not subject to our local acceleration. When we are not in free fall, e.g. standing on the surface of the earth, we feel weight, just as according to the equivalence principle when we are being accelerated by a force (e.g. a rocket engine) in the absence of a gravitational field. These are indeed equivalent because in each case we are moving relative to space according to our own frame of reference. When we are accelerated by a rocket we say that we move in the stationary reference of external space; but when we are accelerated standing on the surface of the earth, it is space itself that is displaced (by time) relative to us. Either we move through space, or space moves through us. That is the experience of weight. A question remains about the global character of spacetime. Gravitational fields are locally positively curved, but Einstein and his philosophical successors evidently expected that spacetime as a whole would be positively curved, since a finite but unbounded universe is aesthetically more satisfying--and it answers Kant's Antinomy of Space. Now, however, the geometry of cosmological spacetime is usually tied to the dynamical fate of the expanding universe. Open, ever expanding universes, are regarded as having Lobachevskian or even Euclidean geometry and only closed universes, headed for ultimate collapse, positive Riemannian curvature. The observational evidence at the moment is for an open universe, and "inflationary" models even have reasons to prefer a Euclidean over a Lobachevskian geometry. These possibilities, however, introduce considerable trouble; for Euclidean and Lobachevskian spaces are both infinite, and it is a much different proposition to say that an infinitely dense Big Bang starts at a finite singularity, into which a finite positively curved space can be packed, than it is to say that an infinite homogeneous and isotropic universe, which must have begun infinite, starts from an infinitely dense Big Bang. An infinitely dense singularity can have a finite mass, but an extended infinite density, even in a small finite region of space, cannot. In a recent cosmological article in Scientific American, "Textures and Cosmic Structure" (March 1992), the authors, Spergel and Turok, speak of the universe (they do not say "the observable universe") starting from an "infinitesimally small point" or of the universe being at one time the size of a "grapefruit," as though that would hold true for all model universes. The infinite universes are not even considered, and so the questions about density can be happily ignored. The closest thing to confronting this conflict that I have seen is a passage in The Matter Myth by Paul Davies and John Gribbin (Touchstone, 1992): I suppose infinity always dazzles us, and I have never been able to build up a good intuition about the concept. The problem is compounded here because there are actually two infinities competing with each other: there is the infinite volume of space, and there is the infinite shrinkage, or compression, represented by the big bang singularity. However much you shrink an infinite space, it is still infinite. On the other hand, any finite region within infinite space, however large, can be compressed to a single point at the big bang. There is no conflict between the two infinities so long as you specify just what it is that you are talking about. Well, I can say all this in words, and I know I can make mathematical sense of it, but I confess that to this day I cannot visualize it. (p.108) The problem here, however, is not visualization, it is the hard logical truth that an infinite space remains infinite and that the big bang for an infinite space, although it can be described as a singularity in relation to any finite region of space, cannot be a finite singularity. Einstein himself introduced his Cosmological Constant to preserve a static universe, before Hubble's evidence of the red shift. He thus seems to have been thinking that a global positively curved geometry for spacetime was not necessarily tied to some dynamical evolution of the universe. This is still a possibility. Three dimensional space can still be conceived as having an inherent hetero-curvature apart from the gravitational fate of the universe: non-Euclidean without the need to regard time or anything else as a fourth dimension into which space needs to be extrinsically curved. This makes for a finite Big Bang regardless of the dynamical fate of the universe, where that fate is tied to the effect of the curvature of time, locally positively curved but globally possibly Lobachevskian or Euclidean. However, a theory of global hetero-curvature then stands separate from the mathematical Relativistic theory of gravity and becomes a theory in metaphysical cosmology more than a theory in physical cosmology. A positively hetero-curved universe happens to suit the most commonly used cosmological model of all: the inflating balloon, where motion is added to our spherical model of non-Euclidean geometry. The surface of the balloon remains spherical regardless of whether the balloon is blown up forever or whether it eventually is allowed to deflate. As a model the balloon therefore actually posits five dimensions, with the surface representing the three dimensions of space, time as the fourth, but as a fifth the third spatial dimension into which the surface is curved and through which the surface moves in time. A positively hetero-curved universe, however, does not need that fifth dimension. Space would be non-Euclidean without higher dimensions, even while it moves along a temporal axis that is locally ortho-curved into an apparently hetero-curved spacetime because of the curvature of time. The balloon model therefore can represent a different kind of theory than it was intended to, but a most suggestive one, where the global structure of the isotropic and homogeneous universe may allow us to avoid an infinite Big Bang independent of the dynamical fate of the universe and fulfill the hope of the philosophers that Einstein answered Kant's Antinomy of Space. 4. Conclusion Just because the math works doesn't mean that we understand what is happening in nature. Every physical theory has a mathematical component and a conceptual component, but these two are often confused. Many speak as though the mathematical component confers understanding, this even after decades of the beautiful mathematics of quantum mechanics obviously conferring little understanding. The mathematics of Newton's theory of gravity were beautiful and successful for two centuries, but it conferred no understanding about what gravity was. Now we actually have two competing ways of understanding gravity, either through Einstein's geometrical method or through the interaction of virtual particles in quantum mechanics. Nevertheless, there is often still a kind of deliberate know-nothing-ism that the mathematics is the explanation. It isn't. Instead, each theory contains a conceptual interpretation that assigns meaning to its mathematical expressions. In non-Euclidean geometry and its application by Einstein, the most important conceptual question is over the meaning of "curvature" and the ontological status of the dimensions of space, time, or whatever. The most important point is that the ontological status of the dimensions involved with the distinction between intrinsic and extrinsic curvature is a question entirely separate from the mathematics. It is also, to an extent, a question that is separate from science--since a scientific theory may work quite well without out needing to decide what all is going on ontologically. Some realization of this, unfortunately, leads people more easily to the conclusion that science is conventionalistic or a social construction than to the more difficult truth that much remains to be understood about reality and that philosophical questions and perspectives are not always useless or without meaning. Philosophy usually does a poor job of preparing the way for science, but it never hurts to ask questions. The worst thing that can ever happen for philosophy, and for science, is that people are so overawed by the conventional wisdom in areas where they feel inadequate (like math) that they are actually afraid to ask questions that may imply criticism, skepticism, or, heaven help them, ignorance. These observations about Einstein's Relativity are not definitive answers to any questions; they are just an attempt to ask the questions which have not been asked. Those questions become possible with a clearer understanding of the separate logical, mathematical, psychological, and ontological components of the theory of non-Euclidean geometry. The purpose, then, is to break ground, to open up the issues, and to stir up the complacency that is all too easy for philosophers when they think that somebody else is the expert and understands things quite adequately. It is the philosopher's job to question and inquire, not to accept somebody else's word for somebody else's understanding. 5. Postscript, 1999 The logjam of conformity and complacency that irritated me for so many years before originally writing this paper, and since, may now be breaking. A new article in Scientific American, "Is Space Finite?" [Jean-Pierre Luminet, Glenn D. Starkman, Jeffrey R. Weeks, April 1999, pp. 90-97] finally divorces the geometry of the universe from its dynamics. A teaser at the beginning of the article says, "Conventional wisdom says that the universe is infinite" [p. 90]. Really? This is "conventional wisdom" now? What does Stephen Hawking have to say about that? The text says, "The question of a finite or infinite universe is one of the oldest in philosophy. A common misconception is that it has already been settled in favor of the latter" [p. 91]. Perhaps it is time for a new edition of A Brief History of Time! Acknowledging that the density of matter in the universe does appear to be too low to "close" the universe gravitationally, which means it is dynamically open, and so, as we have been given to understand, infinite, the article says: One problem with the conclusion is that the universe could be spherical yet so large that the observable part seems Euclidean, just as a small patch of the earth's surface looks flat [a common idea in "inflationary" theories]. A broader issue, however, is that relativity is a purely local theory [!]. It predicts the curvature of each small volume of space -- its geometry -- based on the matter and energy it contains. Neither relativity nor standard cosmological observations say anything about how those volumes fit together to give the universe its overall shape -- its topology. [p. 92, comments added] So all this time, all of the angst about the dynamics of the universe wasn't necessarily about the large scale structure of the universe at all. At little bit of this in the 70's would have been quite nice. If Scientific American had actually printed the letter, as they said they might, that I wrote them in 1976, I would be in an excellent position to say "I told you so" (although my concern was the difference that an extra dimension would make, not the kind of topological questions now opened). In fact, I did tell them so, but I fear it did not make it into the public record. Nor was this essay noticed by anyone in particular when it was posted on the Web in 1996. It probably still won't be, to dampen or affect any of the latest enthusiasms. Another point in the article may be worth noting. Luminet et al. say that the universe may be finite because of Mach's argument about the source of inertia. Grappling with the causes of inertia, Newton imagined two buckets partially filled with water. The first bucket is left still, and the surface of the water is flat. The second bucket is spun rapidly, and the surface of the water is concave. Why? The naive answer is centrifugal force. But how does the second bucket know it is spinning? In particular, what defines the intertial reference frame relative to which the second bucket spins and the first does not? Berkeley [!] and Mach's answer was that all the matter [which Berkeley didn't believe in] in the universe collectively provides the reference frame. The first bucket is at rest relative to distance galaxies, so its surface remains flat. The second bucket spins relative to those galaxies, so its surface is concave. If there were no distant galaxies, there would be no reason to prefer one reference frame over the other. The surface in both buckets would have to remain flat, and therefore the water would require no centripetal force to keep it rotating. In short, there would be no inertia. Mach inferred that the amount of inertia a body experiences is proportional to the total amount of matter in the universe. An infinite universe would cause infinite intertia. Nothing would ever move. [p. 92, comments added] Whatever the "naive" explanation may be, it is not the one used by Newton. The argument made by Luminet et al., Berkeley, and Mach is actually the argument originally made by Leibniz (and just recycled by Berkeley, who believed in space less than in matter) against Newton's idea that space was real. For Newton, the rotating bucket was rotating in relation to space itself. Evidently, it is now such "conventional wisdom" that space itself provides no inertial frame of reference, since Einstein, that it doesn't occur to anyone that the kind of reference it provides vis vis rotation is rather different from what it fails to provide to establish absolute linear motion. The argument that, in empty space, with no "distant galaxies," there would be no centrifugal force in the bucket and the water in one would be just as flat as in the other is not a necessary conclusion, but only a theory. And not a theory easily tested without an empty universe available. On the other hand, the question can still be asked how the bucket can "know" that the "distant galaxies" are out there. There must be a physical interaction for that (the range of gravity is infinite); yet Einstein, again, said that no physical interaction can travel faster than the velocity of light, and in an "inflationary" universe (which Mach didn't know about) light can have reached us from only a finite part of the universe, even in an infinite universe. Thus the argument of Luminet et al. fails, for a infinite universe would make for infinite inertia only if the whole universe could physically affect a location. If only a finite part of the universe, infinite or otherwise, affects a location, then there will still only be finite inertia. Apart from a shake-up over the geometry of space, there has been another surprise in recent cosmology. An article in the January 1999 Scientific American, "Surveying Space-time with Supernovae" [Craig J. Hogan, Robert P. Kirshner, and Nicholas B. Suntzeff, pp. 46-51], discusses observational data that seems to indicate that the expansion of the universe has accelerated over time, not decelerated as it should under the influence of gravity alone. This implies the existence of Einstein's "Cosmological Constant" or some other exotic force that would override the attraction of gravity. It also may clear up another pecularity about "standard" cosmology that had been swept under the rug. That is, all closed universes, where decleration would be enough to produce a colapse into the "Big Crunch," prefered by cosmologists like Stephen Hawking, would have to be younger than 2 3 of the Hubble Time (1 H). This would also mean that no objects in the universe could have a red shift larger than 2 3 of the velocity of light (c), since the red shift gives us the distance in proportion to the Hubble Radius (c H), and also the age in proportion to the Hubble Time. Thus, in the diagram at right, all the universes under the green curve are closed, and all those above the green curve are open. Now, many quasars have red shifts larger than 2 3 c. Many are even over 90% of c. This has been prima facie evidence since the 70's that the universe was open, but nobody of any influence seems to have noticed. Now, however, if the universe is accelerating, then all possible universes are above the straight red line in the diagram which indicates the Hubble Constant. They will all be older than the Hubble Time. This suddenly makes it quite reasonable that very old objects, like many quasars, would have very, very large reshifts. Indeed, the Big Bang itself would appear to be receding faster than the velocity of light -- it would have an infinite red shift. So again we have an object lesson in the history of science, that a careful examination of the implications of a theory is sometimes neglected by professional science. Inconsistencies can be revealed by even a lay examination. Philosophy of Science Metaphysics Home Page Copyright (c) 1996, 1998, 1999 Kelley L. Ross, Ph.D. All Rights Reserved The Ontology and Cosmology of Non-Euclidean Geometry, Note The reader is thus put on notice that it is the empiricist "skeptic" Hume, not the rationalistic Kant, who thinks that the axioms of geometry are self-evident, and Hume who denies that their truth owes anything to experience. Return to quote
Book List on Non-Euclidean Geometry
From theTreasureTroves collection.
Non-Euclidean Geometry -- from Eric Weisstein's Encyclopedia of Scientific Books Non-Euclidean Geometry see also Non-Euclidean Geometry Anderson, James W. Hyperbolic Geometry. New York: Springer-Verlag, 1999. 230 p. $?. Bonola, Roberto. Non-Euclidean Geometry, and The Theory of Parallels by Nikolas Lobachevski, with a Supplement Containing The Science of Absolute Space by John Bolyai. New York: Dover, 1955. 268 p., 50 p., and 71 p. Borsuk, Karol. Foundations of Geometry: Euclidean and Bolyai-Lobachevskian Geometry. Projective Geometry. Amsterdam, Netherlands: North-Holland, 1960. 444 p. Carslaw, H.S. The Elements of Non-Euclidean Plane Geometry and Trigonometry. London: Longmans, 1916. Coxeter, Harold Scott Macdonald. Non-Euclidean Geometry, 6th ed. Washington, DC: Math. Assoc. Amer., 1988. 320 p. $30.95. Greenberg, Marvin J. Euclidean and Non-Euclidean Geometries: Development and History, 3rd ed. San Francisco, CA: W.H. Freeman, 1994. $?. Iversen, Birger. Hyperbolic Geometry. Cambridge, England: Cambridge University Press, 1992. 298 p. $?. Manning, Henry Parker. Introductory Non-Euclidean Geometry. New York: Dover, 1963. 95 p. Out of print. Martin, George E. The Foundations of Geometry and the Non-Euclidean Plane. New York: Springer-Verlag, 1975. $?. Ramsay, A. and Richtmeyer, R.D. Introduction to Hyperbolic Geometry. New York: Springer-Verlag, 1995. 287 p. $39. Sommerville, D.M.Y. The Elements of Non-Euclidean Geometry. London: Bell, 1914. Stillwell, John. Sources of Hyperbolic Geometry. Providence, RI: Amer. Math. Soc., 1996. 153 p. $39. Sved, Marta. Journey into Geometries. Washington, DC: Math. Assoc. Amer., 1991. 192 p. $29.95. Trudeau, Richard J. The Non-Euclidean Revolution. Boston, MA: Birkhuser, 1987. 257 p. $49.50. 1995-2005 Eric W. Weisstein 2003-10-07 http: www.ericweisstein.com encyclopedias books Non-EuclideanGeometry.html
Differential geometry of curves
Answers.com encyclopedia entry on Differential Geometry.
differential geometry of curves: Information From Answers.com Business Entertainment Games Health People Places Reference Science Shopping Words More... On this page: Wikipedia Best of Web Mentioned In --------------- Or search: - The Web - Images - News - Blogs - Shopping differential geometry of curves Wikipedia differential geometry of curves In mathematics , the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space ) using differential and integral calculus . For example, circle in the plane can be defined as the curve where the vector (t) is always perpendicular to the tangent vector (t). Or written as an inner product The differential properties of many classical curves have been studied thoroughly: see the list of curves for details. The main contemporary application is in physics as part of vector calculus . In general relativity for example a world line is a curve in spacetime . To simplify the presentation we only consider curves in Euclidean space , it is straightforward to generalize these notions for Riemannian and Pseudo-Riemannian manifolds . For a more abstract curve definition in an arbitrary topological space see the main article on curves . Definitions Let n be a natural number, r an natural number or , I be a non-empty interval of real numbers and t in I. A vector valued function of class Cr (i.e. is r times continuously differentiable ) is called a parametric curve of class Cr or a Cr parametrization of the curve . t is called the parameter of the curve . (I) is called the image of the curve. It is important to distinguish between a curve and the image of a curve (I) because a given image can be described by several different Cr curves. One may think of the parameter t as representing time and the curve (t) as the trajectory of a moving particle in space. If I is a closed interval [a, b] we call (a) the starting point and (b) the endpoint of the curve . If (a) = (b) we say is closed or a loop. Furthermore we call a closed Cr-curve if (k)(a) = (k)(b) for all k r. If :(a,b) Rn is injective, we call the curve simple. If is a parametric curve which can be locally described as a power series, we call the curve analytic or of class C . We write - to say the curve is traversed in opposite direction. A Ck-curve is called regular of order m if are linearly independent in Rn. Examples Main article: curves in differential geometry Reparametrization and equivalence relation Given the image of a curve one can define several different parametrizations of the curve. Differential geometry aims to describe properties of curves invariant under certain reparametrizations. So we have to define a suitable equivalence relation on the set of all parametric curves. The differential geometric properties of a curve (length, frenet frame and generalized curvature) are invariant under reparametrization and therefore properties of the equivalence class .The equivalence classes are called Cr curves and are central objects studied in the differential geometry of curves. Two parametric curves of class Cr and are said to be equivalent if there exists a bijective Cr map :I1I2 such that and 2 is said to be a reparametrisation of 1. This reparametrisation of 1 defines the equivalence relation on the set of all parametric Cr curves. The equivalence class is called a Cr curve. We can define an even finer equivalence relation of oriented Cr curves by requiring to be (t) 0. Equivalent Cr curves have the same image. And equivalent oriented Cr curves even traverse the image in the same direction. Length and natural parametrization The length l of a smooth curve : [a, b] Rn can be defined as The length of a curve is invariant under reparametrization and therefore a differential geometric property of the curve. For each regular Cr-curve : [a, b] Rn we can define a function Writing we get a reparametrization of which is called natural, arc-length or unit speed parametrization. s(t) is called the natural parameter of . We prefer this parametrization because the natural parameter s(t) traverses the image of at unit speed so that In practice it is often very difficult to calculate the natural parametrization of a curve, but it is useful for theoretical arguments. For a given parametrized curve (t) the natural parametrization is unique up to shift of parameter. The quantity is sometimes called the energy or action of the curve; this name is justified because the geodesic equations are the Euler-Lagrange equations of motion for this action. Frenet frame A Frenet frame is a moving reference frame of n orthonormal vectors ei(t) which are used to describe a curve locally at each point (t). It is the main tool in the differential geometric treatment of curves as it is far easier and more natural to describe local properties (e.g. curvature, torsion) in terms of a local reference system than using a global one like the Euclidean coordinates. Given a Cn+1-curve in Rn which is regular of order n the Frenet Frame for the curve is the set of orthonormal vectors called Frenet vectors . They are constructed from the derivatives of (t) using the Gram-Schmidt orthogonalization algorithm with The real valued functions i(t) are called generalized curvature and are defined as The Frenet frame and the generalized curvatures are invariant under reparametrization and therefore differential geometric properties of the curve. Special Frenet vectors and generalized curvatures The first three Frenet vectors and generalized curvatures can be visualized in three dimensional space. They have additional names and more semantic information attached to them. Tangent vector At every point of a C1 curve we can define a tangent vector. If is interpreted as the path of a particle then the tangent vector can be visualized as the path that the particle takes when free from outer force. The tangent vector is the first Frenet vector e1(t) and is defined as If has a natural parameter then the equation simplifies to The scalar magnitude of the tangent vector is called the speed v of at point t. If has a natural parameter the speed is 1. Since it points along the forward direction of the curve (the direction of increasing parameter), the unit tangent vector introduces an orientation of the curve. Normal or curvature vector The normal vector, sometimes called the curvature vector, indicates the deviance of the curve from being a straight line. It is the second Frenet vector e2(t) and defined as The tangent and the normal vector at point t define the osculating plane at point t. Curvature The first generalized curvature 1(t) is called curvature and measures the deviance of from being a straight line relative to the osculating plane. It is defined as and is called the curvature of at point t. The reciprocal of the curvature is called the curvature radius A circle with radius r has a constant curvature of whereas a line has a curvature of 0. Binormal vector The binormal vector is the third Frenet vector e3(t) It is always orthogonal to the unit tangent and normal vectors at t, and is defined as In 3 dimensional space the equation simplifies to Torsion The second generalized curvature 2(t) is called torsion and measures the deviance of from being a plane curve. Or, in other words, if the torsion is zero the curve lies completely in the osculating plane. and is called the torsion of at point t.. Main theorem of curve theory Given n functions with then there exists a unique (up to transformations using the Euclidean group ) Cn+1-curve which is regular of order n and has the following properties where the set is the Frenet frame for the curve. By additionally providing a start t0 in I, a starting point p0 in Rn and an initial positive orthonormal Frenet frame {e1, ..., en-1} with we can eliminate the Euclidean transformations and get unique curve . Frenet-Serret formulas The Frenet-Serret formulas are a set of ordinary differential equations of first order. The solution is the set of Frenet vectors describing the curve specified by the generalized curvature functions i For a proof of the 3-dimensional case see Frenet-Serret formulas . 2-dimensions 3-dimensions n-dimensions (general formula) See also Osculating circle Curve Curvature Torsion Arc Parameter , parametrization Implicit function Tangent , contact , subtangent Frenet-Serret formulas Envelope , evolute , involute , pedal curve , roulette Four-vertex theorem Geodesic geodesic curvature Isoperimetry Moving frame List of curve topics List of curves This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer ) Donate to Wikimedia Best of the Web Some good "differential geometry of curves" pages on the web: Math mathworld.wolfram.com Mentioned In differential geometry of curves is mentioned in the following topics: evolute torsion involute roulette (curve) list of curve topics pedal curve List of differential geometry topics differential geometry and topology curve Areas of mathematics More Copyrights: Wikipedia information about differential geometry of curves This article is licensed under the GNU Free Documentation License . It uses material from the Wikipedia article "Differential geometry of curves" . More from Wikipedia Your Ad Here Your Ad Here Jump to: Wikipedia Best of Web Mentioned In --------------- Or search: - The Web - Images - News - Blogs - Shopping Send this page Print this page Link to this page Tell me about: Home Webmasters Site Map About Legal Help Advertise
Non-Commutative Geometry
A short historical review is made of some recent literature in the field of noncommutative geometry
[gr-qc 9906059] Noncommutative Geometry for Pedestrians General Relativity and Quantum Cosmology, abstract gr-qc 9906059 From: J. Madore [ view email ] Date ( v1 ): Wed, 16 Jun 1999 09:38:45 GMT (27kb) Date (revised v2): Sun, 25 Jul 1999 14:45:20 GMT (29kb) Noncommutative Geometry for Pedestrians Authors: J. Madore Comments: Lecture given at the International School of Gravitation, Erice, 28 pages, Latex Report-no: LMU-TPW 99-11 A short historical review is made of some recent literature in the field of noncommutative geometry, especially the efforts to add a gravitational field to noncommutative models of space-time and to use it as an ultraviolet regulator. An extensive bibliography has been added containing reference to recent review articles as well as to part of the original literature. Full-text: PostScript , PDF , or Other formats References and citations for this submission: SLAC-SPIRES HEP (refers to , cited by , arXiv reformatted); CiteBase (autonomous citation navigation and analysis) Which authors of this paper are endorsers? Links to: arXiv , gr-qc , find , abs ( - + ), 9906 , ?
Classical Curve Theory and Frenet Equations
A systematic overview of classical curve theory, including Frenet equations and their known solutions, some results on moving frames (relationship between Frenet and Bishop Frame), spherical curves and surface curves.
Classical Curve Theory Classical Curve Theory - Solutions to Frenet's Equations Ueber die Darstellung von Raumkurven aus ihren Invarianten (The Representation of Space Curves by their Invariants) Toni Menninger, master thesis 1996 (revised 2001), Universitaet Wuerzburg The thesis as pdf file (550 kB) This thesis on curve theory explores a classical question: for which types of curves can the Frenet equations be explicitly solved, in other words: for which pairs of curvature and torsion functions is the parameterization of the corresponding curve known? The known answers are: Plane curves- = 0 Helical curves- = const. Curves of constant precession1- = cos( s), = sin( s) (, = const.) The thesis presents - for the first time to my knowledge - a generalization of curves of constant precession ( "Kreisellinien", theorems 26 and 27 ) and the solution of the respective Frenet differential equations; they are characterized by the following relations: = cos(), = sin(), ' = const. ((s) differentiable, (s) continuous) I also present a method to construct a series of curve classes with solutions for the Frenet equations (with plane curves, helical curves and generalized curves of constant precession as the first three classes in the series). Moreover, I attempt a complete and systematic overview of classical curve theory, including some results on moving frames (relationship between Frenet and Bishop Frame2), spherical curves and surface curves. Comments are welcome: toni.menninger@sympatico.ca A curve of constant precession (Scofield 1995) with it's tangent (left), the arcs of which represent spherical helices. 1 Paul D. Scofield: Curves of Constant Precession, The American Mathematical Monthly, Volume 102, Number 6, June-July 1995 2 Bishop or Parallel Transport frames are characterized by the fact that the derivatives of both normal components are tangential. In other words, the moving frame twists only "as much as necessary", it comes as close to parallel translation as possible. Existence of a family of Bishop frames is guaranteed for any regular C2curve. For a discussion of applications and numerical algorithms see Hui Ma: Curve and Surface Framing for Scientific Visualization and Domain Dependent Navigation, 1996 ( Ph.D. Dissertation ) geovisit();
Finite Canonical Commutation Relations
A working paper on FCCR nxn matrices as local kinematical replacement for CCR, and representations by pxp matrices over Galois fields.
FCCR Finite Canonical Commutation Relations You can never solve a problem on the level on which it was created. -- Albert Einstein (1879-1955) "Silence will save me from being wrong (and foolish), but it will also deprive me of the possibility of being right. -- Igor Stravinsky, composer (1882-1971) A working paper on FCCR nxn matricies as local kinematical replacement for CCR, and construction of REPS of CCR by pxp matricies over Galois fields. Some remaining text of the chapters is still being translated to hypertext. An abstract and a brief summary of the essentials are available. The central object of interest here is a chain of algebras given in terms of nxn complex matricies that connect the quantum mechanical Canonical Commutation Relations (CCR) for n unbounded with the Canonical Anticommutation Relations (CAR) for n=2, which seem to provide a local, finite, discrete quantum theory. Finite dimensional representations of CCR by matricies over Galois fields are constructed in appendix J. The detail of mathematics, physics and philosophy presented here is far more than what would be usual in an any professional journal. I believe it to be sufficient that any mathematician physicist should be able to reproduce and confirm (or correct!) every detail in the exposition. Copyright Notice Abstract A summary of FCCR Notational Conventions Symbols FCCR Table of Contents About Author Email: Author Bill Hammel CONTENTS I. Motivation and Introduction II. Truncated Creation and Annihilation Operators III. QM and Limits of unbounded n IV. Higher Commutators V. Algebra of Observables Polynomial functions of the generators Q(n) and P(n) Canonical Transformations VI. P(n) and Q(n) as Algebra Generators Irreducibility Solvability v. Unitarity VII. Discrete n-Fourier Transforms Fr(n), connecting Q(n) and P(n) and UPSILON(n) relating N(n)-eigenbasis with the eigenbasis of the phase operator F(n) VIII. FCCR Structural Theorems and Matrix Element Calculations IX. Eigenvalue Problems P(n), Q(n) and Sine and Cosine Operators X. Diagonalizing Transformations XI(n) and PI(n); Diagonalizing UPSILON(n) XI. Invariance Group of G(n) Right and Left G(n)-Hermiticity. Adjoint and Coadjoint actions - Orbits. Analogy to the Lorentz Group. XII. Homogeneous Complex Spaces Complex Lorentzian hypercones. XIII. Uncertainty Relations Comparison with CCR, Exceptional cases where proof method fails in FCCR and CCR. XIV. IRREPS of SU(2), SU(1,1) SL(2,C) Quantized Noncommutative Geometry XV. Relativstic Structure XVI. Elementary Systems XVII. More Limits XVIII. Conclusions Appendix A: Some Topological and Algebraic Definitions Appendix B: Lie Groups and Lie Algebras Clifford Algebras and Spin Representations of Orthogonal and Pseudo-orthogonal Groups Appendix C: Lie Algebras and Lie Groups su(2), su(1,1) and sl(2,C) Appendix D: G(n)-Hermiticity G(n) Invariance Group Hyperbolic Complex Manifolds Appendix E: Some Polynomial Formulae, Hermite and related polynomials, their roots and algebraic approximations of them Appendix F: Analytic Functions, Fourier and Hilbert Transforms and Dispersion Relations for Causal Functions Appendix G: Hilbert Subspace Calculations Appendix H: Asymptotics of Rotation Matricies Proof of Theorem 14.5. Appendix I: Trace Formulae Appendix J: FCCR Algebra over Finite Galois Fields and Finite dimensional Representations of CCR in non-normable algebras over Galois fields. Appendix K: Necessary Multiple Concepts of "Time". References Go to Physics Page Go to Home Page Email me, Bill Hammel at bhammel@graham.main.nc.us READ WARNING BEFORE SENDING E-MAIL The URL for this document is http: graham.main.nc.us ~bhammel FCCR fccr.html Created: 1996 Last Updated: May 28, 2000 Last Updated: November 28, 2002 Last Updated: February 24, 2004
Alfred Gray Differental Geometry
Graphics of suface geometry. Mathematica code and gallery of images.
Alfred Gray's Home Page This Differential Geometry web site is maintained in memory of Professor Alfred Gray (1939 - 1998) Costa's minimal surface The Mathematica miniprograms and Compressed sample notebooks for Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition, CRC Press (1998), (See CRC Catalog) Curvas y Superficies, Addison-Wesley Ibero-Americana (1994), Differentialgeometrie, Spektrum-Verlag (1994), Japanese Edition (1996), Italian Edition, 1998. are available SurfaceExplorer , a Mathematica application for viewing surfaces from differential geometry, is now available. Visit the space curve gallery, plane curve gallery, constant curvature surface gallery minimal surface gallery and the general surface gallery. For comments on this page, contact Michael Mezzino by email mezzino@math.cl.uh.edu. or his home page http: math.cl.uh.edu ~mezzino Files are also available via anonymous ftp at ftp: math.cl.uh.edu
John Oprea's Home Page
This site includes references to the author's papers and books, including Differential Geometry and its Applications and The Mathematics of Soap Films: Explorations with Maple. There are also Maple files available for downloading.
John Oprea's Home Page Herr Professor Doktor Oprea's Home Page John Oprea, Professor of Mathematics Graduate Program Director Department of Mathematics Cleveland State University Cleveland, Ohio 44115 OFFICE: 1529 Rhodes Tower PHONE: 687-4702 E-MAIL: j.oprea AT csuohio.edu Utah State University Prospects in Mathematics Speaker Monograph on Lusternik-Schnirelmann Category AMS-IMS-SIAM Summer Research Conference 2001; Lusternik-Schnirelmann Category in the New Millennium Research Interests Textbook on Differential Geometry Soap Film Page Teaching Links Graduate Program If you have questions, doubts, comments, suggestions, or desire additional information, send E-mail to: oprea AT math.csuohio.edu Return to List of Math Dept. Faculty Return to Math Dept. Home Page
ArXiv Front: DG Differential Geometry
Differential geometry section of the mathematics e-print arXiv.
DG Differential Geometry Thu 17 Nov 2005 Search Submit Retrieve Subscribe Journals Categories Preferences iFAQ DG Differential Geometry Calendar Search Authors: All AB CDE FGH IJK LMN OPQR ST U-Z New articles (last 12) 17 Nov math.DG 0511398 Quasi-local mass and the existence of horizons. Yuguang Shi , Luen-Fai Tam . DG ( MP ). 16 Nov math.DG 0511387 Bending the Helicoid. William H. Meeks III, Matthias Weber . 17 pages. DG . 16 Nov math.DG 0511377 New structures on the tangent bundles and tangent sphere bundles. Marian Ioan Munteanu . 17 pages. DG . 15 Nov math.DG 0511350 Spinor functions of spinors and the concept of extended spinor fields. Ruslan Sharipov . 56 pages. DG ( MP ). 15 Nov math.DG 0511314 Weakly hyperbolic actions of Kazhdan groups on tori. Benjamin Schmidt . DG ( DS ). Cross-listings 17 Nov math.AG 0511415 Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces. Steven B. Bradlow (University of Illinois), Oscar Garcia-Prada (CSIC, Madrid), Peter B. Gothen (Universidade do Porto). 28 pages. IHES Preprint M 05 51. AG ( DG ). 17 Nov math.CV 0511395 The first coefficients of the asymptotic expansion of the Bergman kernel of the spin^c Dirac operator. Xiaonan Ma , George Marinescu . 21 pages. CV ( DG ). 16 Nov hep-th 0511144 A topological sigma model of biKaehler geometry. Roberto Zucchini . 46 pages. DFUB 05-11. ( MP DG ). Revisions 17 Nov math.DG 0411559 Generalized Bergman kernels on symplectic manifolds. Xiaonan Ma , George Marinescu . 48 pages. DG ( SG CV ). 16 Nov math.DG 0511303 On the topology of compact affine manifolds. Mihail Cocos . 8 pages. DG ( DS ). 15 Nov math.DG 0511242 N-flat connections. Mauricio Angel , Rafael Daz . DG ( MP ). 15 Nov math.DG 0510618 Hodge theory on nearly Kaehler manifolds. Misha Verbitsky . 20 pages. DG ( AG ). Recent Calendar 2005 550+229 November 31+11 October 56+25 September 48+21 August 49+17 July 34+24 June 55+27 May 64+26 April 66+22 March 50+11 February 47+24 January 50+21 2004 592+274 December 43+31 November 58+28 October 42+17 September 50+21 August 39+19 July 50+24 June 64+30 May 57+28 April 41+20 March 45+30 February 65+13 January 38+13 2003 485+240 December 31+18 November 63+20 October 57+23 September 29+14 August 30+16 July 39+18 June 40+27 May 38+29 April 44+18 March 38+19 February 41+19 January 35+19 2002 381+268 December 22+21 November 49+40 October 43+22 September 38+14 August 23+22 July 24+11 June 21+24 May 26+26 April 35+23 March 35+15 February 28+17 January 37+33 2001 336+197 December 36+28 November 35+23 October 33+19 September 24+17 August 22+11 July 15+16 June 32+10 May 33+18 April 36+18 March 17+14 February 24+13 January 29+10 2000 319+173 December 19+11 November 32+15 October 33+20 September 37+12 August 20+8 July 25+13 June 22+12 May 30+21 April 17+16 March 25+16 February 28+20 January 31+9 1999 264+156 December 22+9 November 30+16 October 26+15 September 26+12 August 19+14 July 16+13 June 14+12 May 20+20 April 16+10 March 35+8 February 23+12 January 17+15 1998 230+112 December 26+10 November 14+7 October 22+11 September 14+11 August 22+9 July 28+13 June 16+11 May 20+6 April 14+12 March 18+9 February 18+6 January 18+7 1997 249+67 December 22+9 November 21+4 October 38+8 September 22+10 August 15+2 July 29+2 June 19+2 May 10+4 April 18+5 March 23+7 February 20+8 January 12+6 1996 159+71 December 18+11 November 14+8 October 20+8 September 13+4 August 11+3 July 6+9 June 14+4 May 10+3 April 9+8 March 18+5 February 13+3 January 13+5 1995 130+34 December 12+2 November 18+4 October 11+1 September 6 August 17+4 July 5+5 June 13+2 May 8+2 April 6 March 18+7 February 10+3 January 6+4 1994 64+29 December 9+2 November 15+2 October 7+2 September 5+3 August 5 July 16+7 June 7+3 May 0+3 April 0+3 March 0+2 February 0+1 January 0+1 1993 4+17 December 0+2 November 0+2 October 1+3 September 1+1 July 1+1 June 1 May 0+1 April 0+2 March 0+4 January 0+1 1992 19+7 December 0+1 October 3+3 September 1 July 3 April 3+1 March 1 February 3 January 5+2 1991 0+1 October 0+1 Total: 3782+1875 articles (primary+secondary) Search Author Title ID Anywhere Cat MSC articles per page Show Help AC AG AP AT CA CO CT CV DG DS FA GM GN GR GT HO KT LO MG MP NA NT OA OC PR QA RA RT SG SP ST Authors: All AB CDE FGH IJK LMN OPQR ST U-Z Home Search Submit Retrieve Subscribe Journals Categories Preferences iFAQ - for help or comments about the Front - for help about submissions or downloading arXiv articles
Riemannian Geometry
Mostly a definition with a few equations.
Riemannian Geometry -- From MathWorld INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News (RSS) New in MathWorld MathWorld Classroom Interactive Entries Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Topology Manifolds Topology Bundles Riemannian Geometry The study of manifolds having a complete Riemannian metric . Riemannian geometry is a general space based on the line element with for a function on the tangent bundle . In addition, is homogeneous of degree 1 in and of the form (Chern 1996). If this restriction is dropped, the resulting geometry is called Finsler geometry . SEE ALSO: Non-Euclidean Geometry , Riemannian Metric . [PagesLinkingHere] REFERENCES: Besson, G.; Lohkamp, J.; Pansu, P.; and Petersen, P. Riemannian Geometry. Providence, RI: Amer. Math. Soc., 1996. Buser, P. Geometry and Spectra of Compact Riemann Surfaces. Boston, MA: Birkhuser, 1992. Chavel, I. Eigenvalues in Riemannian Geometry. New York: Academic Press, 1984. Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994. Chern, S.-S. "Finsler Geometry is Just Riemannian Geometry without the Quadratic Restriction." Not. Amer. Math. Soc. 43, 959-963, 1996. do Carmo, M. P. Riemannian Geometry. Boston, MA: Birkhuser, 1992. CITE THIS AS: Eric W. Weisstein. "Riemannian Geometry." From MathWorld --A Wolfram Web Resource. http: mathworld.wolfram.com RiemannianGeometry.html 1999 CRC Press LLC, 1999-2005 Wolfram Research, Inc. | Terms of Use
Differential Geometry Page
Contains several figures which are the result of easy codes using Mathematica, including Enneper's surface.
Differential Geometry Page Differential Geometry Page This page contains a few figures which are the result of easy codes using Mathematica. Enneper's Surface Torus
Super Shapes in 3D
Gives explanation of the superformula used to create various images, which is based on equations by Johan Gielis. Page includes galleries and shareware software downloads.
3D Supershape Superformula Supershape in 3D (Also known as the Superformula) Written by Paul Bourke July 2003 Based upon equations by Johan Gielis Intended as a modelling framework for natural forms. See also: Creating Supershapes in POVRay See also: Supershapes in 2D Table of contents Software Examples Non integer m 2D Extrusion Stability Classical forms Shells Toroidal mapping Renderings Physical model Definition of supershape in 2 dimensions. Extension to 3D using the spherical product. Software The following shows the interface developed to explore 3D supershapes. It is based upon X-Windows and OpenGL and as such is interactive. It is currently available for Mac OS-X (as a UNIX application) and Linux upon payment of a US$30 shareware fee. Use of the program is straightforward, edit fields and hit return. The left mouse button rotates the model in two axes, the middle mouse button rotates about the third axis. The right mouse button brings up a menu. A limited version of the program can be downloaded free. In the case of the Mac OS-X version it is necessary to have X11 running. In both cases it is expected OpenGL capable graphics card and drivers are installed. These versions also allow you to check the software works on your platform before getting the full version. supershape_mac.zip -- supershape_linux.gz Click for full size image Command line interface Many settings can be intialised from the command line, others control is given by single key strokes. To see a complete list type the program name at the command prompt with a "-h" option. At the time of writing these are given below. supershape -h Usage: supershape [command line options] Command line options -h this text -f full screen -s active stereo -ss dual screen stereo -a auto rotate -rw wireframe -rs flat shaded -rp specular shaded Mouse buttons left camera rotate shift left camera pan middle camera roll shift middle camera forward, reverse Key Strokes arrow keys camera rotate , camera forward, reverse +,- camera zoom in, out [,] camera roll r toggle window recording w capture window to TGA file a toggle auto rotation of camera 1,2,3 different rendering mode h camera to home position f1 to f6 axis aligned camera positions ESC,q quit Features Exports geometry as DXF, POVRay, and LightWave (the later includes texture coordinates). Rendering modes include wireframe, flat shaded, and specular shading. Rendering is based upon OpenGL. Colour ramps can be applied to both longitude and latitude. Parameter space exploration by either single parameter ranges or simply choosing a random parameter set. A supershape can be morphed into another. Images can be saved to a single TGA file or repreated saving can be turned on for animations. TGA files and the naming convention used are supported by QuickTime Pro. Examples Non integer values of m 2D The 2D supershapes (but with added small thickness) can be created using this software by setting the second supershape parameters as m=0, n1=n2=n3=1, a=b=1 and using a small z axis scale factor. Indeed this can often be the preferred approach for using a 2D form within a 3D model, after all, real world objects do have some thickness. The example on the left has a thickness of 0.02 units (x and y axis dimensions of 1). Or even set the depth to 0 and vary latitude from 0 to 90 degrees as in the example on the right. Extrusion Extrusions of 2D supershapes can be created by setting the second supershape parameters as m=4, a=1, b=1, and high equal values for each of n1,n2,n3. For example n1=n2=n3=200 gives relatively sharp edges. Stability A significant portion of the parameter space results in surfaces with various types of numerical problems (powers of negative numbers, divide by 0, underflow, overflow, etc) as well as issues related to the representation of 3D graphics. The software that has been developed and created the images shown above, can display the edges where numerical problems have arisen. These regions are shown in pick as shown in the following two images. Classical shapes Cube Cone Diamond Sphere Prism Cylinder Hexagon Pentagon Shells The supershape function can obviously be modulated by another function. For example to create the shell like structures below, the radius (r1) of the superformula that varies the longitude is scaled by either a logarithmic or Archimedes spiral. In some cases the z coordinate is additionally made a linear function of longitude. Toroidal mapping In the above the two supershapes were mapped onto a topological sphere. One could map supershapes onto other forms as well, for example, a torus. In the following images the mapping is as follows: Renderings Contribution by Luc Benard Contributions by Albert Kiefer Physical models
Triple Rotation Curves (3R-curves)
Generalization of 2R-curves (hypotrochiods and epitrochoids). Create line-art images by setting variables for a hypothetical planet, moon and satellite.
3Rcurves Enter The World of ... . . . 3 R c u r v e s
Math Art Gallery
Knots, surfaces and fractals.
Math art gallery Math art gallery Singularity knots. Singularity knot animations. Pictures generated by vort. A few 3D fractals. A g=1 surface, a g=2 surface, and a g=3 surface. An other g=2 surface , one with a different shape, one with a different color and one with three holes . Click here to obtain the source files for generating these pictures. Interactive art in Minnesota. All kinds of beautiful knots.
Un-disassemble-able Object by Jack Snoeyink
A sculpture representing research results on the number of hands that a robot would need to assemble collections of simple geometric objects.
Objects that cannot be taken apart with two hands
Tom Lechner's Sculptures
Polyhedral sculptures for sale. Also includes his memoirs of making polyhedral models, math references, and his art school thesis.
Tom Lechner's Sculptures My newest work Shots of my Great Big Math Object Gallery 1 - 2 - 3 - 4 - 5 Gallery of work for sale Solids Jigs My Art School Thesis Math References Home - Drawings - Prints - Murals - Consumption - Stuff - Other Copyright 2003, Tom J. Lechner
Religious Beliefs Made Visual: Geometry and Islam
A lesson plan. Constructs a geometric motif from Islamic art, and gives its cultural context.
Islamic Patterns Religious Beliefs Made Visual: Geometry and Islam By Jane Norman, consultant, education department, Metropolitan Museum of Art From: Focus on Asian Studies: Asian Religions, New Series, Vol. II, No. 1. Copyright the Asia Society Materials: A good quality compass is recommended. Use any straight-edge. If a ruler is used, please point out that it is being used only as a tool to make a straight line; that measuring with numbers has no relevance in this type of geometric construction. Classroom Exercise: Construction of an Islamic Pattern, part A. By following steps 1 - 6, the student will have the experience of constructing the star-hexagon pattern, a popular Islamic all-over pattern. Construction of an Islamic Pattern, part B. Steps 7 - 9 show how a triangle grid becomes a hexagonal grid. As students learn about the art produced by people of an unfamiliar society, they discover that it tells them many things about what these people did, knew, and believed. Examining the geometric patterns that characterize so much of Islamic art can provide students with important insights into the technology, scientific knowledge, and religious beliefs of Moslems. At first, an American child may question the value of studying art to understand a distant culture, but connections soon become apparent. Appreciation for a basic relationship between the art and the religion of Islam increases with familiarity. Careful observation of the illustrations here will provide an introduction to Islamic religious beliefs through its art. Geometric motifs were popular with Islamic artists and designers in all parts of the world, at all times, and for decorating every surface, whether walls or floors, pots or lamps, book covers or textiles. As Islam spread from nation to nation and region to region, Islamic artists combined their penchant for geometry with pre-existing traditions, creating a new and distinctive Islamic art. This art expressed the logic and order inherent in the Islamic vision of the universe. Although the shapes and structures are based on the geometry of Euclid and other Greek mathematicians, Islamic artists used them to create visual statements about religious ideas. One explanation of this practice was that Mohammad had warned against the worship of idols; this prohibition was understood as a commandment against representation of human or animal forms. Geometric forms were an acceptable substitute for the proscribed forms. An even more important reason is that geometric systems and Islamic religious values, though expressed in different forms, say similar things about universal values. In Islamic art, infinitely repeating patterns represent the unchanging laws of God. Moslems are expected to observe strict rules of behavior exactly as they were orginally set forth by Mohammad in the seventh century. These rules are known as the "Pillars of Faith": 1) pronouncing the creed (chantingan affirmation of the existence of one God and that God is Allah) 2) praying, in a precisely defined ritual of words and motions, five times a day 3) giving alms 4) fasting during the month of Ramadan (time varies according to lunar calendar) 5) making, during a lifetime, at least one pilgramage to the city of Mecca in Arabia The strict rules for construction of geometric patterns provide a visual analogy to religious rules of behavior. The geometric patterns used in Islamic art are aggressively two-dimensional. Artists did not want to represent the three-dimensional physical world. They preferred to create an art that represents an ideal, spiritual truth. Ideals are better represented as two-dimensional than three- dimensional. The star was the chosen motif for many Islamic decorations. In Islamic iconography the star is a regular geometric shape that symbolizes equal radiation in all directions from a central point. All regular stars -- whether they have 6, 8, 10, 12, or 16 points -- are created by a division of a circle into equal parts. The center of the star is center of the circle from which it came, and its points touch the circumference of the circle. The center of a circle is an apt symbol of a religon that emphasizes one God, and symbol of the role of Mecca, the center of Islam, toward which all Moslems face in prayer. The rays of a star reach out in all directions, making the star a fitting symbol for the spread of Islam. Many of the patterns used in Islamic art look similar, even though they decorate different objects. Islamic artists did not seek to express themselves, but rather, to create beautiful objects for everyone to enjoy. It takes considerable experience in analyzing Islamic patterns before discovering that seldom are two designs exactly alike. That is worrisome to Westerners because of the premium placed in the West on originality in evaluating an artist. Not so in Islam; there the artist sees himself as a humble servant of the community, using his skills and imagination to express awe of Allah, the one God, eternal and all-powerful.
Art Geometry and Abstract Culture
Computer Graphics, Art, Math, Geometry, and Abstract Sculpture are closely related. These activities are trying to transcend the boundaries between the fields. With images of models and sculptures.
Art, Geometry, Astract Sculpture Art, Geometry, and Abstract Sculpture Computer Graphics, Art, Math, Geometry, and Abstract Sculpture are closely related. In the activities below we are trying to transcend the boundaries between these fields. ART-MATH Conferences Art Exhibits Abstract Sculptures A small sampling of great contemporary sculptures. "Rainbow Bit" Sculpture The story of the CD-ROM sculpture in Soda Hall Scherk-Collins Sculpture Generator My original parameterized program to generate Scherk-Collins towers and toroids. Art by Carlo Squin A pictorial gallery of my own efforts with physical structures. FermiLab Art Exhibit Sculpture Models exhibited at the FermiLab Art Gallery from May 13 till July 7, 1998. Science - Art Exhibit Sculpture Models exhibited at "Science in the Arts -- Art in the Sciences", June, 1999. 2nd Generation Sculpture Generator An emerging library of modules providing more generality, built by Jordan Smith. Minimum-Variation Surfaces Work with Henry Moreton to find a functional that encodes the fairness of surfaces. Geometry and Topology Klein Bottles, Boy's Surface, and Zonohedra also have an aesthetic appeal. Links to Other Art Sites Galleries, Organizations, Individual Artists. Page Editor: Carlo H. Squin
Stewart Dickson's Art Page
Includes image galleries of 3-D images, a paper on tactile geometry, resume, virtual reality project and diversions.
MathArt.org, Stewart Dickson, Principal - Top Level Welcome to MathArt.org (This is the Mirror Site at http: emsh.CalArts.edu ~mathart ) Please visit the following areas: (Site map) About MathArt.org About Sculpture using Automated Fabrication About 3-D Color Prints About Tactile Mathematics Non-Spherical Geodesic Structures About Virtual Reality About Our Sponsors About Stewart Dickson Stewart Dickson's Portfolio Interactive Diversions: MathArt Lotto , Visual Dictionary , Body Talk , Sign , Visual Poetry and FractalMUD Software Related Links Please take a moment to complete our Reader's Response Form. Contact: Stewart Dickson, Sculptor 127 Claremont Road Oak Ridge, TN 37830 USA (865)241-3748(tel.) (865)220-5580(fax) mathart(a)emsh.calarts.edu (e-mail) Thank you for visiting MathArt.org.
Stix 'n Strings Mathematical Sculpture
Mathematical balance between the forces of compression and tension expressed as sculpture [site needs Java].
Search Directory Page Sponsored Links FretsOnly.com Mail order guitar music. Extensive range of Books, CDs, DVDs Videos! www.FretsOnly.com Elixir Strings Retailers Where you get guitar strings that deliver great tone for a long time www.elixirstrings.com Play Amazing Acoustic Play Acoustic Guitar like the Pro's In Under 1Hr Guaranteed. Amazing! www.fret2fret.com Martins Musik-Kiste Saiten fr alle Instrumente Musikzubehr, Noten DVDs www.martinsmusikkiste.de Saitenkatalog.de Saiten und Musikzubehr unglaublich gnstig und schnell saitenkatalog.de Search These Related Topics Guitar Strings John Pearse Acoustic Guitar Gibson Guitars Lead Guitar Bass Guitars Lacrosse Piano Strings Classical Guitar Martin Guitars Harp Strings Classical Guitars Banjo Strings Guitars Electro Acoustic Guitar Popular Categories Travel Cheap Flights Travel Insurance Hotels Financial Planning Loans Credit Cards Debt Consolidation E Commerce Web Hosting Broadband Domain Names People Search Dating Personals Background check Real estate Mortgages Home Insurance Home Equity Loans Try a Search: Trademark Free Zone , Review our Privacy Policy , Service Agreement , Legal Notice Copyright 2005 Network Solutions. All rights reserved. Buy a Domain as Low as $14.99 www. .com .net .org .biz .info .us FREE Search FREE 24x7 Support FREE online Account Manager Build Your Business: Get listed in top search engines Special web site hosting offer Forward visitors to your web site Incorporate Your Business Trademark Your Company's Name
Polyhedra and other Interesting Structures
Home made polyhedra and other constructions with interesting structural properties, specifically tension and rigidity.
www.pendred.net Your browser doesn't appear to support frames - please click here to view the site.
Ken Snelson
The art and tensegrity sculptures of Ken Snelson.
Preview Gallery: Kenneth Snelson Above: 1992 (click for 2000 ) Needle Tower II, Kroller Museum Museum in Otterlo, Holland Kenneth Snelson Kenneth Snelson's sculptures have done more to popularize the concept of tensegrity than anyone's. His large scale constructs show how compression members can provide rigidity while remaining separate, not touching one another, held in stasis only by means of tensed wires. By means of discontinuous compression and continuous tension, Snelson's multi-story towers and large scale amorphous exoskeletons of wire and steel, give dramatic, visible expression to the idea that tension and compression are the eternally complementary elements in any structure, and that great economy in materials may be achieved through strategies which rely on tension primarily, compression secondarily. In Fuller's synergetics, tensegrity becomes a metaphor for how Universe itself is constructed. 1949 Snelson crossed paths with Fuller in 1948 at Black Mountain College, where Fuller delivered one of his Dymaxion Seminars. Deeply inspired by Fuller, but working on his own in the fall of 1948 in Oregon, Snelson came up with his first prototype structures employing discontinuous compression. Snelson returned to Black Mountain College and shared his discovery with Fuller. Fuller saw deep implications in Snelson's discovery for his evolving Energetic Geometry and coined the term 'tensegrity' soon thereafter. Over time, Fuller stopped crediting Snelson for making a crucial contribution to synergetics, which left Snelson feeling that this major discovery had been snatched away by an egomaniac. The resulting rift between Fuller and Snelson never healed. Snelson felt his treatment by Fuller was symptomatic, part of a life-long pattern which revealed a flaw in Fuller's character that explains much of the resistence his ideas have encountered, in the scientific community and elsewhere (see Further Readings below). Snelson later turned his focus to the atom , finding in our growing understanding of the atom's structure a source of inspiration for a new series of dynamic models, sculptures and computer graphics. More pictures: Group picture at Black Mountain College, 1949 (53K) Mozart I (stereogram, sculpture 73K) Forest Devils' Moon Night (stereogram, computer graphic -- 55K) Easy K II (stereogram, sculpture -- 35K) Glass Atom (stereogram, computer graphic -- 32K) Needle Tower (stereogram -- 30K) Equilateral Quivering Tower (photograph -- 67K) Trigonal Tower (photo + archived communications) Gifts to the webmaster (magnets) Further readings: Portrait of an Atom by Ken Snelson Kenneth Snelson writes to R. Motro on the origins of "tensegrity" (November, 1990), International Journal of Space Structures Kenneth Snelson on Fuller (excerpt from Not in My Lifetime, an autobiography) Chris Fearley's Fuller FAQ sections 4.3.2 and 5.14 Other links: Official Kenneth Snelson home page -- since early June, 2002 Bob Burkhardt's pages re tensegrity Gerald de Jong's elastic interval geometry (EIG) Ken Snelson via email at k_snelson@mindspring.com maintained by Kirby Urner webmaster@grunch.net
GeomeTricks
Explore complex geometric structures based upon the 'basic joint'; an interconnectable joint composed of four intersecting sets of three triangular prisms. Includes assembly instructions, sculpture photographs and interactive virtual 3D models.
geometricks Select An Option Virtual 3D Forum Gallery Make Triangles Twelves Burning Man Commissions Contact ( ViewPoint 3D ) Requirements: Netscape 4.x Explorer 5.x Modem: Cable DSL Processor: 450MHz + Resolution: 800x600 + Color Depth: 16 bit + Explore complex geometric structures based upon the geometrick basic joint; an interconnectable joint composed of four intersecting sets of three triangular prisms. The site includes instructions for assembling the basic joint, photographs of real-world geometrick sculpture, and interactive virtual 3D models. The geometricks are organized in levels of complexity to challenge visitors of all ages. 3D, three dimensional, geometry, geometric, geometricks, construct, construction, triangle, triangular, prism, symmetry, symmetrical, math, mathematics, gallery, art, sculpture, interactive, virtual, VRML, metastream, viewpoint
Islamic and Geometric Art
A personal journey through aspects of geometric art with particular reference to Islamic design. The site includes full descriptions and methods for building often complex structures.
Islamic and Geometric Art by Tim Backhouse Islamic and Geometric Art by Tim Backhouse Welcome, thanks for calling. You have reached a site that is dedicated to the beauty that is Islamic and Geometric design. It has been designed with both entertainment and enlightenment in mind. You are free to wander around the site and simply view the delights that geometric design offers, or if, like the author, you are fascinated by the construction process an explanation of each design is also available. Thumbnail copies of the images are used on the main pages with larger versions available by clicking on them. The larger versions are usually accompanied by a description of the design process. The site features a number of projects together with some additional information, as follows:- The Seasons is an exploration of a single pattern, using differing colour systems to bring out a familiar image. The Gallery contains a collection of images, many of which are concerned with the latent 3-dimensionality often found in 2-D patterns. Borders relates to common Islamic borders and the ways in which they can be coded into HTML documents. Geometrical design as 'Art' - a discussion Deconstruction - analysing an Islamic Illumination About the artist - a short biography. Shop - buy some of the artists work. Feedback - send a message. Except where otherwise stated the copyright for all images on this site is retained by Tim Backhouse. You are free to use any of them for personal use, though an acknowledgement would be appreciated. Use of images for commercial purposes should be arranged with the artist. See The Shop for contact details or complete the Feedback Form .
Synergetic Geometry
Richard Hawkins' digital archive.
Richard Hawkins' Digital Archive A B MODULE HYPERTOONS PREVIEWS Tensegrity Amoeba TimeStar Tensegrity Jitterbug Archex Variation Miscellaneous ClockTet NEXT PAGE
Geometry Gallery
Spatial designs by Vedder Wright.
Gallery [ Home Page | Who am I? | Work | Resume | ] [ Polymorph | Geo-Gallery | Links | Parents | Comments | ] Forksphere Vann Twist Icosa Palmer Flower Tower Woven Stellated Icosa Five Tetra Palmer Flower Tricon Lamp Zonohedron Lamp Last modified on Tuesday, February 06, 2001. Copyright2000 by Vedder Wright ( vedder@TheWorld.com ).
Tips and Tricks to Gothic Geometry
Full explanatory diagrams for constructing your own rose window, ogee arch, and trifoil tracery.
Tips Tricks to Gothic Geometry Search Stone carving , architecture, art...and the Middle Ages Get Gothic. Design your own cathedral. HOME Feature Articles Stone Carver's Tour Virtual Cathedral Cathedral Tours Gothic Field Guide GOTHIC GEOMETRY Virtual Abbey Medieval Art Tours Castle Tours THE CARVING SHOP The Poster Store Gothic Greetings Screen Saver NEWSLETTER Links Resources HOMEWORK HELP Medieval Timeline About The Site FAQ Got a question about medieval art, history, or need solid direction on the road to the Middle Ages? Click HERE , or... TELL US and have your story appear at New York Carver... Front cover | Introduction | Sample Pages Introduction Ideal geometric shapes in architecture have imparted a feeling of order and harmony since the Greeks. The Romans, using only geometry and the repeated use of the semicircular arch, later built an empire. New innovations followed in the Middle Ages. The medieval flying buttress was born from the desire for building higher; and the pointed arch arose from the necessity of efficiently transferring the extra weight from above. Surprisingly, "Gothic" was first used as a term of derision by Renaissance critics who scorned the architectural style's lack of conformity to the standards of classic Greece and Rome. A closer look, however, reveals that the underpinnings of medieval architecture were firmly rooted in the ancient use of geometry and proportion. It's seen in the overall cruciform shape of a cathedral; in the rhythmic, intricate patterns found in stained glass windows; and in the rib vaulting that criss-crosses the ceiling. Up ahead you'll discover several hallmarks of Gothic design along with tips and tricks to their construction. For most of these designs, you will need paper, pencil, ruler and compass. A Zen-like contemplation of line and curve will naturally follow. From there, you can build on your experience to construct something real from the plans that you've mastered! Sacred or Gothic geometry serves as a door into the minds of Gothic masters, so it's strongly recommended that you give this historical reenactment a try - on paper at least, and where all great design begins... Sample Pages Trifoil Rose Window Ogee Arch Equilateral Arch Advertise with us NEW! Order your copy of Tips Tricks to Gothic Geometry from New York Carver. More than 30 pages of clear diagrams to Gothic design... ...with examples of the origins of historic medieval architecture... PLUS links that you can follow to further resources online... Learn more . All contents copyright 2003
Hart, George W. Geometric Sculpture of George W. Hart
Geometric sculptures displaying the beauty of mathematical forms in various media. The geometric sculpture of George W. Hart displays the beauty of mathematical forms in various media.
Geometric Sculpture of George W. Hart, mathematical sculptor Geometric Sculpture George W. Hart As a sculptor of constructive geometric forms, my work deals with patterns and relationships derived from classical ideals of balance and symmetry. Mathematical yet organic, these abstract forms invite the viewer to partake of the geometric aesthetic. I use a variety of media, including paper, wood, plastic, metal, and assemblages of common household objects. Classical forms are pushed in new directions, so viewers can take pleasure in their Platonic beauty yet recognize how they are updated for our complex high-tech times. I share with many artists the idea that a pure form is a worthy object, and select for each piece the materials that best carry that form. In one series of pieces, familiar objects are arranged in engaging configurations, displaying an essential tension between mundane individual components and the strikingly original totality. Because my works invite contemplation, slowly revealing their content, some viewers see them as meditation objects. A lively dancing energy moves within each piece and flows out to the viewer. The integral wholeness of each self-contained sculpture presents a crystalline purity, a conundrum of complexity, and a stark simplicity. Recent Work by George W. Hart This page shows some of my own favorite pieces. Click on any image for a larger image and mathematical description. As with all sculpture, 2D images can't really convey the presence they have in person, and it can be remarkable how differently they appear from different angles, so you'll have to extrapolate from the images here. Most everything shown below will fit through a doorway, but I am seeking sites and commissions for larger works. If you are interested, let me know... Fire and Ice oak and brass 18" currently at Vorpal Gallery I'd like to make one thing perfectly clear acrylic plastic 18" currently at Vorpal Gallery Battered Moonlight paper mache over steel 21 inches No Picnic spoons, knives, forks variable dimensions, up to 16 feet in length destroyed at Hofstra University, student center Yin and Yang walnut and basswood 7" diameter Disk Combobulation 3.5 inch floppy diskettes 12 inches at Goudreau Museum, New Hyde Park, NY The Color-Matched Dissection of the Rhombic Enneacontahedron acrylic plastic shell, acid-free card stock 20 inches 72 pencils 72 pencils 8 inches Gazmogenesis copper 12 inches Chiral Quartet installation at 2000 A.D. at the Vanderbilt museum, Centerport, NY, November 1997 - January 1998 Music of the Sphere steel 10 inches diameter, 36 tall The Plastic Tableware of Damocles 180 plastic knives 26 inches Rorschachohedron wood covered in synthetic fur, base of epoxy composite over steel, painted with pink enamel. 16 inches Stretto 204 CDs 26 inches Fat and Skinny walnut, maple, and brass 23 inches Chronosynclastic Infundibulum 150 CDs 34 inches Long Island Museum of Science and Technology Whoville aluminum 35 inches Labia 360 CDs 33 inch diameter Roads Untaken exotic hardwoods (yellowheart, paela, and padauk) with walnut "grout," 17 inch diameter Propello-octahedron 150 CDs 26" Princeton University Dept. Mathematics Leonardo Project Cherry reconstructions of Leonardo da Vinci's Models The Susurrus of the Sea transparent blue acrylic plastic 16" Septvaginta Duarus Planus Vacuus Cherry 16" currently at Vorpal Gallery Loopy painted aluminum 8 foot at Steinway Gallery , Chapel Hill, NC Rainbow Bits 642 CDROMs 77 inch diameter U.C. Berkeley, Dept. Computer Science Rainbow Bits Construction Millennium Bookball Wood and bronze, 5 foot diameter Commission for Northport Public Library Northport, NY. Community Assembly Event Gonads of the Rich and Famous 3D printing two balls, 3" each Five-Legged-Bee Hive 3D printing 3" Deep Structure 3D printing 4" Just Two Cavities 180 Toothbrushes 20" Twisted Rivers, Knotted Sea steel 4 foot diameter Tangled Reindeer solid freeform fabrication 3 inch This End Up acrylic plastic 9 inch Deep Sea Tango acrylic plastic 13 cm Quintessence of Hedgehog acrylic plastic 13 cm Bouquet acrylic plastic 9 inch Frabjous wood 11 inches The Triangles Which Aren't There acrylic plastic 12 inches Knot Structured wood 7 inches Compass Points wood 14 inches Compass Points (larger version) wood 46 inches Salamanders wood 30 inches Nessie wood 7 inches Gem wood 12 inches Star Corona acrylic 8 inches Aardvarks wood 12 inches Cagework 1 acrylic 7 inches Some of the above one-of-a-kind pieces are available for purchase, plus some limited edition acrylic sculptures . Contact me for more information or if you are interested in commissioning something special. You can also buy a set of four postcards of my sculpture. Public and Corporate Artworks Stony Brook University Northport Public Library (1999) Long Island Museum of Science and Technology (1999) U.C. Berkeley, Dept. Computer Science, Soda Hall (1999) Princeton University, Mathematics Dept. common room (1998) Goudreau Museum of Mathematics in Art and Science, New Hyde Park, NY (1998) Recent Group Shows Steinway Gallery , Chapel Hill, NC, Dec 2001-? Vorpal Gallery, Soho, New York City, (1999-2003) Towson College, Maryland, Bridges Exhibition (June-July 2002) Northport Museum, Northport, NY (May-August 2002) Art-trium, Melville NY, (Nov. 2001-Feb. 2002) Art Math 2001, Berkshire Community College, Pittsfield , Massachusetts, (Feb-March 2001) Art Mathematics 2000, The Cooper Union, NY, Nov. 7 - Dec. 15 2000. Colloquium on Math and Arts Exhibition, Maubeuge, France (Sept. 2000) Elaine Benson Gallery, Bridghampton, NY. (July 2000) Walt Whitman National Historic Birthplace Gallery, Huntington, NY (March-May 2000) Northport Museum, Northport, NY (May-July 2000) International Society of Art, Math, and Architecture, Exhibit at SUNY Albany (June 2000) Digital Soup II, Fayetteville Arts Council, N.C. (April-May 2000) b.j. Spoke Gallery, 299 Main St., Huntington, NY (Sept-Oct 1999) Mather Hospital Annual Outdoor Sculpture Show, Port Jefferson, NY (June-August 1999) Clayton Liberatore Gallery, Montauk Hwy., Bridgehampton, NY (May 1999) Northport Museum, Do Art: Summer Exhibition, Northport NY (May-July 1999) Soundview Gallery, Port Jefferson, NY (Nov. 1998-March 1999) 10th Annual Juried Fine Arts Exhibition at Chelsea Center, East Norwich, NY, Award of Excellence (Oct. 1998) Huntington Station Arts, Huntington, NY (Oct.-Nov. 1998) SOMA Gallery, Northport, NY, (Sept.-Oct. 1998) Math and Art 1998 Exhibition, U.C. Berkeley, California (August 1998) Bridges Exhibition, Southwestern College, Winfield, Kansas (July 1998) Northport Museum, Northport, NY (June-July 1998) Suffolk County Vanderbilt Museum, Centerport, NY (Dec. 1997-Jan. 1998) Hofstra University, Student Center (1997-2000) Math and Art 1997 Exhibition, SUNY Albany, NY (June 1997) Grants Awards Individual Artist Award, New York State Council for the Arts. 1999, to create Book Ball, a geometric construction of books celebrating the millennium. Award of Excellence, 10th Annual Juried Fine Arts Exhibition at Chelsea Center, 1998. Recent sculpture-related papers and presentations "In the Palm of Leonardo's Hand," Nexus Network Journal . "Rapid Prototyping of Geometric Models," Proceedings of Canadian Conference on Computational Geometry," University of Waterloo, August, 2001. ( online version ) "Computational Geometry for Sculpture," Proceedings of ACM Symposium on Computational Geometry, Tufts University, June 2001, pp. 284-287. ( PDF version ) "Loopy," Humanistic Mathematics, June, 2002, pp. 3-6. ( online version ) "The Geometric Aesthetic," to appear in The Visual Mind 2, Michele Emmer (ed.), MIT Press. Craig S. Kaplan and G. Hart, "Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons," Proceedings of Bridges 2001: Mathematical Connections in Art, Music, and Science, Southwestern College, Winfield, Kansas, July 2001, pp. 21-29. ( PDF version ) Douglas Zongker and G. Hart, "Blending Polyhedra with Overlays," Proceedings of Bridges 2001: Mathematical Connections in Art, Music, and Science, Southwestern College, Winfield, Kansas, July 2001, pp. 167-174. ( PDF version ) "Solid-Segment Sculptures," Colloquium on Math and Arts, Maubeuge, France, 20-22, Sept. 2000, and in Mathematics and Art, Claude Brute ed., Springer-Verlag, 2002. ( online copy ) "Sculpture based on Propellorized Polyhedra," Proceedings of MOSAIC 2000, University of Washington, Seattle, Aug 21-24, 2000. ( online copy ) "The Millennium Bookball," Proceedings of Bridges 2000: Mathematical Connections in Art, Music and Science, Southwestern College, Winfield, Kansas, July 28-30, 2000, and in Visual Mathematics Vol. 2 No. 3, 2000. ( US copy ) "Geometric Sculpture," ISAMA 2000, SUNY Albany, NY, June 24-28, 2000. "Reticulated Geodesic Constructions," Computers and Graphics 24(6), Dec. 2000. ( online version ) "Computer Modeling and Construction of Geometrical Sculpture," U.C. Berkeley, Feb. 1999. "Geometric Sculpture," Carpenter Center for the Visual Arts, Harvard University, Nov. 2, 1998. "Geometric Sculpture," New York Academy of Sciences, October, 1, 1998. "Polyhedra and Art," Art and Math '98, U.C. Berkeley, August 3-7, 1998. ( online version ) "Paper Prototype of a Geometric Sculpture: Whoville," (Invited workshop presentation) Art and Math '98, U.C. Berkeley, August 3-7, 1998. "Icosahedral Constructions," in Proceedings of Bridges: Mathematical Connections in Art, Music and Science, Southwestern College, Winfield, Kansas, July 28-30, 1998, pp. 195-202 (invited presentation). ( online copy ) "Zonish Polyhedra," Proceedings of Mathematics and Design '98, San Sebastian, Spain, June 1-4,1998. ( online copy ) Math awareness week, keynote speech, Rhode Island College, Providence, RI, April, 1998 "Polyhedra Models over the Internet", MAA Mathfest, Atlanta GA, August, 1997. "Calculating Canonical Polyhedra," Mathematica in Research and Education, Vol. 6 No. 3, Summer, 1997, pp. 5-10. ( online version ) "Zonohedrification," The Mathematica Journal, vol. 7 no. 3, 1999. ( online version for subscribers ) "Applications of Virtual Reality and Java for Illustrating Polyhedral Geometry over the Internet," Conference on Electronic Communication of Mathematics, Geometry Center, U. Minn. June 1997. "A Color-Matching Dissection of the Rhombic Enneacontahedron," Art and Math conference, S.U.N.Y. Albany, N.Y., June, 1997. ( online version ) "Virtual Reality Polyhedra," Art and Mathematics Conference, SUNY Albany, NY, June, 1996. Book about the Beauty of Geometry George W. Hart and Henri Picciotto, Zome Geometry: Hands-on Learning with Zome Models, Key Curriculum Press. ( more information ) More Check out a nice photo with one of my models which made the NY Times. Read an interview by Math Cats . Read an interview that appeared in studioNOTES . Read an interview that appeared in Ubiquity . See my entry at sculpture magazine . For background on the historical relationships between polyhedra and art, see the Polyhedra and Art section of my online Encyclopedia of Polyhedra . This was the topic of my talk at the 1998 Art and Math conference at Berkeley . I also make paper polyhedron models which are more at the mathematics end of the scale than the art end. Somewhere in between, I put my " geometric constructions ," which blend math and art more evenly. Finally, you might also like to look at some of my 2D computer-generated images , which are concepts for sculptures too difficult to realize physically, or some of my early plotter drawings . If you like this stuff, here are some links to some other geometric sculptors you may like: Brent Collins , Helaman Ferguson , Robinson Fredenthal , Bathsheba Grossman , Jean-Pierre Hbert , Chris Palmer , Charles O. Perry , John Robinson , Carlo Sequin , Arthur Silverman , Ken Snelson , Simon Thomas , Keizo Ushio , Koos Verhoeff . For more, see the International Society for Art, Math, and Architecture . There is also general information for about sculptors at Richard Collins' www.sculptor.org site. Search my pages. Copyright 1997, 1998, 1999, 2000, 2001, George W. Hart. All rights reserved.
The world of Escher
An online collection of the works of Escher, along with an online store.
World of Escher - Secure Shopping, Artwork Gallery, Tesselations Contests The place for everything Escher View Cart Checkout $ 0.00 Help Home Gallery Search Forum Newsletter Reading Contest Select an Artwork Another World II Ascending and Descending Balcony Belvedere Bond of Union Circle Limit IV Concave Convex Continuous Knot Cycle Day Night Division Double Planetoid Dragon Drawing Hands Encounter Eye First Day of Creation Fish and Scales Gravitation Hand with Reflecting Sphere Hell High Low Horseman; Reg Div Plane III House of Stairs Liberation Magic Mirror Man with Impossible Box Metamorphose I Metamorphose II Mobius Strip II Mosaic II Mummified Priests Periodic Design A13 Print Gallery Puddle Rabbit Regular Division of Plane I Relativity Reptiles Rind Sky Water I Snakes Snow Stars Still Life and Street Still Life with Sphere Sun and Moon Symmetry E105; Pegasus Symmetry E106; Bird Symmetry E110; Bird Fish Symmetry E114; Frog Fish Symmetry E117; Crab Symmetry E118; Lizards Symmetry E124; Lizard Symmetry E127; Bird Symmetry E128; Birds Symmetry E12; Butterfly Symmetry E21; Imp Symmetry E25; Lizards Symmetry E28; Three Birds Symmetry E32; Fish Symmetry E34; Bird Fish Symmetry E42; Shells Starfish Symmetry E47; Two Birds Symmetry E55; Fish Symmetry E59; Two Fish Symmetry E69; Fish Duck Lizard Symmetry E6; Camel Symmetry E70; Butterflies Symmetry E71; Twelve Birds Symmetry E76; Horse and Bird Symmetry E78; Unicorn Symmetry E84; Bird Fish Symmetry E88; Sea Horse Symmetry E89; Fish Symmetry E91; Beetle Symmetry E92; Two Birds Symmetry E9; Bird Three Spheres II Three Worlds Tower of Babel Verbum Waterfall White Cat Shop Online Posters Adult Shirts Kids Shirts Puzzles Silk Ties Books 2006 Calendars Mouse Pads Software Magnets Bookmarks Videos Miscellaneous What's New Gift Items Sale Items Clearance Items Send an E-Card Newsletter Sign up for our low volume newsletter. We give away a poster every month. We at the World of Escher are proud to be here to tell you stories, discuss M.C. Escher's works, provide insight, and offer our high quality products promoting the intriguing work of Escher. If you already know of Escher and his work you'll have a great time just looking around, otherwise it's time to get ready to explore a world as fascinating as the Internet; The World of Escher! T-Shirts Posters Puzzles Silk Ties Books Calendars Computer Misc. M.C. Escher was a Dutch graphic artist, most recognized for spatial illusions, impossible buildings, repeating geometric patterns (tessellations), and his incredible techniques in woodcutting and lithography. M.C. Escher was born June 1898 and died March 1972. His work continues to fascinate both young and old across a broad spectrum of interests. M.C. Escher was a man studied and greatly appreciated by respected mathematicians, scientists and crystallographers yet he had no formal training in math or science. He was a humble man who considered himself neither an artist or mathematician. Intricate repeating patterns, mathematically complex structures, spatial perspectives all require a "second look". In Escher's work what you see the first time is most certainly not all there is to see. Featured Three Worlds, 1955 Mini Poster Retail $7.95 Our Price $6.95 Sale Price $5.95 Concave Convex, 1955 Poster Retail $10.95 Our Price $9.59 Sale Price $8.95 To order by Phone, call 1-800-237-2232 Privacy and Security Our Policies Contact Us Terms of Use About Us FAQ 1995-2005 Worldofescher.com All Rights Reserved
Sculptures by Carlo H. Squin
Includes photos of knots and tangles, mathematical models of surfaces, and stereolithography models.
sculptures by Sequin SCULPTURE DESIGNS and MATH MODELS by Carlo H. Squin Recent Work (since July 2004) "Hilbert_512_3D" (July 2005) - Stainless Steel and Bronze, 5"-cube. "Borromean Torus" (June 2005) - 3D-Print, 4" diam. "DodecaHamCycle2" (June 2005) - 3D-Print, 5" diam. "K18_19HexTangle" (June 2005) - 3D-Print, 5" diam. "Icosahedral Volution Shell W-6T" (June 2005) - FDM, red, 3" diam. "Icosahedral Volution Shell W-4T" (June 2005) - 3D-Print, 4" diam. "Icosahedral Volution Shell J711-2T" (June 2005) - 3D-Print, 4" diam. "DodecaHamCycle2" (May 2005) - FDM yellow, 3" diam. "DodecaPentafoil Tangle" (April 2005) - Sintered Metal, 3.5" diam. "Arabic Icosahedron" (April 2005) - 3D-Print, 5" diam. "OctaTrefoil Cluster" (March 2005) - 3D-Print, 4" diam. "Alter-Alterknot" (Feburary 2005) - FDM Yellow, 4" diam. "TetraTrefoil Cluster" (Feburary 2005) - 3D-Print, 4" diam. "TetraTrefoil Tangle" (Feburary 2005) - FDM Black, 3.5" diam. "Triply Split Trefoil" (Feburary 2005) - FDM Yellow, 3.5" diam. "Tangle of Two Trefoils" (Feburary 2005) - FDM red, 3.5" diam. "Borromean Torus" (February 2005) - FDM, multiple colors, 4" diam. "Triply Twisted Moebius Space" (February 2005) - FDM, red + black paint, 4" diam. "Knot Divided" (January 2005) - Snow Sculpture, 12feet tall "Hex-split Torus" (January 2005) - FDM white, 4" diam. "Costa Surface in a Cube" (December 2004) - Bronze, 2 patinas, 5"x5"x5" "Icosahedral Volution Shell W-4T" (November 2004) - FDM, green, 3.5" diam. "Icosahedral Volution Shell - Scher4T" (October 2004) - FDM white, 3" diam. "Hamiltonian Bisections of the Platonic Solids" (Octobber 2004) - FDM, multiple colors "Double Hamiltonian Cycle" (September 2004) - 3D-Print, 8" tall "Dodecahedral Hamiltonian Cycle" (September 2004) - FDM Black, 3" diam. "Icosahedral Volution Shell Genus 2" (September 2004) - FDM, blue-white 3" diam. "Dodecahedral Volution Shell Genus 2" (September 2004) - 3D-Print, 4" diam. "Dodecahedral Volution Shell Genus 0" (September 2004) - FDM red, 2" diam. "Totem 3" (September 2004) - Bronze, 13" tall "Volution's Evolution" (August 2004) - 3 bronzes, 5" cubes Work 2003-2004 "K12 Graph on Genus-6 Surface" (June 2004) - 3D print, painted, 5"diam. "Genus-6 Kandinsky Surface" (June 2004) - 3D print, painted, 5"diam. "Congruent Hamiltonian forming Double Volution Shell" (May 2004) - 3D-Print, 5" diam. "3 Congruent Hamiltonian Paths on 4D Cross Polytope" (May 2004) - 3D-Print, 5" diam. "Congruent Hamiltonian Paths on 4D Simplex" (May 2004) - 3D-Print, 3" diam. "Congruent Hamiltonian Paths on Dodecahedral Double Shell" (May 2004) - 3D-Print, 3.5" diam. "Sphere Eversion - Stage 5" (March 2004) - 3DcolorPrint, 6" wide {with Alex Kozlowski} "Sphere Eversion - Stage 3" (March 2004) - 3DcolorPrint, 7" tall {with Alex Kozlowski} "Defying Gravity" (February 2004) - FDM, white, 5"x5"x7.5"tall. "Solar Arch" (December 2003) - Bronze, 12" diam. "Whirled White Web" (December 2003) - Bronze, 6" tall "Volution_5" (December 2003) - Bronze, 5" cube "Volution_0" (December 2003) - Bronze, 5" cube "Altamont" (December 2003) - Bronze, 4.5" diam. "Introverted Snowball" (November 2003) - 3DPrint, 6" tall "MorinMesh-Red Green" (October 2003) - 3DcolorPrint, 5" tall {with Alex Kozlowski} "MorinGrid" (September 2003) - 3DPrint, 4" tall "Volution_5" (September 2003) - FDM white, 5" cube "Volution_0" (August 2003) - FDM white, 5" cube "Maloja" (August 2003) - Bronze, 4.2" diam. "Infinit Duality" (August 2003) - another view - Bronze, 3" diam Selected Pieces "Volution_1" (June 2003) - Bronze, 5" cube "Gabo2X" (April 2003) - FDM white, 4" diam. "Antipodal Split Trefoil Knot" (April 2003) - FDM yellow, 3" diam. "Perspective Projection of the 600 cell" (March 2003) - 3DPrint, 8" diam. {with Mike Pao} "Cohesion" (2002) - Bronze, 12" "Galapagos-6" (2001) - FDM, white, 8" tall "Bonds of Friendship" (2001) - FDM, 10" tall "Moebius Space" (2000) - FDM black + silver paint, 4.2" diam. "Solar Arch" (1998) - FDM white, 10" diam. More Work, Sorted by Type Roads on the Globe : Max Bill Spheres : Moebius Structures : Scherk Collins Toroids : Polytopes : Math Surfaces : Knots and Tangles : Escher Spheres : Large Knots, Pre_1997:
Dunfield, Nathan
Caltech. 3-dimensional topology, geometry, and related topics.
Nathan Dunfield Nathan Dunfield My research interests are 3-dimensional topology, geometry, and related topics. I arrived here at Caltech in the summer of 2003. Previously, I spent four years at Harvard , and before that I was a graduate student at the University of Chicago . Research: Vita: PDF or Postscript . Publications, Preprints, and Slides. Data on the Virtual Haken Conjecture. Program to compute the boundary slopes of a 2-bridge or Montesinos knot. Additions to SnapPea, and some related tables . t3m: A box of tinker toys for topologists. Miscellaneous software . Interesting links: CompuTop: Links for computation in low-dimensional topology. Mark Brittenham's Low-dimensional topology links , including a list of home pages. The Geometry Junkyard . Beautiful knot pictures from Knot Plot and Morwen Thistlethwaite . Fun web games on tori and Klein bottles from Jeff Weeks . The sadly defunct Geometry Center . I'm a proud owner of an ACME Kleinbottle , as well as their Kleinbottle hat . Lego Escher . My father Chris . My brother Joshua . Julie Cidell: My spouse. It's a carpet that fires plastic darts! Classes, etc.: Nothing: Fall 2005 194c: Spring 2005 Geometry Seminar N + 2nd SCTC Old classes Office: Sloan 258 Phone: (626) 395-4339 Fax: (626) 585-1728 No email w o Javascript. Mathematics 253-37 Caltech Pasadena, CA 91125 PGP public key
DeLaVina, Ermelinda
University of Houston Downtown. Computational geometry - Graffiti. Publications and software.
index Dr. E. DeLaVina Associate Professor Office: S712 Phone: (713) 226-5241 Fax: (713) 221-8086 email: delavinae@uhd.edu Department of Computer and Mathematical Sciences University of Houston- Downtown One Main Street, Houston, TX 77002 Math 1301 Math 1306 Math 2305 2005 Schedule Senior Project Students and Other Activities Curriculum Vita Research Links of Interest
Ballmann, Werner
Rheinische Friedrich-Wilhelms-Universitt Bonn. Differential geometry; geometric topology.
Werner Ballmann Werner Ballmann Mathematisches Institut Rheinische Friedrich-Wilhelms-Universitt Bonn Beringstrae 1 D-53115 Bonn E-mail: hwbllmnn at obvious domain Fax: +49-228-737298 Sprechstunde: nach der Vorlesung Prfungstermine: nicht per email oder Telefon Lehre Teaching Skripten Lecture Notes Forschung Research Differential Geometry in Bonn Oberseminar Differentialgeometrie
Glazebrook, James F.
Eastern Illinois University and University of Illinois at Urbana-Champaign. Differential Geometry and its Applications to Mathematical Physics; Index Theory and Foliations; Holomorphic Vector Bundles; Noncommutative Geometry. Books, articles and preprints.
James F. Glazebrook James F. Glazebrook Professor: Department of Mathematics Eastern Illinois University 600 Lincoln Ave. Charleston, Illinois 61920-3099 Office: Old Main 338 (217) 581-5328 e-mail: cfjfg@eiu.edu Adjunct Professor: Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, Illinois 61801-2975 Office: Coble Hall B3 (217) 244 3288; FAX: (217) 333-9576 e-mail: glazebro@math.uiuc.edu General Information Ph.D. University of Warwick , 1985 Research Interests Differential Geometry and its Applications to Mathematical Physics; Index Theory and Foliations; Holomorphic Vector Bundles; Noncommutative Geometry Appointments and Visits Publications Books Journal Articles In Preparation Recent Presentations Membership
Anderson, Jim
University of Southampton. Hyperbolic geometry, mostly in dimensions 2 and 3, and its connections to other areas, such as the geometry and topology of 3-manifolds and Riemann surfaces. Preprints and teaching material.
Jim Anderson's Home Page Jim Anderson at work Professional I'm a Senior Lecturer in the School of Mathematics at the University of Southampton . My contact information, including snail-mail address, phone numbers, and e-mail, is here . I have links to my recent mathematics preprints (in PDF), as well as a list of papers published, with links to the journals where possible. I am a member of both the American Mathematical Society and the London Mathematical Society , both of which have informative web sites. There is also the information service site run by the European Mathematical Society . I am also currently co-editor of the Bulletin of the London Mathematical Society . My area of research interest these days is hyperbolic geometry , mostly in dimensions 2 and 3 (though I am occasionally tempted to explore higher dimensions every once in a while), and its connections to other areas, such as the geometry and topology of 3-manifolds and Riemann surfaces. Personal All my personal stuff has moved to my internet domain, defenestrati.org . Last modified: Thu Mar 3 14:20:20 GMT 2005
Sormani, Christina
Lehman College and CUNY Graduate Center. Riemannian reometry: manifolds with Ricci curvature bounds, their Gromov-Hausdorff limits and metric spaces.
Christina Sormani's Home Page Christina Sormani Department of Mathematics C.U.N.Y. Graduate Center 365 5th Avenue (34th St), NY NY 10016 Graduate Center Office: 4217.02 sormani (at) comet.lehman.cuny.edu Grad Center Office Hours: Thursdays 10-11 am Department of Math Computer Science, Lehman College , City University of New York Bronx, New York 10468 Lehman College Office: Gillet Hall 200B Lehman College Fax: (718) 960-8969 Lehman Office Hours: Monday Wednesday 12-1:30 pm I am a Riemannian Geometer and specialize in the study of manifolds with Ricci Curvature bounds, their Gromov-Hausdorff limits and Metric Spaces . At Lehman College, I am teaching Precalculus , chairing the Calculus Committee . and maintaining the webpage of the Math and Computer Science Club which has links to sites related to careers, graduate school and faculty members. At the C.U.N.Y. Graduate Center I am teaching Metric Geometry . I'm also coorganizing the upcoming CUNY Geometric Analysis Conference and the Differential Geometry and Lie Theory Seminar I'm the AWM web moderator for the forum on the Diverse Personal Lives of Mathematicians . I was not elected to the position of Member at Large in the AMS Council although I was defeated by only 10 votes. Research: What is Riemannian Geometry? A description for the nonmathematician The abstract of my NSF current grant. A list of my papers and how they can be downloaded. My Curriculum Vitae in pdf . Conferences and Seminars that I've organized and coorganized. The Poincare Conjecure (for undergraduates) and A webpage with notes on Perelman's papers Teaching: I created a webpage, Explore Geometry for Bronx High School teachers in conjunction with teaching MAT 636 in Fall 2002. I am chair of the Calculus Committee and maintain its webpage for use by faculty and students who want to consult the official syllabi. In 2002-2004, I coorganized the Lehman College Math Circle for high school students. Our top student has just been accepted at MIT! Maple Projects that I designed for Calculus I Laboratory at Lehman College. Intermediate Calculus I , MAT 226, Lehman College, Spring 2005. Intermediate Calculus II , MAT227, Spring 2003. Geometry , MAT346, Lehman College, Fall 2002 Noneuclidean Geometries , MAT636, Lehman College, Fall 2002 Intermediate Calculus I , MAT226, Lehman College, Fall 2002 Elements of Linear Algebra , Lehman College, Fall , 2000 (using Kolman). Calculus I Laboratory , MAT155, Lehman College, Fall, 2000 (using Larson). Calculus I , MAT175, Lehman College, Spring, 2000 (using Stewart). Calculus I Laboratory , MAT155, Lehman College, Spring, 2000 (using Stewart). Analysis II , Johns Hopkins, Spring, 1999. Analysis I, Johns Hopkins, Fall, 1998. Introduction to Differential Geometry , Johns Hopkins, Fall, 1998. Partial Differential Equations, Johns Hopkins, Spring, 1998. Riemannian Geometry , Johns Hopkins, Fall-Spring 1998. Some Projects for Calculus II that I designed for use at New York University. Introductory Lessons for Real Analysis in dvi , ps , pdf (sans pictures) that I designed for Math112 at Harvard University and used at Johns Hopkins as well. It was used for 1 month. See course description above. Family: My kids and just me and Kendall in 2005. My family in Fall, 2003. E-mail: sormani (at) comet.lehman.cuny.edu
Hales, Thomas C.
University of Pittsburgh. Kepler conjecture (announced a computer-aided proof), other space tiling conjectures, Langlands theory.
University of Pittsburgh: Thomas C. Hales Thomas C. Hales PhD, Princeton University Mellon Professor Representation theory, motivic integration, discrete geometry, honeycombs and foams. Thackeray 416 412-624-8375 hales@pitt.edu I suppose you are two fathoms deep in mathematics, and if you are, then God help you, for so am I, only with this difference, I stick fast in the mud at the bottom and there I shall remain. -Charles Darwin Don't use manual procedures -Andrew Hunt and David Thomas and ... don't rely on social processes for verification -David Dill Math 2810 syllabus Errata to Hulek's EAG homework (Sep 13) License This web site is part of the creative commons Research A list of publications The Kepler Conjecture , 2002 code update , 2003 update , 2004 update . (What is the densest arrangement of spheres in space?) The Dodecahedral Conjecture (2003 update) (Sean McLaughlin proved Fejes Tth's dodecahedral conjecture.) The Flyspeck Project (Become involved in the Formal Proof of the Kepler Conjecture.) Flyspeck Google Group Jordan Curve Theorem (tar of the formal proof) The honeycomb conjecture (What is the most economical way to partition the plane into equal areas?) Langlands theory and the Fundamental Lemma Links Google search on the Kepler Conjecture Google search on the Honeycomb Conjecture Google search on Flyspeck Mathematics Home | Pitt Home | Finding People | Top of Page
Rollin, Yann
Low-dimensional geometry.
Yann Rollin Yann ROLLIN Moore Instructor Royal Society University Research Fellow (starting oct. 2005) MIT Department of Mathematics 77 Massachusetts Avenue Building 2 Room 269 Cambridge MA 02139, USA E-mail: rollin (@math.mit.edu) Office: 2-269 Tel: +1 617 253 3778 Fax: +1 617 253 4358 Mathematical contents of this page : My Papers and Preprints Seminar on Seiberg-Witten Floer homology My Research interests Differential geometry, gauge theory and low dimensional geometry. In particular Seiberg-Witten theory, knot theory, and metrics with special holonomy. The department of mathematics at MIT Teaching Fall 2004: recitations for 18.02 (Multivariable Calculus) MW 10,1,2 Office hours W3-5 pm If you want to see the level curves of a two variables function, here is a nice example at Grand Canyon Alternatively, you can use this applet This picture shows the flux of a vector field through a pretzel Spring 2005: Introduction to Analysis (18.100A) Other links: Application material My PGP public key This is where you could meet me in a near future. Bookmarks hits to this page since october 1st 2004
Sullivan, John M.
Optimal geometries.
John M. Sullivan John M. Sullivan Professor in the Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, IL, USA 61801-2975 Campus Mail: 250 Altgeld Hall (MC-382) Office: 326 Illini Hall tel: +1-217-244-5930 fax: +1-217-333-9576 e-mail: jms@uiuc.edu Since July 2003, I have been on leave at TU Berlin . Please see also my homepage there. Research : Optimal geometry Annotated Bibliography: chronological , or by subject Online preprints Grant support Curriculum Vitae Graphics: Still images of bubbles, foams, etc. Videos : " The Optiverse ", " Knot Energies " Teaching: Teaching philosophy Spring 2003: Math 317 , Abstract Algebra, 10am MWF Previous semesters Further teaching information "Democracy abhors undue secrecy." Judge Marrero, 2005 Sep 29 Quotes and sources Adelphi Charter Leo Amer. Herit. Dict: Other Activities: Snow Sculpture 2004 , 2005 The International Society of the Arts, Mathematics, and Architecture ( ISAMA ) Essay on Optimal Geometry as Art , for Math Awareness Month 2003 . Essay on the beginning of the new millennium . MAA Session on Mathematical Connections in Art, Music, and Science , January 2003, Baltimore AMS Special Session on Optimal Geometry of Curves and Surfaces , October 2002, Madison AMS Special Session on Optimal Geometry , March 1999, Urbana Presentation on " Equal-tempered musical scales with few notes " at Bridges 2000. Public Lecture, Dec 8, 1998: " The Geometry of Foams ", in the series Mathematics in Science and Society Software: Java applets: stereographic projection ; parallel transport ; tight clasp ; panoramic photos . Interactive K-12 math: Eggmath UIUC Math Seminar Calendar Older software My GnuPG public key Outside hobbies , Photos
Kimberling, Clark
Triangle centers, integer sequences, mathematical history and biography.
Clark Kimberling Home Page Clark Kimberling Professor of Mathematics, University of Evansville, 1800 Lincoln Avenue, Evansville, IN 47722 Welcome to my home page. Here you'll find various mathematical and historical topics, in many cases matched with biographical sketches. The mathematical topics include triangle geometry and number theory. The section on triangle geometry probably contains the world's largest web collection of triangle centers. The number-theoretic topics deal mostly with integer sequences and arrays. The historical collection started out with 19th century scientists associated with the famous town, New Harmony, Indiana, located about 30 miles from Evansville. The collection has grown to include educators, writers, and artists. HAVE YOU SEEN from Key College Press: GEOMETRY IN ACTION (Click!) A pdf of The Shape and History of The Ellipse in Washington, D. C. (Click!) ENCYCLOPEDIA OF TRIANGLE CENTERS - ETC Triangle Centers: Classical and Recent Euler Line, including 102 triangle centers Algebraic Highways in Triangle Geometry Triangle Centers and Central Triangles (book) INTEGER SEQUENCES AND ARRAYS Neil Sloane's Encyclopedia of Integer Sequences Unsolved Problems and Rewards Interspersions and Dispersions Continued Fractions Fractal Sequences BIOGRAPHICAL STUDIES New Harmony Scientists, Educators, Writers Artists Emmy Noether, Her Mentors Colleagues Fibonacci Number-Theorists Triangle Geometers
Keith, Sandra Zaroodny
St. Cloud State University MN. Interests in visualisation and education.
Sandy Keith's Webpage SANDRA KEITH SANDRA ZAROODNY KEITH TWENTY QUESTIONS THAT STUDENTS COMMONLY ASK! CLICK HERE ADVISING SUGGESTIONS FOR WOMEN CLICK HER E Sandra Zaroodny Keith or Sandra Keith Mathematics , St. Cloud State U. , MN Engineering and Computer Center, Rm. 157 320-308-2282(office), 255-3001 (secr.), 253-9419 (home) FAX : 320-255-4269 St. Cloud State University, St. Cloud MN 56301 mailto:szkeith@stcloudstate.edu For the Humanistic Math Network Journal, for which I was managing editor, please click http: www2.hmc.edu www_common hmnj BACKGROUND: 1971 MA, Ph.D, University of Pennsylvia, Algebra 1966 BA, Brown University 1989-present St. Cloud State University, Professor 1975-76 University of Virginia: Visiting Lecturer in Applied Math 1973-75 Washington and Lee University, VA.: Assistant Professor 1974 Summer Employment at Aberdeen Proving Ground, MD. MY INTERESTS Mathematical: With co-authors Bonnie Gold and Bill Marion I've edited an MAA Notes volume, Assessment Practices in Undergraduate Mathematics . This book is the basis of an NSF grant which the MAA won for 2002-5, during which period the MAA will be pushing the idea of model assessment projects in math depts. nationally. I've published a study guide to the Ellis-Gulick Calculus text, Proceedings of the (national) Conference on Women in Mathematics and the Sciences (1989) and VISUALIZING LINEAR ALGEBRA WITH MAPLE (Prentice Hall). Partial Resume (sample articles) excerpts from book--see link above for more Personal: Besides teaching, I enjoy piano (Beethoven and ALL Russian composers), watercolors ("without a paddle" ) and portraits, jamming on music (family = piano quintet) with my 3 children all of whom are musicians and muses. Phil, (English prof. and VIOLIST) is my delightful husband, shown here relaxing... Jun 05 anyone who can web page, please save me....
Cherowitzo, Bill
Finite geometry. Department of Mathematics. University of Colorado at Denver.
Bill Cherowitzo's WWW Home Page Welcome Professional Research Teaching Contact Math Links Flocks of Cones Hyperoval Page Presentations Bill Cherowitzo's Home Page Welcome to Bill Cherowitzo's home page! william.cherowitzo@cudenver.edu Professor of the Mathematics Department , University of Colorado at Denver I'm so glad you've dropped by. I've organized my home page as follows: General Information Professional : My current curriculum vitae Research : Research interests and list of my research papers Teaching : My current teaching activity, gateway to all course pages Contact : Other ways to contact me (addresses, phone numbers, etc.), schedule, conference plans Math Links : Some useful Discrete Combinatorics Math Links Educational Policy and Planning Committee of Faculty Assembly Seminars UCD Geometry Seminar : Run by me and Stan Payne Algebraic Combinatorics Seminar : The Fort Collins seminar that Stan and I attend Research Web Pages Flocks of Cones : A research oriented web page, still under construction Hyperoval Page : Another research oriented page concerned with Hyperovals in Desarguesian Planes Last Updated: August 18, 2001 by Bill Cherowitzo
Hang, Fengbo
Veblen Research Instructor, Department of Mathematics, Princeton University. Subjects: geometric analysis, nonlinear partial differential equations, geometric measure theory.
Fengbo Hang Welcome to Fengbo Hang's Homepage Current address Dept. of Math Princeton University Fine Hall, Washington Road Princeton, NJ 08544-1000 Office : Fine 808 Phone : 609-258-6467 Fax : 609-258-1367 fhang@math.princeton.edu Research Interests Nonlinear partial differential equations Geometric analysis Publications Math Links Teaching MAT202, Fall 2005. Office hours: 1:30-2:30 p.m. Visitor Information Campus Map Google Mapquest NJ Transit Newark Airport Weather
Palais, Richard
Differential geometry, mathematical visualisation.
Richard Palais' Home Page Home Page of Richard S. Palais With my wife and frequent co-author, Chuu-lian Terng at the dedication of a memorial bust of Sophus Lie,at Lie's birthplace in Nordfjord, Norway. PhotobyS.Helgason In 1997, after 37 years as a member of the Brandeis Department of Mathematics , I retired to leave myself more time to work in the area of Mathematical Visualization and more specifically to continue the development of my Macintosh program 3D-Filmstrip (now called 3D-XplorMath ). In the Fall of 2004, my wife, Chuu-lian Terng , resigned from Northeastern Univ. to accept a position in the mathematics department at UCI (where she holds the Advance Chair) and we have now moved permanently to Irvine. I am continuing to work on mathematical visualization and in particular I am cooperating with David Eck of Hobart and William Smith College, helping with the design of his Java port of 3D-XplorMath, that will be called VMM---for The Virtual (or Visual) Mathematical Museum. However I have also partially "unretired" and I am now an Adjunct Professor of Mathematics at UCI , which means I will be teaching one or two courses per year. My long term research interests have been in the areas of: Compact Differentiable Transformation Groups Nonlinear Global Analysis Critical Point Theory (in particular Morse Theory) Submanifold Geometry Integrable Systems and Solitons In recent years I have become interested in mathematical visualization, and one of my major ongoing projects is the development and continued improvement of a program called 3D-XplorMath for MacOS X. This is a tool for aiding in the visualization of a wide variety of mathematical objects and processes. Based on what I have learned from my experience in writing this program, I wrote an essay called " The Visualization of Mathematics: Towards a Mathematical Exploratorium" that appeared in the June July 1999 issue of the Notices of the American Mathematical Society. With the help of Xah Lee , I have created a Gallery of visualizations produced using 3D-XplorMath. I have also recently been thinking about integrable, one-dimensional wave equations---an area that is usually referred to as soliton mathematics. I wrote an expository article for the Bulletin of the AMS (October 1997 issue) called The Symmetries of Solitons . My Curriculum Vitae and Bibliography of Published Works. Home Office 45 Murasaki Street Department of Mathematics Irvine, CA 92617 103 MSTB Cell: 949 468 7102 Irvine, CA 92697-3875 Vonage: 949 608 7367 Voice: 949 842 3151 Fax: 949 8427553 Click here to send me email at palais@uci.edu Created: May 25, 1995, Updated: August 6, 2005
Groe-Brauckmann, Karsten
Differential geometry, especially surfaces of constant mean curvature.
Karsten Grosse-Brauckmann: Research www.math.uni-bonn.de people kgb research.html K. Grosse-Brauckmann: Research Projects Much of my research is devoted to constant mean curvature (cmc) surfaces, in particular the construction of examples. Constant mean curvature surfaces appear in nature, in particular when the area of an interface is minimized under a volume constraint. Soap bubbles are the most popular example: The photos show Tom Noddy at the International Congress 1998 (courtesy of J. Sullivan ). Mathematicians have used the following methods to construct constant mean curvature surfaces: Kapouleas produced surfaces close to some degenerate well known surfaces with a singular perturbation approach; Pinkall, Sterling, and many others found tori as solutions of an integrable system (a more general approach by Dorfmeister, Pedit and Wu remains to be exploited); and the Lawson-Karcher conjugate cousin method yields sufficiently symmetric surfaces. Moduli Spaces of Embedded Constant Mean Curvature Surfaces with Finite Topology In this current project, which is joint with R. Kusner and J. Sullivan, constant mean curvature surfaces with ends are studied. Using the Lawson cousin construction we have a complete analysis of all embedded surfaces of genus 0 with 3 ends (triunduloids); it probably extends to surfaces with arbitrary many ends which admit a symmetry. Our main theorem is that the moduli space of triunduloids is homeomorphic to an open 3-ball. The trousers decomposition of arbitrary embedded constant mean curvature surfaces makes our result significant for the general case: indeed, we can understand each trouser as a truncated triunduloid, and the trouser is approximately described by a complete triunduloid. For the example displayed on the left 30 trousers compose to a cmc surface of genus 1 with 30 ends. Its special feature is that all ends are asymptotic to cylinders so that the total curvature is finite. Here is an interesting consequence of our description: A bubble generating loop in the moduli space of the 3-ended surfaces lets us expect the moduli spaces of the surfaces with k ends and genus 0 is connected. The Gyroid and its Constant Mean Curvature Deformations The gyroid is a triply periodic minimal surface discovered by A. Schoen (here is a translational fundamental domain ). It is contained in the associate family of the P- or D-surface, discovered in the 19th century by H.A. Schwarz. Currently the gyroid attracts the attention of material scientists. Various materials develop interfaces with the symmetry group Ia3d of the gyroid (the respective black white group is I4132). For modelling purposes, constant mean curvature surfaces are widely used, and so the following question arises: Can the minimal gyroid be deformed into constant mean curvature? This question was originally posed to M. Wohlgemuth and myself by E.L. Thomas (Dept. of Mat. Science, MIT). Constant mean curvature gyroid families were determined with rigorous methods, as well as in computer experiments using Ken Brakke 's Surface Evolver , Triply periodic minimal and constant mean curvature surfaces For many applications modelled with triply periodic surfaces (minimal, constant mean curvature, elastic, etc.) it is important to know about all surfaces with a given symmetry group, at least for low genus. It seems clear that genus can systematically be increased by adding handles. For instance, minimal surfaces with genus 3+12k , having all the symmetries of the primitive lattice Z3, seem to exist for k=0,1,2,... Indeed, B. Oberknapp's computations confirmed this for many cases (Oberknapp's surface shown at left belongs to a similar sequence). -- For a talk at MSRI in April 99 ( video , transparencies.ps ) I studied systematically the feasible ways of adding handles to generate a complete list of minimal surfaces with a given symmetry. In the example of the previous symmetry group, I suspect there are exactly seven minimal surfaces with genus at most 14 . A proof in terms of the bounding curves on a generating tetrahedron should be easy. Moreover, continuing the list further, it seems likely that all larger genera, except for genus 20, are actually attained by minimal surfaces. -- Each such minimal surface comes as a member of a one-parameter family of constant mean curvature surfaces. Indeed, this is a mathematical fact provided the surface is nondegenerate, i.e. the operator Laplacian+|A|2 (where A is the second fundamental form) has only the zero function in its kernel. As long as the latter is true, the constant mean curvature family continues. However, when the assumption fails we cannot say what happens in general; in the few known such cases a branching occurs. As an example, I computed two one-parameter families of constant mean curvature IWP-surfaces , having one surface in common (an existence proof with the conjugate surface method should be straightforward). Compact Constant Mean Curvature Surfaces Jointly with K. Polthier I conducted computer experiments to study symmetric compact cmc surfaces. We used the program S3 written by B. Oberknapp and K. Polthier, which is part of the software package grape , developed under the former Sonderforschungsbereich 256 at Bonn. While the existence of large surfaces with thin necks was proved in much generality by Kapouleas we concentrated on surfaces consisting of few bubbles with possibly large necks. In addition we assumed a large symmetry group. In the dihedrally symmetric case the simplest such surfaces turned out to exist only for finitely many genera, an example is the genus 5 surface to the right. We also find compact cmc surfaces with the symmetry group of the Platonic polyhedra, such as tetrahedral symmetry (left) or dodecahedral symmetry (here is part of and the almost complete surface of genus 30). Links Websites for images of constant mean curvature surfaces: M. Heil (TU Berlin): CMC tori J. Sullivan (U Illinois): compound soap bubbles and foams N. Schmitt (U Amherst) CMC surfaces created by an implementation of the Dorfmeister Pedit Wu method Some links to minimal surface images: U Bonn: Minimal surface library of the SFB 256 Ken Brakke (U Susquehanna): Triply periodic minimal surfaces and their Surface Evolver input files D. Hoffman, J. Hoffman (MSRI, Berkeley): Scientific Graphics Project Konrad Polthier (TU Berlin): Surface Images Francisco Martin (Granada): Surface Images An interactive program to create surface images: (TU Berlin): Java View Back to main page K. Grosse-Brauckmann. Last update 9 02. For permission to use my images, please contact kgb@math.uni-bonn.de
Doran, Charles
Columbia University. Geometry, mathematical physics, number theory.
Charles Doran. Mathematics. University of Washington, Seattle Assistant Professor of Mathematics University of Washington Padelford Hall C-417 Seattle, Washington 98105 Office: (206) 543-7386 Email: doran@math.washington.edu
Dodson, C.T.J. (Kit)
UMIST, Manchester. Differential geometry, stochastic geometry and applications.
Professor C.T.J. Dodson Professor C.T.J. Dodson School of Mathematics , University of Manchester , Sackville Street, Manchester M60 1QD, UK Welcome to Kit Dodson's homepage Research and preprints: differential geometry and stochastic geometry ; books . Other interests: walking , windsurfing , sailing , and local history . Email: ctdodson@manchester.ac.uk Some of my other webpages: PDF LaTeX Tutorial for scientific wordprocessing Notes On Making Mathematical Notes For Your Course Differential Geometry course notes Curves and Surfaces course notes Mathematics for Mechanical Engineering Students course notes Introduction to Algebra and Calculus course notes; On-line Courses and Courseware; School of Mathematics Homepage Our Klein Bottle Mural Research Interests + Preprints Differential geometry : Global differential geometry of manifolds; spaces of connections; universal connections; Banach manifolds and bundles; harmonic lifts and maps. Applications: pseudo-Riemannian geometry and general relativistic cosmology; geometry of parametric statistical models; information geometry and information topology. Books Stochastic geometry : Characterization of spatial statistics of assemblages of discrete objects like lines, rectangles discs, cylinders; quantification of small departures from random or chaotic states. Applications: Structure of stochastic porous media and its fluid transport properties. Links: London Mathematical Society + American Mathematical Society + Mathematics in the Media European Mathematical Society + Biographies of Great Mathematicians + Journal of Geometry and Physics General Relativity Geometry + Mathematical Moments--Mathematics in Science, Technology and Society Eric Weisstein's World of Mathematics + Geometry Center Experiments + Luis Cordero's Geometry Gallery
Chang, Sun-Yung Alice
Director of Graduate Studies, Department of Mathematics, Princeton University. Subjects: geometric analysis, algebraic geometry, differential geometry.
Sun-Yung Alice Chang Sun-Yung Alice Chang Department of Mathematics, Princeton University email: chang@math.princeton.edu Office Phone: 609-258-5114 MathSciNet Home Page Recent preprints (Differential Geometry) Recent preprints (Analysis of PDE ) Preprints: Sun-Yung A. Chang , Jie Qing and Paul Yang, ``On finiteness of Kleinian groups in general dimension," preprint 2002.[ pdf] Sun-Yung A. Chang, Zheng-Chao Han and Paul Yang, ``Classification of singular radial solutions to the $\sigma_k$ Yamabe equation on annular domains'', preprint 2004.[ pdf ] Sun-Yung A. Chang, Jie Qing and Paul Yang, ``On the renormalized volumes for conformally compact Einstein manifolds'', preprint, 2004. ______________________________________________________________________________________________________________________ Lecture Notes: Sun-Yung A. Chang, " Conformal Invariants and Partial Differential Equations", Colloquium Lecture Notes, AMS 2004, Pheonix meeting.[ pdf ] Sun-Yung A. Chang, " Conformal Invariants and Partial Differential Equations", Colloquium Lecture Transparencies, AMS 2004, Pheonix meeting. [ pdf ] [ pdf ] [ pdf ] Sun-Yung A. Chang, "Non-linear Elliptic Equations in conformal Geometry", Nachdiplom Lectures Course Notes, ETH, Zurich. To be published by Springer-Birkh\"auser.[ pdf ] List of Publications S.Y.A. Chang, "A characterization of Douglas subalgebras," Acta Math. 137 (1976) pp. 81-89. S.Y.A. Chang, "On the structure and characterization of some Douglas subalgebras," Amer. J. Math. 99 (1977) pp. 530-578. S.Y.A. Chang, "Structure of subalgebras between $L^\infty$ and $H^\infty,"$ Trans. Amer. Math. Soc. 227 (1977) pp. 319-332. S.Y. Chang and D.E. Marshall, "Some algebras of bounded analytic functions containing the disk algebra," in "Banach Spaces of Analytic Functions, J. Baker, C. Cleaver and J. Diestel (eds.), Lecture Notes in Mathematics, Vol. 604, Springer-Verlag, (1977) pp. 12-20. S.Y.A. Chang and J.B. Garnett, "Analyticity of functions and subalgebras of $L^\infty$ containing $H^\infty,$" Proc. Amer. Math. Soc. 72 (1978) pp. 41-45. S. Axler, S.Y.A. Chang and D. Sarason, "Products of Toeplitz operators," Integral Equations and Operator Theory 1 (1978) pp. 285-309. S.Y.A. Chang, "Structure of some subalgebra of $L^\infty$ of the torus," in "Harmonic Analysis in Euclidean Spaces", Proceedings of Symposia in Pure Mathematics, Vol. 35, Amer. Math. Soc., (1979) pp. 421-425. S.Y.A. Chang, "Carleson measure on the bi-disc," Ann. of Math. 109 (1979) pp. 613-620. S.Y.A. Chang and R. Fefferman, "On a continuous version of duality of $H^1$ with BMO on the bidisc," Ann. of Math. 112 (1980) pp. 179-201. S.Y.A. Chang, "A generalized area integral estimate and applications," Studia Math. 69 (1980) pp. 109-121. A. Chang, M.M. Schiffer and G. Schober, "On the second variation for univalent functions," J. Analyse Math. 40 (1981) pp. 203-238. S.Y.A. Chang and R. Fefferman, "The Calder\'on-Zygmund decomposition on product domains," Amer. J. Math. 104 (1982) pp. 455-468. S.Y.A. Chang, "Two remarks about $H^1$ and BMO on the bidisc," in "Proceedings of Conference on Harmonic Analysis in Honor of Antoni Zygmund", Vol. II, Beckner, Calder\'on, Fefferman and Jones (eds.), Wadsworth, 1983, pp 373-393. S.Y.A. Chang and Z.Ciesielski," Spline characterization of $H^1,"$ Studia Math. 75, (1983) pp. 180-192. S.Y.A. Chang and R. Fefferman, "Some recent developments in Fourier analysis and $H^p$ theory on the product domains," Bull. Amer. Math. Soc. 12, No. 1 (1985) pp. 1-43. S.Y.A. Chang, M. Wilson and T. Wolff, "Weighted norm inequalities related to Schroedinger operators," Comm. Math. Helv. 60, (1985) pp. 217-246. S.Y.A. Chang and D. Marshall, "On a sharp inequality concerning the Dirichlet integral," Amer. J. Math. (1985) pp. 1015-1033. T. Carbery, S.Y.A. Chang and J. Garnett, "Weights and L log L," Pacific J. Math. 120, (1985) pp. 33-45. L. Carleson and S.Y.A. Chang, "On the existence of an extremal function for an inequality of J. Moser," Bull. Sci. Math. 110, (1986) pp. 113-127. B. Berndtsson, S.Y.A. Chang and K.C. Lin, "Interpolating sequences on the polydisk," Trans. Am. Math. Soc. 302, (1987) pp. 161-169. S.Y.A. Chang and P.C. Yang, "Prescribing Gaussian curvature on $S^2$", Acta Math. 159 (1987), pp 215-259. S.Y.A. Chang, "Extremal function in a sharp form of Sobolov inequality", in Proceeding of International Congress of Mathematician, I.C.M. 1986, pp. 715-723. S.Y.A. Chang and P.C. Yang, "Conformal deformation of metrics on S$^2$," J. Diff. Geometry 27, (1988) pp. 259-296. S.Y.A. Chang and P. C. Yang, "Compactness of isospectral conformal metrics on S$^3$", Comm. Math Helv., 64, (1989) pp. 363-374. S.Y.A. Chang and P.C. Yang, "The conformal deformation equation and isospectral set of conformal metrics", Contemporary Mathematics, Vol. 101, Vol. 101, 1989, pp 165-178. S.Y.A. Chang and P.C. Yang, "Isospectral Conformal Metrics on 3-manifolds" J. of AMS 3, (1990) pp. 117-145. Sun-Yung A. Chang and Paul C. Yang, "A perturbation result in prescribing scalar curvature on $S^n$," Duke J. of Math. 64 (1991) 27-69. Thomas Branson, Sun-Yung A. Chang and Paul C. Yang, "Estimates and extremal problems for the log-determinant on 4-manifolds,". Comm. Math.Physics, Vol. 149, No.2, 1992, pp 241-262. Sun-Yung A. Chang and Paul C. Yang, "Spectral invariants of conformal metrics," Proceedings of Satellite Conferences of the International Congress of Math., Kyoto, (1990), pp 51-60. Sun-Yung A.Chang, Matt Gursky and Paul C.Yang, " Prescribing scalar curvature on $S^2$ and $S^3$", Calculus of Variation, 1, 1993, pp 205-229 [ ps ]. Sun-Yung A. Chang, Matt Gursky and Thomas Wolff, "Lack of compactness in conformal metrics with $L^(d 2)$ curvature", Journal of Geometric Analysis, Vol 4, No.2, 1994, pp 143-154. Sun-Yung A. Chang and Paul C.Yang, Addendum to " A Perturbation result in prescribing scalar curvature on $S^n$", Duke Math J. 71, 1993, pp 333-335. Sun-Yung A. Chang, "Moser Trudinger inequality and applications to some problems in conformal geometry", Nonlinear Partial Differential Equations, Hardt and Wolf editors, AMS IAS Park city Math series, vol 2, pp 67-125 [ ps ]. Sun-Yung A. Chang and Paul C.Yang," Extremal metrics of zeta function determinants on 4-manifolds", Annals of Math. 142 (1995), pp 171-212. Sun-Yung A. Chang, Xing-Wang Xu and Paul C.Yang, "Perturbation Results for Prescribing Mean Curvature ", Math. Annalen 310 (1998) pp 473-496. Sun-Yung A. Chang, Matt Gursky and Paul C.Yang, " Remarks on a fourth order invariant in conformal geometry", Aspects of Mathematics, edited by N. Mok, Hong Kong University, Proceedings of a conference in Algebra, Geometry and Several Complex Variable, (June, 1996) pp. 353-372. Sun-Yung A. Chang and Jie Qing, "Zeta functional determinants on manifolds with boundary ", Research announcement, Math. Research Letters, 3 (1996), {\bf 3} (1996) pp. 1-17. Sun-Yung A. Chang and Jie Qing, "The Zeta functional determinants on manifolds with boundary I--the formula", JFA 147, No. 2, (1997), pp 327-362 Sun-Yung A. Chang and Jie Qing, "The Zeta functional determinants on manifolds with boundary II--Extremum metrics and compactness of isospectral set", JFA 147, No. 2, (1997), pp 363-399 Sun-Yung A. Chang and Paul C.Yang, " Extremal metrics of zeta functional determinants", Survey article, Geometry from the Pacific Rim, Eds.: Berrick Loo Wang, Walter de Gruyter Co., Berlin, New York 1997, pp 37-57. Sun-Yung A. Chang, "On zeta functional determinants", Lecture notes in the Proc. of Summer Lecture Series in Analysis, Fields Institute, Toronto, 1995 [ ps ]. Sun-Yung A. Chang and Paul C. Yang, "On uniqueness of solution of a n-th order differential equation in conformal geometry", Math. Research Letters, 4 (1997) pp. 91-102.[ pdf ] Sun-Yung A. Chang, " On Paneitz operator--a fourth order differential operator in conformal geometry", Harmonic Analysis and Partial Differential Equations; Essays in honor of Alberto P. Calderon, Editors: M. Christ, C. Kenig and C. Sadorsky; Chicago Lectures in Mathematics, 1999, Chapter 8, pp 127-150 [ ps ]. Sun-Yung A. Chang, Matt Gursky and Paul C.Yang, " Regularity of a fourth order PDE with critical exponent", American Journal of Math, 121 (1999), pp 215-257 [ ps ]. Sun-Yung A. Chang, Lihe Wang and Paul C. Yang, "Regularity of harmonic maps", CPAM, LII (1999), pp. 1099-1111 [ ps ]. Sun-Yung A. Chang, Lihe Wang and Paul C. Yang, " Regularity of bi-harmonic maps to spheres", CPAM, LII (1999), pp. 1113-1137 [ ps ]. Sun-Yung A. Chang and Paul C. Yang, "On a fourth order curvature invariant ", Comp. Math. 237, Spectral Problems in Geometry and Arithmetic, Ed: T. Branson , AMS, (1999) pp 9-28 [ ps ]. Sun-Yung A. Chang, Jie Qing and Paul C. Yang, "On the Chern-Gauss-Bonnet integral for conformal metrics on $R^4$ ", Duke Math Journal, vol 103, No. 3 (2000), pp 65-93 [ ps ]. Sun-Yung A. Chang, Jie Qing and Paul C. Yang, "Compactification for a class of conformally flat 4-manifold ", Inventiones Mathematicae, 142 (2000), pp 65-93 [ ps ]. Sun-Yung A. Chang and Paul C. Yang, "Fourth order equations in conformal geometry", Seminaires et Congres 4 (2000), Global Analysis and Harmonic Analysis, J.P. Bourguignon, T. Branson, O. Hijazi (Eds); pp 155-165 [ ps ]. Sun-Yung A. Chang and Wenxiong Chen, "A note on a class of Higher order conformally covariant equations,'' Discreet and Continuous Dynamical systems, Vol. 7, No. 2, (April 2001), pp. 275-281. [ ps ]. Sun-Yung A. Chang, Matt Gursky and Paul Yang, ``An equation of Monge-Ampere type in conformal geometry, and four-manifolds of positive Ricci curvature,'' {\it Annals of Math.}, Vol. 155, No. 3, May (2002) pp. 711-789. [ ps ]. Sun-Yung A. Chang and Paul Yang, ``Partial Differential Equations related to the Gauss-Bonnet-Chern integral on 4-manifolds,'' Conformal, Riemannian and Lagrangian Geometry, {\it AMS}, University lecture series Vol 27. The 2000 Barrett Lectures, pp 1-29. [ ps ] Sun-Yung A. Chang, " An article in memory of Thomas Wolff", {\it Notices AMS}, Vol. 48, No. 5, (2001) pp. 485-486. Sun-Yung A. Chang , Jie Qing and Paul Yang, ``On finiteness of Kleinian groups in general dimension," preprint 2002, to appear in J. fur die Reine und Andgewandte Math.[ pdf ] Sun-Yung A. Chang, Matt Gursky and Paul Yang, "A prior estimate for a class of nonlinear equations on 4-manifolds," Journal D'Analyse Journal Mathematique, special issue in memory of Thomas Wolff, Vol 87 (2002) , pp. 151-186.[ ps ] Sun-Yung A. Chang, C.C. Chen aand C. S. Lin, ``Extremal functions for a mean field equation in two dimension,'' Lecture on Partial Differential equations in honor of Louis Nirenberg's 75th birthday, Editors S.-Y. A. Chang, C.-S. Lin and H.-T. Yau,Chapter 4, International Press, 2003, pp 61-94. [ pdf ] Sun-Yung A. Chang, Matt Gursky and Paul Yang ``Entire solutions of a fully non-linear equation,'' Lecture on Partial Differential Equations in honor of Louis Nirenberg's 75th birthday, Editors S.-Y. A. Chang, C.-S. Lin and H.-T. Yau, Chapter 3, International Press, 2003, pp 43-60.[ pdf ] Sun-Yung A. Chang and Paul Yang, ``The inequality of Moser and Trudinger and applications to conformal geometry,'' Comm. Pure and Applied Math., Vol LVI, no. 8, August 2003, pp 1135-1150; Special issue dedicated to the memory of Jurgen K. Moser. [ pdf ] Sun-Yung A. Chang , Matt Gursky and Paul Yang , ``A conformally invariant spheretheorem in four dimensions'', Publications de l'IHES, no, 98, 2003, pp. 105-143.[ pdf ] Sun-Yung A. Chang and Paul Yang, ``Non-linear Partial Differential equations in Conformal Geometry", Proceedings for ICM 2002, Beijing, volume I, pp 189-209.[ pdf ] Sun-Yung A. Chang, Jie Qing and Paul Yang, ``On the topology of conformally compact Einstein 4-manifolds", Noncompact Problems at the intersection of Geometry, Analysis and Topology; Contempoary Math. volume 350, 2004, pp 49-61. [ pdf ] Sun-Yung A. Chang , Fengbo Hang and Paul Yang , `` On a class of locally conformallyflat manifolds", IMRN 2004, no. 4, pp 185-209.[ pdf ] Sun-Yung A.Chang, J. Qing and P. Yang, ``On the renormalized volumes for conformally compact Einstein manifolds'', preprint, 2003. Sun-Yung A. Chang , ''Nonlinear elliptic equations in conformal geometry'', Nachdiplom lectures in Mathematics, ETH, Zurich, To be published by Springer-Birkh\"auser. [ pdf ] Sun-Yung A. Chang, ''Conformal Invariants and Partial Differential Equations'', Colloquium Lecture notes, AMS, Phoenix 2004. [ pdf ] Sun-Yung A. Chang, J. Qing and P. Yang, ``On a conformal gap and finiteness theorem for a class of four manifolds", preprint 2004. Sun-Yung A. Chang , Z.C. Han and P. Yang, ``Classification of singular radial solutions to the $\sigma_k$ Yamabe equation on annular domains'', preprint 2004.[ pdf ]
Calegari, Danny
Specializes in topology and classical geometry. Department of mathematics. California Institute of Technology.
Danny Calegari's Home Page Danny Calegari's Home Page Danny Calegari 251 Sloan Department of Mathematics California Institute of Technology Pasadena CA 91125 (626) 395 4360 Contents Curriculum Vitae Publications and work to appear Papers in progress and submission Book project Miscellaneous mathematics and programs Course and seminar home pages Non-mathematical Links Schedule Curriculum Vitae My Curriculum Vitae is available in TeX , pdf , or HTML . Publications and work to appear Short Stories (excerpts) A Green Light - short story, winner of The Age short story competition, published January 1993 The Rubbernecks - short story, published in Quadrant magazine, January 1994 Pantopia(tm) - short story, published in Southerly quarterly journal, June 1994 The Intermediary - short story, published in Overland magazine, Summer 1993 Mathematical Papers Note: Papers are listed in the order in which they were accepted for publication. .tex files are LaTeX; .tar files contain .eps figures. Uncompress .gz files with gunzip filename.tar.gz and then .tar files with tar -xvf filename.tar Strong geometric isolation in 3-orbifolds Bull. Austral. Math. Soc. 53 (1996), no.2, 271-280 Foliations transverse to triangulations of 3-manifolds Comm. Anal. Geom. 8 (2000), no.1, 133-158 A degree one Borsuk-Ulam theorem Bull. Austral. Math. Soc. 61 (2000), no.2, 267-268 R-covered foliations of hyperbolic 3-manifolds Geom. Topol. 3 (1999), 137-153 Distortion of leaves in product foliations Top. Appl. 124 (2002), no.2, 205-209 Napoleon in isolation Proc. Amer. Math. Soc. 129 (2001), no.10, 3109-3119 Geometry and topology of R-covered foliations Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 31-39 The Gromov norm and foliations Geom. Func. Anal. 10 (2000), no.6, 1423-1447 The Geometry of R-covered foliations Geom. Topol. 4 (2000), 457-515 With N. Dunfield - Commensurability of 1-cusped hyperbolic 3-manifolds Trans. Amer. Math. Soc. 354 (2002), no. 7, 2955-2969 Leafwise smoothing laminations Algebr. Geom. Topol. 1 (2001), 579-585 Almost continuous extension for taut foliations Math. Res. Lett. 8 (2001), no. 5-6, 637-640 Problems in foliations and laminations of 3-manifolds Proc. Symp. Pure Math. 71 (2003) 297-335 Foliations with one-sided branching Geom. Dedicata 96 (2003), 1-53 Every orientable 3-manifold is a BG Algebr. Geom. Topol. 2 (2002), 433-447 With N. Dunfield - Laminations and groups of homeomorphisms of the circle Invent. Math. 152 (2003) no. 1, 149-204 Dynamical forcing of circular groups Trans. Amer. Math. Soc. to appear version 5 24 2004 Circular groups, planar groups, and the Euler class Geom. Topol. Mon. 7 (Proceedings of the Casson Fest) (2004), 431-491 With N. Dunfield - An ascending HNN extension of a free group inside SL(2,C) Proc. Amer. Math. Soc. to appear version 6 6 2005 With D. Gabai - Shrinkwrapping and the taming of hyperbolic 3-manifolds Jour. Amer. Math. Soc. to appear version 10 22 2005 Papers in progress and submission Gershenfeld's Law on Writing: Good papers are never finished, just abandoned. Promoting essential laminations version 8 20 2004 Bounded cochains on 3-manifolds version 11 26 2001 Groups with non-negative combinatorial Ricci curvature version 4 7 2003 Universal circles for quasigeodesic flows version 5 4 2005 NEW May 4th 2005 With M. Freedman, and an appendix by Y. de Cornulier - Distortion in transformation groups version 10 7 2005 NEW October 7th 2005 Real places and torus bundles version 10 27 2005 NEW October 27th 2005 Slides from my talk at the Ahlfors-Bers Colloquium. Notes: The last section of Useful branched surfaces has been re-written and expanded as Bounded cochains. I have proved conjecture 4.9, the "coarse Scott core theorem" in Bounded cochains; the paper is being revised to reflect this. update 4 7 2003: My plan is to write up this result eventually. The material in my PhD thesis is somewhat dated. A more up to date version of (most of) the original content is available in my published and accepted papers. A more polished version of the background notes on foliations is available here . The paper on non-negative combinatorial Ricci curvature answers a question of Yau. The result is almost trivial, but short, and is similar enough in spirit to the Cheeger-Gromoll splitting theorem to make me think it was worth writing it up. Circular groups, Planar groups and the Euler class is a complete rewrite of Planar groups and circular groups. The stuff on quasigeodesic flows and applications to 3-manifolds has been taken out, and is contained in Universal circles for quasigeodesic flows. What the press are saying . My Dissertation: Foliations and the Geometry of Three-Manifolds - postscript file of my dissertation for a PhD at UC Berkeley, Spring 2000 Book project I am writing a book, tentatively called Foliations and the geometry of 3-manifolds, which will be published eventually by Oxford University Press in their Mathematical Monograph series. The aim of the book is to give an exposition of the so-called "pseudo-Anosov" theory of foliations of 3-manifolds, and their relation to the underlying geometry of the manifold, especially hyperbolic geometry. The content of the book will include an exposition of some important work of people such as Candel, Fenley, Gabai, Mosher, Thurston and others, as well as some of my own work. A preliminary table of contents is available. Miscellaneous mathematics and programs "No problem is too small or too trivial if we can really do something about it." Richard Feynman Lecture notes for a short course given at Melbourne University in January 1999 on Foliations and 3-Manifolds has been included, with substantial corrections, in my dissertation. Lecture notes for a short course given at Tokyo Institute of Technology in September 2000 is here A short proof of Craig Hodgson's theorem that surgeries on a cusped manifold produce incommensurable manifolds for all but finitely many surgeries is here Notes on Chern-Simons invariants of hyperbolic manifolds can be found here Notes on Scissors Congruence and Dehn invariants are here How to perturb a taut foliation to a symplectically fillable contact structure here The Qual. Questions page is now to be found here Jason Manning maintains the seminar page for the geometry topology seminar here at Caltech. Miscellaneous mathematical notes here . Java applets and other programs Foliate the plane Java applet Mandelbrot set explorer Java applet Thin Obstacle Problem Java applet Invariant lamination calculator Java applet iknot X11 program. lamination X11 program. Course and seminar home pages Harvard 2000-2002: Math. 138 Classical Geometry Fall 2000 Math. 139 Classical Geometry and Low-Dimensional Topology Spring 2001 Geometrization seminar Spring 2001 Math. 277x Foliations and the topology of 3-manifolds Fall 2001 Math. 278x Progress towards Geometrization Spring 2002 Caltech 2003-: Math. 191e Geometry of infinite groups Winter 2003 Math. 157b Introduction to 3-manifolds Spring 2003 2003 Southern California Topology Conference Math. 191a Foliations and 3-manifolds Fall 2003 Math. 2b Differential equations (prac.) Winter 2004 Math. 157b Riemannian geometry 2 Spring 2004 2004 Southern California Topology Conference Complexofcurvesfest Winter 2005 Math. 131 Algebraic Geometry of Curves Winter 2005 Math. 157a Riemannian geometry 1 Winter 2005 Non-mathematical "I don't think you can hold anybody accountable for a situation that maybe if you had done something different, maybe something would have occurred differently." Vice Admiral Albert T Church III, explaining why Rumsfeld et. al. are not responsible for abuses at Abu Ghraib and elsewhere Lisa 's home page. NEW PICTURES October 18th, 2005 Anna 's home page. NEW PICTURES October 18th, 2005 I bought a house! We've had some renovations done, including painting the outside, and installing some nice lights. Look here . Here 's a picture of Boadicea, my late pussy cat. (RIP 2003) I am an associate professor at Caltech . If you're considering me as a PhD advisor, look here . I see a lot of movies. This is what I think of some of them. I wrote a little puzzle that runs under Linux called rubix square . It's put together with graphical macros from the qt library. I think it uses .gif functions, which are deprecated in later verions of qt, so you might need to hack it. Here's Carlotta 's home page, including Carlotta's Memory Game. I used to live in Cambridge Massachusetts. Now I live in Pasadena California with my wife Tereez . Here 's a nice picture of the two of us. My best friend William Webber is a programmer in Melbourne. Here is a representative picture, exemplifying our relationship. If you want to see a picture of me and Tereez (nee Therese Walsh ) at our wedding, click here . A dinner party at the house of Gretchen and Barry Mazur. Gretchen is in the background. Here is a publicity photo of Buster Keaton, promoting the film Our Hospitality. Here is my OuLiPo page. I'm a big fan of Japanese comics, including Urusei Yatsura and Maison Ikkoku , of course. I like a lot of music. Here is a silly piece of music I wrote. It's in mp4 format, sorry if you can't hear it. And here 's a little fugue. Some words . Links Journals and Reviews Math. Reviews Zentralblatt mathematics database Front for the Lanl Mathematics Archives AMS Journals Annals of Mathematics Inventiones Mathematicae Journal of Differential Geometry Geometric and Functional Analysis Geometry and Topology Algebraic and Geometric Topology Geometriae Dedicata JSTOR journal storage Mathematics resources The American Mathematical Society History of Mathematics Mark Brittenham's Low-Dimensional Topology Page Steve Hurder's Foliations Page Jon McCammond's Geometric Group Theory Page John Baez' stuff MSRI's online movie collection Dipankar's Moduli of Curves wiki Sloane's online Encyclopaedia of integer sequences Knot a braid of links The SnapPea home page Home page for Snap Computop Archive Science U Online mathematics texts Ian Agol's low dimensional topology blog My mathematical history The mathematics genealogy project Australian Mathematics Trust Melbourne University mathematics UC Berkeley mathematics Mathematical Sciences Research Institute Harvard mathematics Caltech mathematics Linux resources TeX-related documentation KDE home page Trolltech home page (makers of QT) Gnome home page Linux home page Mozilla XFree86 Web building tutorials RPM repository SourceForge Science and Politics Hubbert Peak Intergovernmental Panel on Climate Change Real Climate Nonviolence Secular Web Noam Chomsky Post-Autistic Economics Commercial Free Childhood Google News Slashdot NPR homepage Friends and relatives My little brother Frank My cousin Nick and family David BenZvi Matthew Emerton Daniel Kelson Nathan Dunfield Mark Kisin Eric Antokoletz William Webber Jason Behrstock Diane Maclagan Saul Schleimer Ben Davis Stephen Bigelow Dylan Thurston Cartoons Jim Woodring Mark Martin Sammy the graduate student Scott McCloud Crippling Depression Rumic World Industrial Lum and Magic Hayao Miyazaki web OuBaPo America Language and literature La page OuLiPo Palindromes Grown Dodo Rikai Project Gutenberg Online books The New Yorker Jim Breen's Japanese Page Kanji alive Music Sheet Music Archive Chalkhills: the XTC site Steve Vai website Classical guitar tablature Rosegarden Miscellaneous Edge foundation Julian Jaynes society NYPL Digital Gallery Exult! xu4 SJIHBOR Dr Who Blake's 7 Live cricket coverage The Onion Dave Barry Administrative details There have been N visitors to this page. last updated: 4th May, 2005 Copyright notice Optimized for use with Mozilla Source code for the Julia applet is released under the terms of the GNU GPL . Become a registered GNU Linux user
Bestvina, Mladen
Geometric group theory. Includes a problem list.
Mladen Bestvina Mladen Bestvina Research Wasatch Topology Conference Photos from a hike in the Wasatch during the August 2000 WTC Favorite links Problem list . I updated the list in July 2004 by adding references and pointers to solutions, but there are no new questions. A future edition is planned with a streamlined list of questions that will also include new questions. Here is the old version. 1060 6510 Isabella Department of Mathematics University of Utah 155 South 1400 East, JWB 233 Salt Lake City, Utah 84112-0090 Tel: 801 581 6851, Fax: 801 581 4148
Banchoff, Tom
Brown University. Geometry, visualisation; Popularisation.
Thomas Banchoff's Home Page Thomas F. Banchoff is a geometer, a professor at Brown University since 1967. During the fall semester 2005, he is teaching Math 54 , Honors Linear Algebra, and Math 106 , Differential Geometry. His current project is a Professional Enhancement Program PREP workshop on Internet-Based Course in Multivariable Calculus. Participants in this workshop can access the course webpage here . Participants in the Georgia PREP workshops August 16, 17, and 19 can access the workshop webpage here . Visit the web site for the TFBCON2003 , the conference held at Brown on October 31st and November 1st, 2003, in celebration of Professor Banchoff's 65th birthday year. LIST OF RECENT COURSES ----------ON-LINE PROJECTS ----------ARTWORK ON-LINE ANNOTATED BIBLIOGRAPHY See also an article "The Best Homework Ever?" VIDEOS AND DVDs To see a curriculum vitae of Thomas Banchoff, select here . If you have comments on any of the material presented here, please send me mail . Thanks to Mark Howison for help with the design of this page.
Lissajous Figures by Geo Doubek
Applet that draws the famous Lissajous Figures, with mathematical explanations about it, by Geo Doubek.
Geo Doubek - Figuras de Lissajous - Lissajous Figures Search: Lycos Angelfire Dating Search Share This Page Report Abuse Edit your Site Browse Sites Previous | Top 100 | Next GEO DOUBEK - LISSAJOUS FIGURES - FIGURAS DE LISSAJOUS VERSO EM PORTUGUS This is an applet that draws the famous Lissajous Figures. It do that preliminarly determining dots : (1) in the vertical axis according to the cos (ang); (2) in the horizontal axis according to the sin (ang). After that, conecting with lines the succesively defined dots. The domain used in the equation is [ 0,2pi [, or [ 0, 360[, that corresponds to a complete round in the trigonometric cincunference. The applet has some ressources that allow it to modify the parameters and produces very singular effects. The first of them is of sampling. By sampling are determinated how many points are defined inside the domain. The minimum are two to be posssible to define a single line. For example: if the number of points is defined as 36, the considered step between them will be 10. The second one is the deflexion x and the deflexion y, that permit to adjust, respectively, the size of the figure in its width and height. The third one is the relation angle (sin) x angle (cos). When the applet is loaded, the values are defined as 2 x 3 . It means that the angle used in the sin(ang) is multiplyed for 2, or, in mathematical language, is sin(2*ang), while the angle used in the cos(ang) is cos(3*ang). It is interesting to see that the coeficients for multiplication were the same, the result is a circular or star figure (since, of course, the criteria of sampling be high enought to permit it). The fourth ressource is the difference of phase between the angle (sin) and the angle (cos). It defines if one angle will be always a little bit "ahead of schedule" ou "delayed" relatively to the other. For example, if the values choosed be -15, it means that the angle use to calculate the sin will be all the time 15 smaller than the one used to calculate the cos. Modifying this value quickly generates the effect of a 3D rotating figure. The fifth and the last ressource is the color of the line, that obviously define the color used to plot the figure. I am studying another Java ressources to make this apllet more interactive and efficient. I leave here my e-mail for any suggestions, critics or doubts. All of them are welcome :-) geodoubek@bol.com.br
UVModeller
A java applet that lets you input a surface function and view the surface interactively in real time.
3D Graph Explorer Goodies 3D Graph Explorer Click here to download 3D Graph Explorer. To run double click, or type "java -jar uvmodeller.jar" in command line after you download. To preview as an applet click on the button: Note: Some features like save load are disabled when 3D Graph Explorer is run as an applet. Note: You need JRE 1.4.2 or higher to run this program. If you don't already have it, get it here . And if you want to check out some results... click here . 3D Graph Explorer 1.00a Remember those functions f(x,y) from highscool? Ellipsoids, hyperboloids, hyperbolic parabolids, and what not... Ever wished you had an utility that could show you those surfaces? Well, your wish is fulfilled! Behold, for here comes 3D Graph Explorer! This program lets you plot, and interactively view 3D surfaces and curves described by mathematical functions. This is my first major project in Java, and unlike Graph Explorer this time I used a parser generator called JavaCC , which I found quite easy to use. 3D Graph Explorer is also known as UVModeller. Features Plots 3D surfaces curves entered as parametric non-parametric functions by the user. Perspective drawing, i.e. objects look bigger when they are nearer. Has optional fog effect. Supports partially transparent colors. Supports a comprehensive list of mathematical functions. Supports creating unlimited number of objects each with its own settings. Automatically creates and displays axes according to your settings. Explore the scene that you have created in real time using mouse or keyboard. Automatically calculates curve length or surface area for each function. Operating System independent since this is a Java application. (Even runs as an applet, but with limited functionality.) Export the image to PNG formatted bitmap. (Not supported in Applet.) Examples Click here to see some examples to get an idea about what can be done with 3D Graph Explorer. Version History V 1.00 Well, this is the first time I gave it a version number... V 1.00a Objects now have a little bit of shinyness. No bugs found... so none fixed! to Goodies geovisit();
Rite Item
Geometrical shareware software. Anyangle (Windows DOS) finds answers to problems involving triangles. Partydot creates patterns.
RITE ITEM Anyangle is no longer available in shareware form. However you can still purchase the program here . You are welcome to browse my web site and read about new Anyangle version 5. Jim Preston Contact Rite Item Privacy Policy Powered by BlueDomino Copyright 2005 Jim Preston
GEUP - Interactive Geometry software.
GEUP is an interactive geometry tool. Lets you dynamically explore Mathematics, build mathematical models of real world or create interactive mathematics presentations.
GEUP - Interactive Geometry software Bienvenido a la pgina web de GEUP (Redireccin) Welcome to GEUP web site (Redirection) Espaol | English
Qhull for convex hulls, etc.
Qhull for computing the convex hull, Delaunay triangulation, Voronoi diagram, and halfspace intersection about a point.
Qhull code for Convex Hull, Delaunay Triangulation, Voronoi Diagram, and Halfspace Intersection about a Point URL: http: www.qhull.org To: News Download CiteSeer Images Manual FAQ Programs Options Qhull Qhull computes the convex hull, Delaunay triangulation, Voronoi diagram, halfspace intersection about a point, furthest-site Delaunay triangulation, and furthest-site Voronoi diagram. The source code runs in 2-d, 3-d, 4-d, and higher dimensions. Qhull implements the Quickhull algorithm for computing the convex hull. It handles roundoff errors from floating point arithmetic. It computes volumes, surface areas, and approximations to the convex hull. Qhull does not support constrained Delaunay triangulations, triangulation of non-convex surfaces, mesh generation of non-convex objects, or medium-sized inputs in 9-D and higher. News and Bugs about Qhull 2003.1 2003 12 30 Download Qhull Examples of Qhull output Development at Savannah www.qhull.org How is Qhull used? CiteSeer references to Qhull Google Qhull, Qhull Images , Qhull in Newsgroups , and Who is Qhull? MATLAB uses Qhull for their computational geometry functions: convhulln delaunayn griddata3 griddatan tsearch tsearchn voronoin . MATLAB R14 upgraded to Qhull 2002.1 and triangulated output ('Qt'). GNU Octave uses Qhull for their computational geometry functions. Mathematica 's Delaunay interface: qh-math Geomview for 3-D and 4-D visualization of Qhull output Introduction Fukuda's introduction to convex hulls, Delaunay triangulations, Voronoi diagrams, and linear programming Lambert's Java visualization of convex hull algorithms LEDA Guide to geometry algorithms MathWorld's Computational Geometry from Wolfram Research Skiena's Computational Geometry from his Algorithm Design Manual. Stony Brook Algorithm Repository, computational geometry Qhull Documentation and Support Manual for Qhull and rbox Programs and Options qconvex -- convex hull qdelaunay -- Delaunay triangulation qvoronoi -- Voronoi diagram qhalf -- halfspace intersection about a point rbox -- generate point distributions Description of Qhull COPYING.txt - copyright notice REGISTER.txt - registration README.txt - installation instructions Changes.txt - change history Qhull functions , macros, and data structures with source Frequently asked questions about Qhull Send e-mail to qhull@qhull.org Report bugs to qhull_bug@qhull.org Related URLs Amenta's directory of computational geometry software BGL Boost Graph Library provides C++ classes for graph data structures and algorithms, Clarkson's hull program with exact arithmetic for convex hulls, Delaunay triangulations, Voronoi volumes, and alpha shapes. Erickson's Computational Geometry Pages and Software Fukuda's cdd program for halfspace intersection and convex hulls Gartner's Miniball for fast and robust smallest enclosing balls (up to 20-d) Google's directory for Science Math Geometry Computational Geometry Software Leda and CGAL libraries for writing computational geometry programs and other combinatorial algorithms Magic Software source code for computer graphics, image analysis, and numerical methods Mathtools.net of scientific and engineering software Owen's Meshing Research Corner Schneiders' Finite Element Mesh Generation page Shewchuk's triangle program for 2-d Delaunay Skorobohatyj's Mathprog@ZIB for mathematical software Voronoi Web Site for all things Voronoi FAQs and Newsgroups FAQ for computer graphics algorithms ( geometric structures) FAQ for linear programming Newsgroup : comp.graphics.algorithms Newsgroup : comp.soft-sys.matlab Newsgroup : sci.math.num-analysis Newsgroup : sci.op-research The program includes options for input transformations, randomization, tracing, multiple output formats, and execution statistics. The program can be called from within your application. You can view the results in 2-d, 3-d and 4-d with Geomview . An alternative is VTK . For an article about Qhull, download from CiteSeer or www.acm.org : Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T., "The Quickhull algorithm for convex hulls," ACM Trans. on Mathematical Software, 22(4):469-483, Dec 1996, http: www.qhull.org Abstract: The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull Algorithm with the general dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and Delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains non-extreme points, and that it uses less memory. Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating point arithmetic, this assumption can lead to serious errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of "thick" facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions. Up: Past Software Projects of the Geometry Center URL: http: www.qhull.org To: News Download CiteSeer Images Manual FAQ Programs Options The Geometry Center Home Page Comments to: qhull@qhull.org Created: May 17 1995 ---
GEUP
An interactive geometry program for Windows. Evaluation version free to download.
GEUP - Interactive Geometry software Home What's new in version 2 More Info Screenshots Download Order Contact Espaol Discover the power of the Interactive Geometry for learning and doing Mathematics GEUP is an interactive, powerful and easy to use software for learning and doing Geometry on the computer. It enables to study graphically a general problem and obtain dynamically many particular cases. This property together with its calculation capability allows to study problems of different areas. It is a versatile and very useful tool in education and practice of Mathematics at any level. With GEUP you can explore mathematics visually and interactively, build mathematical models of real world or create interactive mathematics presentations easily. In the new version 2, GEUP is more powerful, more versatile and more easy to use. What's new in version 2 . To learn more about GEUP see More Info . Download the evaluation version of GEUP. To register GEUP see Order . [ What's new in version 2 | More Info | Screenshots | Download | Order | Contact ] Copyright 2002 Ramn Alvarez. All rights reserved.
GeomNet
Family of geometric computing servers execute a variety of geometric algorithms on behalf of remote clients, which can be either users interacting through a Web browser interface or application programs connecting directly through sockets.
GeomNet Home Page GeomNet WELCOME TO BROWN UNIVERSITY'S GEOMNET SITE Researchers in computational geometry and graph drawing are developing a host of efficient algorithms for solving problems and many are taking the design of these algorithms all the way to working implementations. Unfortunately, there is no standard form for the representation of geometric data and implementations typically define their own formats, making it difficult to use elements from a combination of packages. We feel that practitioners and researchers can take better advantage of the latest developments in computational geometry if implementations of geometric algorithms are made widely available on the Internet and are fully usable without the learning of new file formats, calling sequences, or class hierarchies in software libraries. To address these issues, we envision an ``Internet computing'' framework, which we call GeomNet, where a family of geometric computing servers execute a variety of geometric algorithms on behalf of remote clients, which can be either users interacting through a Web browser interface or application programs connecting directly through sockets. A prototype implementation of part of this vision is available through the links below. GeomNet Home Visit the main GeomNet site, hosted by Johns Hopkins University. Try out the computational geometry algorithms of GeomNet, hosted by Johns Hopkins University. (choose "Use GeomNet", then "GAS") Explore the graph drawing algorithms of GeomNet, hosted by Brown. Algorithm Animation Experiment with the algorithm animation capabilities of GeomNet, hosted by Brown. Comments Please let us know about any comments you might have. ssb@cs.brown.edu
KwikTrig
A free trigonometry solving program for Windows.
Riemann surfaces visualisation
Unifpack is a set of programs and C libraries designed to help in the study of Riemann Surfaces.
Unifweb -- Hyperbolic Riemann Surfaces with Symmetry Unifpack (Web version 2.0) A Package for Studying Riemann Surfaces with Symmetry by Carlos O'Ryan Lira (coryan@mat.puc.cl) Unifweb World Wide Web front end by Paul Burchard at the Geometry Center, using W3Kit . CLICK HERE TO START. Unifpack is a set of programs and C libraries designed to help in the study of Riemann Surfaces. The first version was written in the beginning of 1992 as part of Carlos' undergraduate thesis. See the Unifpack User's Manual for a complete explanation of the program. Here is a brief summary: Basically Unifpack accepts a presentation of a finite group and calculates the Cayley graph of the group for that presentation. If the presentation is ``admissible'' in the sense described below, Unifpack can calculate a fundamental polygon for a uniformization of a Riemann surface that admits this group as a group of symmetries. What the Picture Means The calculated Riemann surface is displayed as a tiling in the hyperbolic plane (Unifpack only works with surfaces of genus 1). The ``pelt'' of the split-apart surface forms a large tile of which a single copy is displayed. This large tile is then split into smaller tiles representing fundamental domains of the action of the symmetry group on the surface. Adjusting Parameters Usually there is a family of Riemann surfaces that admit such a group of symmetries; this maps to a family of fundamental polygons controlled by some parameters. Unifpack lets you change these parameters and thus study the family of Riemann surfaces. Different admissible presentations of the same group generally produce different families of surfaces; the families may even differ in genus. Reflections Although Unifpack normally works with conformal symmetries, in some cases the symmetry group can be extended to include reflection symmetries (which reverse orientation). In these cases, the program offers a boolean ``parameter'' which enables the display of the reflection axes in a lighter color (it's not really a parameter of the Riemann surface). Admissible Presentations In Unifpack, the symmetry groups are specified using generators and relations. The main difficulty with this form of input is that it is difficult to tell if the group you have defined is finite! However, this is required in order for the program to work. The second condition for a presentation to be ``admissible'' is that the relations must be written in a special form, listing in order: a power of each generator; a power of the product of all the generators; and other relations consistent with these (optional). Consistency means in particular that these optional additional relations should not redefine the orders of the generators as specified by the required relations. In Unifpack, writing out the relations in this form picks out a genus for the Riemann surface. The final condition for ``admissibility'' is that this genus be greater than 1 so that the surface can be displayed using a tiling of the hyperbolic plane. The Geometry Center University of Minnesota 400 Lind Hall 207 Church Street S.E. Minneapolis, MN 55455
KSEG Interactive Geometry Software
KSEG is an Open Source program for interactively exploring geometric constructions which can be used to teach geometry in the classroom or for personal enjoyment.
KSEG KSEG Free Interactive Geometry Software Update: May 15, 2005 -- KSEG in Chinese Xu Xianghua kindly contributed Chinese translations of the KSEG UI and the help. They are not in the latest version so you can download the UI translation here and the help here . Update: November 24, 2004 -- KSEG 0.402 This release provides support to export to SVG files (contributed by Bulia Byak). It also includes a real Russian translation by Anton Petrunin and a couple of bugfixes that were required to support it. enjoy! Update: July 3, 2004 -- KSEG 0.401 for Windows! By popular demand, despite my dislike for microsoft, I've ported KSEG to windows using the old Qt noncommercial version 2.3 (I hope I'm not violating anything). The source is uglier and there are some bugs not in the Linux version. Download the whole thing here (it should run out of the box). Sample Output High-quality images (not screenshots--those are below) of a well-known theorem and a strange locus, both exported with KSEG: Description: KSEG is a Free (GPL) interactive geometry program for exploring Euclidean geometry. It runs on Unix-based platforms (according to users, it also compiles and runs on Mac OS X and should run on anything that Qt supports). You create a construction, such as a triangle with a circumcenter, and then, as you drag verteces of the triangle, you can see the circumcenter moving in real time. Of course, you can do a lot more than that--see the feature list below. KSEG can be used in the classroom, for personal exploration of geometry, or for making high-quality figures for LaTeX. It is very fast, stable, and the UI has been designed for efficiency and consistency. I can usually make a construction in KSEG in less than half the time it takes me to do it with similar programs. Despite the name, it is Qt based and does not require KDE to run. KSEG was inspired by the Geometer's Sketchpad, but it goes beyond the functionality that Sketchpad provides. Languages so far: Portuguese -- Jorge Barros de Abreu French -- Jean-Philippe Martin (UI) and Marie-Paule Canou, Jose Goyer, Michle Sidobre (Help) German -- Andreas Goebel Norwegian Bokmal -- Skolelinux Hungarian -- Gabor Nagy Spanish -- Eduardo Dueez Dutch -- Bram Schoenmakers Italian -- Giancarlo Bassi Partial Japanese -- Linux Magazine (ASCII) (UI) Yokota Hiroshi (Help) Welsh -- Kevin Donnelly Turkish -- Vildan Ozturk Russian -- Anton Petrunin Chinese -- Xu Xianghua Requirements: You will need at least Qt 3.x. Qt 2.x may work, but I haven't tested it much. If you don't have it, go to http: www.trolltech.com . If your system is strange, you may need to edit the makefile to compile it (but don't worry, only the first 6 lines are relevant). Although I develop KSEG in Linux, people have compiled it under Mac OS X and FreeBSD. For Mac OS X 10.3 and later, Markus Bongard built a standalone binary installer for KSEG, which also includes Qt. Get it here (about 5 MB download). Download KSEG v0.402: Here is the source . If you look around on the web, you should be able to find RPM's and other packages. Here is the windows executable, version 0.401, and the ported (read: mutilated) source. Current Features: (features in italics are those that make KSEG special :) Fast core which can support large constructions Free GPL-runs on Linux (and probably most other systems which support Qt) Supports multiple languages Fully Documented Construction of points, segments, rays, lines, circles, and arcs Make measurements Transformations (rotation, translation, scaling, reflection) Construction of adaptively sampled loci for better quality and speed Reverse dragging Infinite undo redo Ability to easily redefine points to "edit" drawings Easy to use editable scripting macro with support for recursion Pretty formulas for calculations-with my libkformula Pretty colors, fonts, etc. View panning zooming and multiple simultaneous views Export view to image file, including antialiased option Printing Selection Groups--group a bunch of objects, then select them later with two clicks Suggestions? Never hurts to tell me your thoughts. Screenshots - click to enlarge Click on screenshot 3 to read more about it. History ChangeLog SEG started out as a little DOS program (in DJGPP) back in 1996 because I didn't want to pay $40 for a copy of Sketchpad. It was my first real C++ project and I made many design mistakes. Then I rewrote it under Windows and it was way better-fast, very stable, flexible and easy to use. I have successfully used it to generate the idea and write my highschool senior thesis (on chaotic dynamics of a family of geometrically-defined functions) and have played around with it a lot. Finally, I rewrote SEG a third time for linux (calling it KSEG because initially it was a KDE project--but I found the KDE API's were changing too fast for me to keep up), using my experience with the previous two designs. It has gone from a program for my personal use into a real piece of interactive educational software. Although I know that it is being used in education, I have heard few details about the experience, so please, if you use KSEG for teaching math in a school or college, write me about it. Back to my homepage
WinGCLC
a tool for describing geometric constructions and making digital illustrations in LaTeX and bitmap format.
GCLC page GCLC page About GCLC GCLC (from Geometry Constructions - LaTeX converter is a tool for producing mathematical illustrations and for teaching geometry. It basic functionality is converting formal descriptions of geometric constructions into digital figures. It provides easy-to-use support for many geometrical constructions, isometric transformations, general conics, etc. Making figures is based on the idea of ``describing figures'' rather than of ``drawing figures''. Thus, this approach stresses the fact that geometrical constructions are abstract, formal procedures and not figures. A figure can be generated on the basis of abstract description, in Cartesian model of a plane. Scope: Although GCLC was initially built as a tool for converting formal descriptions of geometric constructions into LaTeX form, nowadays it is much more than that. For instance, there is support for symbolic expressions and for drawing parametric curves. On the other hand, Windows version makes GCLC a tool for teaching geometry, and not only geometry but other mathematical fields as well. Platform: There are command-line versions of GCLC for Windows and for Linux. WinGCLC is the (Microsoft) Windows version of GCLC and provides a range of additional functionalities, including interactive work, animations, traces, ``watch window'', etc. There is no version with graphic user interface for Linux. The main purposes of GCLC WinGCLC: producing digital mathematical illustrations of high quality; use in teaching geometry; use in studying geometry and as a research tool. The main features of GCLC WinGCLC: support for a range of elementary and advanced constructions, isometric transformations, and other geometrical devices; support for symbolic expressions, second order curves, parametric curves, while loops etc. user-friendly interface, interactive work, animations, tracing points, watch window ("geometry calculator"), and other tools; very simple, very easy to use, very small in size; export of high quality figures into \LaTeX{} and bitmap format; versions for command line (DOS Windows and Linux) and the MS Windows version; import from JavaView JVX format; freely available (from http: www.matf.bg.ac.yu ~janicic gclc and from EMIS (The European Mathematical Information Service) servers: http: www.emis.de misc software gclc ). Author: GCLC WinGCLC is being developed at the Faculty of Mathematics, University of Belgrade, by Predrag Janicic and his collaborators. It has had several releases since 1996 and it has been used for producing digital illustrations for a number of books and journal volumes and in a number of different courses. References: More on the background of GCLC WinGCLC can be found in: P.Janicic and I. Trajkovic: WinGCLC --- a Workbench for Formally Describing Figures. In Proceedings of the 18th spring conference on Computer graphics (SCCG 2003), pages 251--256, Budmerice, Slovakia, April, 24-26 2003. ACM Press, New York, USA. M.Djoric and P.Janicic. Constructions, instructions, interactions. Teaching Mathematics and its Applications, 23(2):69--88, 2004. Others about GCLC WinGCLC: ``... program Wingclc ... is a very useful, impressive professional academic geometry program.'' (from an anonymous review for ``Teaching Mathematics and its Applications'') Please send us your comments and or suggestions to janicic@matf.bg.ac.yu (Predrag Janicic). Please send us your GCLC gems and we will put them on this page. Downloads WinGCLC (c) 2003-2005 Predrag Janicic, (the graphical interface made by Ivan Trajkovic); MS Windows version of GCLC, user-friendly multi-document interface, support for animations, traces, export to bitmaps etc. Includes gclc.sty, GCLC manual, sample files. (available since 01.01.2005; version 2005 (GCLC engine 4.0: 01.01.2005) Download GCLC command-line version for DOS Windows (c) 1996-2005 Predrag Janicic; includes gclc.sty, VIEW (simple previewer), GCLC manual, sample files. Download GCLC command-line version for Linux (c) 1996-2005 Predrag Janicic; also gclc.sty, GCLC manual and sample files. Download GCLC system description Download GCLC manual Download jv2gcl converter (for DOS Windows) (c) 2002 Predrag Janicic: converts files from JavaView to GCLC format. Download Samples WinGCLC screenshot GCLC gems Prof. Zoran Lucic (Belgrade): Euclid's construction of dodecahedron (2005) Please send us your GCLC gems and we will put them on this page. Copyright notice GCLC WinGCLC is a copyrighted and cannot be used in commercial purposes. However, you are free to use it in teaching, research and in producing figures and digital illustrations for non-commercial purposes. If you download and use GCLC package, please let me know by sending an e-mail to janicic@matf.bg.ac.yu (Predrag Janicic); we will put you on the GCLC mailing list and inform you about new versions. If you used GCLC for producing figures for your book or a paper, we would be happy to hear about that. Acknowledgements I am grateful to prof. Mirjana Djoric for the initial discussion which led to the first version of GCLC; prof. Neda Bokan and other members of the Group for geometry, education and visualization with applications (mostly based at the Faculty of Mathematics, University of Belgrade) for their invaluable support in developing the WinGCLC package; EMIS (The European Mathematical Information Service) for mirroring this page at http: www.emis.de misc software gclc ) and other EMIS locations. DAAD for funding the visit to Konrad Polthier's group at Mathematical Institute of TU Berlin, which was used for making the JavaView - GCLC converter; Konrad Polthier and Klaus Hildebrandt from TU Berlin for their hospitality and their support; Ivan Trajkovic, the co-author of WinGCLC and Aleksandar Samardzic for their help in making releases of other additional GCLC modules; MNTRS for the research grant 1646 which partly supported the project WinGCLC; the colleagues which gave valuable contributions and suggestions in earlier stages of development of WinGCLC: Nenad Dedic, Milos Utvic, Nikola Begovic, Ivan Elcic, Jelena Grmusa, Aleksandra Nenadic, Marijana Lukic, Srdjan Vukmirovic, Goran Terzic, Milica Labus, and Aleksandar Gogic; All GCLC WinGCLC users for their support, feedback and suggestions. Predrag Janicic [Predrag Janicic's home page]
GANG Software Suite
A set of mathematical environments for computing, visualizing and experimenting with geometric objects; built with the OpenGL Mesa library and the GTK+ user interface.
GANG Software | GANG Software Software | iSight | SurfLab | MinLab | CMCLab | KLab | BubbletonLab | MrBubbleLab | GANG GANG Software GANG Geometry Tools The GANG Software Suite is a set of mathematical environments for computing, visualizing and experimenting with geometric objects. To see what the GANG Software Suite can do, visit the GANG Gallery of Constant Mean Curvature Surfaces , GANG Gallery of Minimal Surfaces and GANG Gallery of Pseudospherical Surface . The GANG Software Suite is built with the OpenGL Mesa library and the GTK+ user interface. The GANG Software Suite was created by Nick Schmitt at the Center for Geometry, Analysis, Numerics and Graphics ( GANG ). S O F T W A R E iSight iSight is a viewer for visualizing, exploring and interacting with geometric surfaces based on the OpenGL Mesa and VRML standards and the GTK user interface. iSight can be used to visualize surfaces in Euclidean and non-Euclidean geometries, and in 3 and 4-dimensional spaceforms. CMCLab CMCLab is an interactive software package for constructing, viewing and experimenting with constant mean curvature surfaces. CMCLab has been effective for understanding known surfaces and creating new examples, including, besides the classical examples, tori, bubbletons, generalized Smyth Surfaces, cylinders with umbilics, and k-noids. KLab KLab is an interactive software package for computing and visualizing pseudospherical surfaces, surfaces with constant negative Gaussian curvature. MinLab MinLab is an interactive software package for computing and visualizing minimal surfaces, surfaces with zero mean curvature arising as soap bubbles. SurfLab Interactive software package for computing and visualizing parametrized surfaces. BubbletonLab Interactive software package for computing and visualizing CMC bubbletons. MrBubbleLab Interactive software package for computing and visualizing CMC Smyth Surfaces. GANG Software Nick Schmitt
Symmeter
An on-line tool for understanding and measuring symmetry.
Symmeter and SymFace - Facial Symmetry, Symmetry Measurement and Analysis SEARCH: What is Symmeter? Symmeter is a web-based system that provides a simple way to measure the symmetry of any person, place or thing that can be rendered through a digital image. July, 2005 Symmeter and SymFace were recently used in a research project on symmetry and handedness. Click here to see the poster presentation. (PDF is ~1 mb) HOME | ABOUT | TRY IT | FAQ | SYM-FUN | CONCEPT | TERMS | 2002-2005, Symmeter.com. All rights reserved.
Atomic Concepts Software
Freeware for the visualization of atomic structure and process in the cosmos.
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Math Forum: Search geometry-software-dynamic
Users of geometry software programs like Cabri Geometry II may share tips and advice at the geometry-software-dynamic discussion list.
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Home Page of Isard
A free dynamic geometry package for Windows, PowerMac and Linux. Like Cabri, but in Smalltalk (and smaller). Source available : you can extend it as needed !
Home Page of Isard [ Retour la page de JS ] Welcome to the home page of Isard. Last update: sep 25, 2002 Cette page existe aussi en Franais ! (parfois obsolte...) What is Isard ? Isard is an interactive geometry software written in Smalltalk by Jean-Sbastien Roy during the years 1996-1997, now being released (and sometimes updated). Isard enable to create and interactively manipulate many kinds of geometric figures. You can download it here: Isard-1.9.zip . This version only works under VisualWorks 7 (which is quite buggy on my Mac as of september 24, 2002. EPS export may not work on a Mac). The previous version ( Isard-1.8.zip ) works under VisualWorks 3.0 NC and VisualWorks 5i and should work under VisualWorks 2.5 too. To get VisualWorks NonCommercial for free, see the Cincom's VisualWorks downloads page. Available on: Linux (X86), Digital Unix (Alpha), IBM AIX, HP-UX, SGI IRIX, Sun Solaris (SPARC), Windows 95 NT (X86) and MacOS (PowerPC)) ! If you want to see how VisualWorks looks like, have a look at this screenshot (1280x960). To export figures as GIF files, VisualWave is needed (included in the download above). Do not hesitate to write me if you encounter any problem. The previous versions are still available. Version history: 1.9 (sep 24, 2002): VisualWorks 7 compatibility. 1.8 (mar 6, 2000): Saving as EPS is improved. Miscelanous improvements. 1.7 (feb 28, 2000): Automatically updated commentary. Some bugs corrected. 1.6 (feb 26, 2000): A few fixes. Automatically generated and editable commentaries. 1.5 (jan 5, 2000): Corrected a bug in the main menu. The main menu is improved. Added two examples. 1.4 (dec 27, 1999): Loading the parcel (without loading VisualWave first) resulted in an error. It's fixed now. 1.3 (dec 26, 1999): Miscellaneous PostScript rendering bugs fixed. 1.2 (dec 23, 1999): Saves as color EPS files (instead of grayscale), Double buffering is optional, prerequisites reduced 1.1 (dec 20, 1999): GIF Export. 1.0 (nov 6, 1999): First version widely available. To do: - HTML export - PDF export - add new examples - comment examples Note: There is a little bug on PC that prevents GIF export (at least on the PC I tried it on). The bug does not show up if the display is in 24bits. You should try various display settings. Acknowledgments: Thierry Fuhs , who taught me Smalltalk. Travis Griggs : optional double buffering, ENVY version, french to english translation of Isard classes. Isard Manual You can read the online manual . A few examples: Isard can save figures as either encapsulated PostScript files or uncompressed GIF files. These are low resolution images. Below each image are links to high resolution images and encapsulated PostScript files. High resolution images are EPS files rendered using MacGhostScript or MacGhostView , then scaled down and saved to PNG by GraphicConverter . (If you do not see the images, your browser do not support the PNG format , which now replace the compressed GIF format, which is not free anymore !) Bissectrices: [ Hi-Res | EPS ] Centre du cercle de Taylor: [ Hi-Res | EPS ] Orthocentre: [ Hi-Res | EPS ] Cercle d'Euler: [ Hi-Res | EPS ] Limaon de Pascal: [ Hi-Res | EPS ] Droite de Newton Thorme de Mlnas: [ Hi-Res | EPS ] Parabole: [ Hi-Res | EPS ]
Geometria
A Java program in solid geometry. Solids can be measured, cut, drawn upon. 60 sample problems with resolutions.
Interactive Geometry With Geometria, learning high school math is easier than ever. Its comfortable graphic interface lets you perform various actions upon solids as if holding them in your hands. Have a look Geometria's extended toolbag enables you to draw lines, perpendiculars, midpoints, bisectors; measure and lay off distances and angles; transform, cut and join solids; build them from scratch. Tell me more Geometria is available for download at the GeoCentral Web store. Is it safe? System requirements I wanna try it first Buy Geometria is being internationalized. Spanish version With Geometria, problems can be composed, solved, emailed to a friend or teacher, reviewed. Geometria is packed with 60 sample problems classified, according to their complexity, into 8 categories. Show me one
The Geometer's Sketchpad Resource Center
This Resource Center supports users of The Geometer's Sketchpad "Dynamic Geometry" software. Contents include a freely-downloadable demo and Java versions; research bibliography; online activity guide; technical support center.
The Geometer's Sketchpad - Resource Center Getting Started Product Information How to Order Curriculum Modules General Resources Classroom Activities Recent Talks Bibliography 101 Project Ideas Advanced Sketch Gallery Sketchpad for TI Calculators User Groups Links Instructor Resources Download Instructor's Evaluation Edition Workshop Guide Professional Development Technical Support FAQ Product Updates Tech Support Request Form Feedback User Survey Mailing List Customer Service JavaSketchpad About Gallery Download Center Developer's Grammar Links Other Key Sites Key Curriculum Press Key College Publishing KCP Technologies Keymath.com Welcome to The Geometer's Sketchpad and to Dynamic Geometry software! The latest version of this award-winning mathematics visualization environment features enhanced support for algebra and calculus as well as geometry; a full set of formatting tools for mathematical notation and styled text; a more flexible user interface for greater ease of use; built-in Web integration; and many, many other enhancements. For a partial list of features new in Version 4, please visit the Product Information page . If you dont already use Sketchpad: Learn more about it. Instructors: Download the Instructor's Evaluation Edition. Students: Find out about the $39.95 Student Edition. Explore online JavaSketches. See what others have to say. Place an online order. If you already use Sketchpad: Search our collections of Sketchpad classroom activities for download or purchase , or browse our advanced sketch gallery . Download the latest update (4.06) for Version 4. (If you're still using version 3, click here to upgrade to version 4 .) Access technical support . Find links to sketches and other resources. Please fill out our User Survey . Teachers, learn about our Student Edition Class Kit Offer . Sign up for the Sketchpad mailing list and receive announcements of updates and new materials. Privacy Policy: The Geometer's Sketchpad Resource Center Web server collects and reports routine information to us, in summary form, about visitors to our site and the pages they visit. In addition, particular forms on this site may request more detailed or personal information from you as a visiting Sketchpad user. Providing such information is in all cases voluntary and is initiated by you as the visitor. We use this information to inform us about the needs and requirements of Sketchpad users; we do not sell or share it with other parties. In particular, if you provide your e-mail address on any form, Key Curriculum Press may respond to you at that address, but will not share that address with other parties. 2005 KCP Technologies. All rights reserved. Recent Developments Exploring Algebra 1 with The Geometer's Sketchpad now available Find out more about the latest addition to the Sketchpad Curriculum Module series. Teaching Geometry with The Geometer's Sketchpad Online Course Enhance your Sketchpad skills for teaching mathematics with this six-week online course . Report from the 2005 NCTM Annual Meeting Learn about Sketchpad events at the NCTM national conference. Report from the 2005 Joint Mathematics Meetings Download presentation materials from the Sketchpad User Group. Download Connected Math Activities for Sketchpad Download field-tested Sketchpad activities for use with the Connected Mathematics Project Grade 6 and 7 curriculum. Version 4.06 ships Sketchpad users: download the free update .
MowMowMow's Cabri Room
Many Cabri files aimed at high school level geometry.
MowMowMow'S CABRI ROOM Mow'S This page is has been visited times since 11 12 97 I am a math teacher at a junior high School in Japan. I am making Geometric Data for "Cabri Geometric II". if you have "CABRI Geometric(even DEMO version)", you can enjoy these data. You can get "CABRI Geometric II DEMO version (for Mac Dos Win) from here. Let's enjoy Geometric World ! [Japanese Page]
Polyhedron
Interactive application for solid geometry with 250 built-in problems. Simulates ruler, compass, protractor and other tools. Solids can be cut, displayed differently, rotated, joined again.
Polyhedron Polyhedron Interactive geometry software Download English Version Download Romanian Version Polyhedron is an MS DOS application based on a practical approach to solid geometry. It allows the user perform various actions upon solids as if holding them in hands. Polyhedron simulates a number of tools, such as ruler, protractor, setsquare, compass, bisector, saw, eraser, all accessible from a menu bar. Solids can be revolved, displayed in different manners, cut. Polyhedron is packed with 250 built-in problems. You may be surprised by what some of them deal with. There are problems about frameworks, toys, beetles, other improbable stuff. The problems of complexity 5-6 (there are 6 complexity levels) have proven difficult enough even for the smartest undergraduate students. Polyhedron was built on the carpenter workshop philosophy. You can read more about it here . The note also contains sample problems with resolutions. Geometria is a Java upgrade to Polyhedron . Copyright 1997-2005 Stelian Dumitrascu
Cinderella: Interactive Geometry
Cinderella is a Java based interactive geometry tool. The only available tool that gives correct solutions to typical geometrical problems.
Cinderella : Cinderella Cinderella. The Interactive Geometry Software Cinderella The Interactive Geometry Software Cinderella Cinderella is a software for doing geometry on the computer, and it is designed to be both mathematically robust and easy to use. Read more about it on our website, check some interactive examples or download a demo to see yourself. Alternatively, you can read the review in MAA online . To the left you see a screenshot of the software. You can run Cinderella on Windows, MacOS, Linux, and many other Unix variants, since it is written in Java. And since Java is built into many browsers, such as Netscape and Internet Explorer, you and others can use your constructions across the Internet. Communicating geometry is easy with Cinderella. Server re-installed on: Mon 05 of Sep, 2005 [09:51 UTC] (376 reads) Our server has been hacked, using a security flaw in xmlrpc.php that comes with the content management system we use. As of today, everything should work again as expected, if not, please contact bugs@cinderella.de . Cinderella.2 is coming... on: Sun 13 of Mar, 2005 [16:28 UTC] (4222 reads) The next version of Cinderella is coming! Cinderella.2 introduces many new geometric features, as well as completely new features like physics simulations and a programming interface. If you want to stay informed about the new version, just subscribe to our low-volume (at most one email Cinderella release) newsletter by sending an email to announce-subscribe@cinderella.de . Cinderella @ CeBit on: Tue 08 of Mar, 2005 [19:54 UTC] (2143 reads) Cinderella.2 at the Future Market of CeBit 2005 ! Visit our booth at the Future Market in Halle 9, A60, of CeBit 2005 in Hannover, Germany. The Technical University of Berlin DFG Research Center Matheon and the Technical University of Munich show the "Classroom of the Future". We present Cinderella.2 and its new features, like physics simulation and electronic whiteboard Tablet PC support. last modification: Friday 24 of June, 2005 [09:06:51 UTC] The content on this page is licensed under the terms of the Licence . refresh similar print Cinderella Home Download FAQ Support Forum Contact Language: en Catalan () esky Dansk Deutsch Greek English English British Espaol Franais Hrvatski Magyar Italiano Nederlands Norwegian Polish Portuguese Portugus Brasileiro Pijin Solomon Slovensk Srpski Svenska Tuvaluan Cinderella@Amazon Buy Cinderella online at Amazon.com for $59.95 (single user). Network licenses are available, please contact Springer NY for details. Last forum posts 1) Cinderella Support (E): Upgrade link doesnt work 2) Cinderella Support (E): Upgrade link doesnt work 3) Cinderella Support (E): Cinderella doesn't run 4) Cinderella Support (E): Cinderella doesn't run 5) Cinderella Support (D): Termin fr Cinderella.2 6) Cinderella Support (E): software dosn't work 7) Cinderella Support (E): software dosn't work 8) Cinderella Support (D): Termin fr Cinderella.2 9) Cinderella Support (D): Termin fr Cinderella.2 10) Cinderella Support (D): Viertelkreise Login user: pass: register
Great Math Programs
A collection of mathematical programs, with particular reference to geometry.
Xah: Great Math Programs If you spend more than 30 minutes on this site, please send $1 to me. Go to http: paypal.com and make a payment to xah@xahlee.org. Or send to: P. O. Box 390595, Mountain View, CA 94042-0290, USA. back to Xah's home Great Math Programs Xah Lee. Page created: 1996. Last major update: 2004. This is a list of fun math programs i have played over the years. They are mostly shareware or freeware. For Mac, Windows, or Linux. A star sign means it is an excellent software that can be enjoyed without any math study. If you think your program should be listed here, please submit a copy for review. (email me first). There are about 700 unique visitors to this page each week. What's New (2005-10) Sokoban (=warehouse keeper) is a classic game invented by Hiroyuki Imabayashi in 1982. It is a puzzle where the player pushes boxes around a maze to designated locations. For more info, see: http: en.wikipedia.org wiki Sokoban Here's a Sokoban game in Javascript: http: michbuze.club.fr Boxworld sokojs.htm (local copy: sokojs.zip ) Curves and Surfaces plotters Graphing Calculatorby Ron Avitzur. Imagine your hand-held graphing calculator with the power of a desktop computer, then you get the idea of what this program is about. Out of 10 or 20 programs that plots curves and surfaces i've tried, I think this is the best. Most versatile and most easy to use. Go download a demo and see for yourself. If you want a plotting program for your highschool or college installation, I suggest this one. URL: http: www.nucalc.com . The website A Visual Dictionary of Famous Plane Curve includes many files of Graphing Calculator. 3D-XplorMath, by mathematician Richard Palais. 3D-XplorMath is a program for visualization of objects and processes in geometry. It can plot and animate surfaces, 2D 3D curves, complex valued functions, differential equations and others. The most eye-catching is its rendering of over 100 curious surfaces in differential geometry. 3D-XplorMath also includes some over one hundred expositions in PDF on various geometry topics on curves, surfaces, comlex mappings, fractals. The flaw with 3DXM is that it is very difficult to use. The menus are overflowing and non-standard, and user must read a big documentation to learn what various parameters mean and how to use them. If you are studying or teaching math beyond calculus, this is a worthwhile program. Mac only. (OS X native). URL: http: rsp.math.brandeis.edu 3D-XplorMath (2003-09) * Mathematica package for converting 3D-XplorMath surface files to Mathematica graphics: 3dxm2mma.zip . Note: i'm a member of 3D-XploreMath Consortium. See here for some more curves and surface plotters. Polyhedra and Polytopes Stella by Robert Webb Stella, by Robert Webb. A polyhera program that the author intends to be the ultimate one, and it truely is an excellent program, and has the most features. http: www.software3d.com Stella.html (2004-01) KaleidoTile does interactive polyhedron software by Jeff Weeks. You can see how some polyhedra can be generated by mirroring tiles in space, and how one transforms into another. All this in real-time with dynamic control. Download at: http: www.geometrygames.org KaleidoTile index.html HyperSpace Polytope Slicer. Mark Newbold has written some beautiful polytope visualization Java appletes. One is called HyperSpace Polytope Slicer, which lets you view any of the 6 4D regular polytopes by slice. Here it is: http: dogfeathers.com java hyperslice.html . Another, more general one called Hyperspace Star Polytope Slicer. This one is more general but much demanding on the computer: http: dogfeathers.com java hyperstar.html Both are beautiful and well-done, and with very detailed explanations. (2003-09) Poly (v.1.05, 1999 12) Poly is another interactive 3D program that does polyhedra. This program contains a fairly complete set of regular and semi-regular solids that includes: Platonic, Archemedean, Prism and Antiprisim, Johnson, Catalan, Dypyramids and Deltohedrons. A special feature of the program is showing the solids as nets (graphs) and flat un-wrapped represenations. (the latter is good for making paper models of the solid.) Poly is a shareware. It's available for Mac and Windows. Download it at Pedagoguery Software . (2001 01) See here for some more polytope programs. Interactive Plane-Geometry Interactive plane geometry software were made popular first by Geometer's Sketchpad at least back to 1994. It is a wonderful tool made possible by technology. Such program allows one to construct plane geometry drawings dynamically, much in the way of Greek's Ruler and Compass. For example, draw a triangle with lines bisecting the 3 angles. They intersect in a point called the incenter of the triangle. (which is also the center of the largest possible circle inside.) Now, you can drag the corners of your triangle to change its shape, and all your drawings change accordingly. Because of such dynamic power, a few theories on plane geometry have been found because of such software. As of today, there are about 5 softwares on plane geometry. They are all similar. They are all of high quality. Almost all of them can be saved as Java appletes so that it can be placed on the web for students to use without needing extra software. Their features differ slightly. One of them is FreeSoftware. The following is a short commentary of each. Compass'n'Ruler screenshot. Parameters can be changed dynamically by dragging points. Compass and Ruler is written in Java by von R Grothmann. This program is excellent. It is very well written, in Java and available for Mac or Windows or Linux, and it is free with source code too. Its home page is at http: mathsrv.ku-eichstaett.de MGF homes grothmann java zirkel index.html (2004-11) von R Grothmann's Compass'n'Ruler is so great in quality, features, price ($0), it basically beats all other commercial makes in all aspects. There is little point of checking out other similar softwares. But if you really like to see options, they are listed at other softwares page . Non-Euclidean Geometry Flat Truncated Octahedra space from Curved Space by Jeff Weeks. Curved Space Jeff Weeks wrote a wondrous program of flying through various "warped" geometric spaces (3-manifolds). Windows only. Source code and tutorial are included for programers. http: www.geometrygames.org CurvedSpaces index.html If you question what is a "curved space", be sure to check out this excellent book The Shapes of Space by Jeffrey Weeks (2nd, 2001). It introduces topology and geometry in the most fascinating way, not like other math popularization books that are trivial in content. For some notes on manifolds, see: http: xahlee.org Periodic_dosage_dir 20031225_shape_space.html . . KriviznaPlus and KriDva by Viktor Massalogin KriviznaPlus and KriDva are "interactive screensavers". That means, it runs like a screensaver but one can control it with the mouse. (press esc exits the program) According to the author Viktor Massalogin, KriviznaPlus draws the stereographic projection of a 3D sphere (x^2+y^2+z^2==0) to the plane. And KriDva draws the stereographic projection of 4D sphere (x^2+y^2+z^2+z^2==0) to the 3D space. These programs are extremely fun and beautiful. I hope the author adds more explanations on the math. As of 2004, he added another version CPace. http: www.hot.ee bntren Program.html . (2003-10) Non-Euclid by Joel Castellanos. NonEuclid by Joel Castellanos. It is a free software for drawing hyperbolic geometry. Its function is similar to Geometer's Sketchpad except it is not dynamic. Elements cannot be dragged around once drawn. The program comes with a wonderful tutorial on hyperbolic geometry. Available versions are: Macs 68k, Windows, and Java. ( old mac version screenshot ). URL: http: cs.unm.edu ~joel NonEuclid . (2003-09) 2002-07-03: Matthew Cook has a Java applet Space Jewels ( screenshot ) that explores hyperbolic space and others. http: www.paradise.caltech.edu ~cook Workshop Java SpaceJewels main.html . He has a write up here . He's website is very interesting. Another DOS program related to hyperbolic geometry is Silicon Alley's NegaNaut at http: www.silicon-alley.com cat neganaut.html . Tilings and Patterns 2004-01. Bob is a Windows program that does Penrose tilings, by Stephen Collins. http: www.stephencollins.net penrose 2002-10. Taprats is a Java applet that generates Islamic patterns. Excellent. Craig S Kaplan! Home page: http: www.cs.washington.edu homes csk taprats Tyler is a Java applet that draws tilings. Superb! It is written by Melinda Green and Don Hatch. http: www.superliminal.com geometry tyler gallery Hyperbolic Tesselations Applet is a Java applet that draws hyperbolic tilings. Superb! By Don Hatch. http: www.plunk.org ~hatch HyperbolicApplet Kali ( Mac screenshot ; Windows screenshot ) is a freeware by Jeff Weeks. It is an interactive program for drawing figures of plane symmetry. The program is so appealing that it is suitable for grade-level kids too. An user selected symmetry pattern automatically appears based on whatever you draw on the screen. Mac and Windows versions at: http: www.geometrygames.org Kali index.html . Artificial Life Cellular Automata This section cover subjects like neuro networks, artificial life, genetic programing, cellular automata. These are sometimes lumped under the heading of Artificial Intelligence. Cellular Automata is having a bunch of cells with simple rules and see them evolve. (see the many intros and tutorials elsewhere below.). The most famous cullular automata is Conway's _Game of Life_. Artificial life relate to things like genetic programing, genetic algoritms, or boids. Conway's Game of Life is a hugely popular on the internet. There any many sites dedicated to it. Check this one by Al Hensel to begin: http: hensel.lifepatterns.net Alex Kasprzyk has a page dedicated to Artificial Life programs for the Mac: http: www.kasprzyk.demon.co.uk www ALHome.html LifeLab by Andrew Trevorrow. This patterns shows a gun battering with obstacles. A glider is generated shooting back at the gun but will miss. LifeLab v.3 ( 1999 screenshot ) by Andrew Trevorrow. LifeLab plays the infamous Conways' cellular automata Game of Life. I've seen quite a few game of life programs on the Mac and I pronouce LifeLab the best. Its main features are: fairly fast, automatic glider deletion, auto repetition and symmetry detection, auto expansion of grid, arbitrary rules, comes with a good library of patterns, and reads patterns of several formats. It also plays 1D CA. LifeLab is shareware. http: www.trevorrow.com index.html . Game of Life demonstrates that a deterministic system could be unpredictable, and suggests that the universe could be discrete (as opposed to continuous. (then, what it means to say that the universe is either discrete or continuous is quite philosphical.)). See Martin Gardner's excellent introduction in Weels, Life, and Other Mathematical Amusements (W.H. Freeman and Co., 1988). Cellebration by Mirek Wojtowicz plays some 12 types of CAs. This Java applet plays many types of CAs. The complete source code and binary are available for download.. http: www.mirwoj.opus.chelm.pl ca mjcell mjcell.html . This site cell-auto.com by Tim Tyler has lots of info on CA, including Java Applets. There's also George Maydwell's collidoscope at collidoscope.com . It is a hexagonal color CA; a screensaver type of software that runs on Windows. The site features CA gallery modernca using Java Appletes. (not interactive). 2002-05: many good cellular java appletes programs written by computer science students on a CA course. Includig cellular automata on triangular and hexagonal grids, as well as some regular tilings. Professor is Carter Bays. http: www.cse.sc.edu ~bays CAhomePage A flock of birds and tapeworms in Ishihama Yoshiaki's Simulations. Simulations by Ishihama Yoshiaki. This program does lots of artificial life simulations. The program is a bit of crude. It works like a screensaver, where you just watch the interactions on the screen, and you can adjust some parameters. The program has simulations like flocks of bird, fish school, worms, swarms, and also simulate centipedes showing its gaits, and then spiders, ants, etc. It's basically a collection of simulations that satisfies the programer's curiosities himself. Not a full-fledged program nor polished, but very inspiring. http: www.asahi-net.or.jp ~hq8y-ishm Boid by Craig Reynolds, is one of the original boid machinenary. Basically, it is computer simulation of the bahavior of flock of birds or school of fishes. The gist is that each move according to the behavior of their nearest neighbors. Fascinating. http: www.red3d.com cwr boids Fractals Dynamic Systems Board Games Puzzles Screen Savers Software for Math Professionals or programers Last Updated: 2005-05. copyright 1995-2005 by Xah Lee . ( xah@xahlee.org ) http: xahlee.org PageTwo_dir MathPrograms_dir mathPrograms.html
Descriptive Geometry
Software for creating and printing drawings, including conicsections, color fillings in Monge's projection and axonometry.
Descriptive geometry Czech page Descriptive geometry What this program can: speedy delineation of a drawing and its printing, saving to formats BMP and WMF assists you in solving the exercises of the descriptive geometry supports Monge's projection, axonometry and perspective user can create new methods and projections more information in User's Guide How it's working: it processes procedures written in the text-form runs under Windows 95 98 NT ME and XP or newer Success of this program: 1. place in the national round of the Czech republic of The Scholarly Activity of Secondary Schools - SOC in the discipline 01. mathematics and mathematical informatics, which was hold 10.-12. 6. 1999. 10 point (of possible 10) in Chip CD 11 99 (Czech Edition) This program is successfully used in practical education at schools in the Czech Republic What's new in version 1.2: easy drawing with mouse New version 1.3: program can be translated to any language, all texts, menu and commands are saved in separate files supports English and Czech Getting the program: Download shareware version 1.31 (1.03 MB, English) Download translator for DG programs (between Czech and English) (166 kB) In the case of interest for the full version of this program see order form or write to author. Price is only $22 including postage. If you want use more copies you can buy multi-licence at better price. The price of the program according to the count of licenses: 1 $22 5 $73 10 $123 20 $208 50 $413 Contact: MP Konsulta, s.r.o. Heyrovskho 974 Trebc 674 01 Czech Republic e-mail: Updated: 12. 9. 2005
TTC
Mathematica package for doing tensor and exterior calculus on differentiable manifolds.
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Dave Wilson's Poincar Notes
A page describing some Cabri macros relating to hyperbolic Geometry in the Poincare disc model.
Poincar Notes A massively disorganised collection of macros and figures for the Poincar Disk model of hyperbolic geometry. These will get slightly more organised (and modified) as I shall be teaching a unit on non-euclidean geometries (August 2001). The first file is a Cabri Menu file containing all the major macros, the remainder mainly duplicate ones macros contained within this. Poincar Menu this one should work on both Mac and Win (The P-distance macro refused to work in WinCabri; I had to create the macro using WinCabri, transport it across to the server. And now I know why! WinCabri cannot construe but only ) Poincar Menu the P-distance macro does not work on WinCabri (damn! merde!) P-Line.mac basic Poincar line P-Lineb.mac Poincar line if the 2 points are on the boundary P-seg.mac Poincar segment P-Circ.mac Poincar circle P-Angle.mac measures the angle formed by 3 points P-Dist.mac distance. I used the log cross-ratio form. P-distance an alternative form which should work on both MacCabri and WinCabri Orthog1.mac orthogonal P-line from a point to a P-line Orthog2.mac orthogonal P-line from a point on the boundary, to a P-line Orthog3.mac orthogonal to 2 P-lines P-mirr.mac constructs the P-line in which 2 points are inverted P-mirrb.mac deals with the cases when the 2 points are on a diameter of the disk And some figures, A test of the 2 P-distance macros Poincar Objects shows some of the stuff in one file if you want a quick look, Rotate.fig shows a rotating triangle: reflection in a pair of P-lines from an intersecting pencil Trans.fig 'free translation': reflections in a pair of P-lines with a common orthogonal -hyperparallels Trans2.fig translation, this time using P-lines form the same pencil, ie all with the same common orthogonal ParDisp.fig parallel displacement: reflections in a pair of P-lines which are parallel ParDisp2.fig parallel displacement: P-lines from the same pencil of parallels. Triangle Angle Sum a Cabri figure Triangle Angle Sum using CabriJava so that you do not need to have a local copy of Cabri. Triangle angle sum segments Old notes for these files: One day (huh) I'll learn to organise and to document stuff as I produce them (I'm horrified looking at the great piles of Cabri files stashed all over my hard disk - 330 according to a quick find file and there are more at work). I did mean to neaten them up and make them more elegant, but term started. I did try at the time to remember to provide help for each macro, which is not my usual practice. They seemed to work at the time of writing. Most of them were created over the Christmas break 96 97, and a couple modified slightly at a later stage - the angle measure was simplified to only require the 3 points (I made you also select the P-lines containing the angle) and the distance measure got an absolute value function added to keep it positive. I half remember creating a reflection (inversion) macro, not realising that the built in inversion construct worked when selecting an arc - I think that I then discarded mine, but it may lie around in a figure or 2. Using the menu file places the major macros onto the menu bar in a separate button. ). Does this work on the PC version? Perhaps giving them a .men suffix is needed? I ought to find out.(Yes it does, at least on the PC next door!) It does not include all the macros - notable exceptions are the orthogonal ones (it does now 14 8). The basic macros rely on the user to create a point and to keep it within the disk! I did experiment with a construction where a point was created on the boundary if a free point was dragged outside the disk, but did not like the effect (One of the macros for a circle exhibits similar behaviour as an outcome of that particular construction and without the side-effect I did not like). If you want to ensure that you can drag a point only up to the boundary, create a p-line and a point on that line (not entirely satisfactory, but ...). There are some macros to deal with certain special cases when you want to involve points on the boundary or at the centre of the Poincare disk. This was the real difficulty: trying to get the macros to be generic. There seemed to be so many special cases! There are still some special cases which fail. Some are trivial to deal with as a user (diameters, P-circles centre at the disk centre) as the Euclidean constructs apply. There are others. I became reluctant to produce a fistful of possibly confusing extras (P-circle centre on the boundary, the horocycle?) In order to explore some of the properties it is useful to be able to create points on the boundary. The macro for p-lines does allow one of the points to be on the boundary, but not both so I feel happy to include the boundary point constructs. I used the cross ratio to measure distance. I have not got round to checking that it works independently of observing its behaviour - the distances don't seem to get large enough to me, but they do seem consistent with, for example, drawing a series of equal radius Poincar- (non equal radius Euclidean-) circles. I conceived the P-circle as being the locus of a point always moving perpendicular to the p-segment joining it to a fixed point (equiangular spiral with a right angle). Coaxal circle stuff quickly gives this as a Euclidean circle and allows for its construction. Copy segment constructs a congruent segment (together with the line containing it). If you want it to go in a particular direction, use the P-circle construction to find all points the same distance from one of the end points of the segment. Dave Wilson Updated August 2001
C.a.R. Geometry Program
A free program allowing Compass and Ruler constructions. Now also available in Java.
C.a.R. C.a.R. Geometry Program This site is outdated. Please correct your bookmarks or links. The current version of C.a.R. runs on Java. Click here for the Java version If you are still interested in one of the old versions, look here .
Edu2000's Visual Geometry Series
A visual learning environment (full-curriculum, Mac Windows) that helps students explore and understand geometry.
Online algebra, geometry, trigonometry software demo Please visit Online algebra, geometry, trigonometry software demo . Thanks.
Cabri Geometry
The home site for Cabri Geometry, a dynamic geometry package .
Projet Cabri
Cabri Java beta version
Create dynamic geometrical figures using Cabri and publish them as Java applets.
CabriJava Cabri Java Project Parlez-vous franais ? Be careful: CabriJava needs a Java 1.1 or better compliant browser like Internet Explorer 5 5.5 6 ou Netscape 6 7 on all plateforms. Take a look at our Compatibility Page before testing CabriJava. Some examples of CabriJava figures in Web pages : Animations Jumping "cabri" Santa Claus Optical Mirror and lens Refraction Hemicylinder Electricity RLC circuit Mechanical Balance Combustion engine 3D Reuterswrd Geometry Pythagore1 Pythagore2 Trigonometry Loci Astroide Cardioide Deltode Conics Carnot Poncelet Thanks to contributors which have constructed all these examples. How you can construct your own examples with Cabri II software and CabriWeb ? All these examples can be downloaded by the Web for local testing at CabriJava.zip (~400 Kb) for Windows or CabriJava.sea.hqx (~420 Kb) for MacOS 9 CabriJava.dmg.gz (~540 Ko) for MacOS X You can choose only to download the latest version of CabriJava applet code CabriJava.jar.zip (~140 Kb) for Windows or Mac OS X or CabriJava.jar.sea.hqx (~160 Kb) for MacOS 9 A handbook can also be downloaded : handbook.zip (~280 Ko) or handbook.sea.hqx (~350 Ko) Cabri-Java Project - version 1.1.0 - 18 10 2004
Open Problems on Discrete and Computational Geometry
Compiled by Jorge Urrutia, University of Ottawa.
OpenProblems Open Problems on Discrete and Computational Geometry. Introduction: This web page contains a list of open problems in Discrete and Computational Geometry. Contributions to the list are invited. To contribute problems, submit them to me by e-mail, in html format. For each problem you pose, you may include one or two figures, in gif or jpg format. Make sure they are not too big, as this slows down their downloading time considerably . If any problem posed here is solved, I would appreciate it if you send me an e-mail to jorge@csi.uottawa.ca . In each problem you pose, include, to the best of your knowledge, who posed the problem first, and relevant references. Try to be short, concise and to the point. This will make your problems more attractive, and may increase the chances someone will read and try to solve them. If you detect inaccuracies regarding references, etc. in the problems posed here, please let me know so that I can correct them. At least until the end of this year, the format of this page will be evolving, until a satisfactory final layout is reached. Sorry for the inconveniences this may create. Jorge Urrutia, November, 1996. Table of contents Art Gallery or Illumination Problems Mirrors Illuminating a polygonal room of mirrors A forest of circular mirrors Mirrors and shadows Illumination of polygons Illuminating polygons with holes using vertex guards Shermer's problem on illuminating orthogonal polygons with holes using vertex guards Hoffman's problem on illuminating orthogonal polygons with holes using vertex guards Guarding a polygon with edge guards The prison yard problem for orthogonal polygons Floodlight illumination problems The stage illumination problem Illuminating polygons with vertex pi-floodlights Illuminating polygons with point pi-floodlights Illuminating polygons with vertex floodlights Optimal floodlight illumination of polygons Illuminating sets of polygons Illuminating families of triangles on the plane Illuminating the free space of a family of quadrilaterals Watchman routes The shortest watchman route with no starting point Optimal watchman routes to guard the exterior of two convex polygons Back to the Department's Home Page Back to my home page Jorge Urrutia, Department of Computer Science, University of Ottawa, 150 Louis Pasteur, P.O. Box 450 Stn A, Ottawa Ontario Canada, K1N 6N5 phone: (613) 562-5800 x6693, fax: (613) 562-5187 E-mail: jorge@csi.uottawa.ca
The Open Problems Project
A project to record open problems of interest to researchers in computational geometry and related fields.
The Open Problems Project Next: Numerical List of All The Open Problems Project edited by ErikD.Demaine - JosephS.B.Mitchell - JosephO'Rourke Introduction This is the beginning of a project 1 to record open problems of interest to researchers in computational geometry and related fields. It commenced with the publication of thirty problems in Computational Geometry Column42[ MO01 ] (see Problems 1-30 ), but has grown much beyond that. We encourage correspondence to improve the entries; please send email to TOPP@cs.smith.edu . If you would like to submit a new problem, please fill out this template . Each problem is assigned a unique number for citation purposes. Problem numbers also indicate the order in which the problems were entered. Each problem is classified as belonging to one or more categories. The problems are also available as a single Postscript or PDF file. To begin navigating through the open problems, you may select from a category of interest below, or view a list of all problems sorted numerically . Categorized List of All Problems Below, each category lists the problems that are classified under that category. Note that each problem may be classified under several categories. arrangements: 3-Colorability of Arrangements of Great Circles (Problem44) art galleries: Vertex -Floodlights (Problem23) coloring: 3-Colorability of Arrangements of Great Circles (Problem44) combinatorial geometry: k-sets (Problem7) Binary Space Partition Size (Problem14) Chromatic Number of the Plane (Problem57) Counting Polyominoes (Problem37) Distances among Point Sets in 2 and 3 (Problem39) Extending Pseudosegment Arrangements by Subdivision (Problem34) Monochromatic Triangles (Problem58) Pushing Disks Together (Problem18) The Number of Pointed Pseudotriangulations (Problem40) Thrackles (Problem30) Union of Fat Objects in 3D (Problem4) Vertical Decompositions in d (Problem19) combinatorial geometry.: Lines Tangent to Four Unit Balls (Problem61) convex hulls: Dynamic Planar Convex Hull (Problem12) Inplace Convex Hull of a Simple Polygonal Chain (Problem36) Output-sensitive Convex Hull in d (Problem15) data structures: Binary Space Partition Size (Problem14) Dynamic Planar Convex Hull (Problem12) Point Location in 3D Subdivision (Problem13) Delaunay triangulation: Flip Graph Connectivity in 3D (Problem28) folding and unfolding: Edge-Unfolding Convex Polyhedra (Problem9) General Unfoldings of Nonconvex Polyhedra (Problem43) Vertex-Unfolding Polyhedra (Problem42) Volume Maximizing Convex Shape (Problem62) graph drawing: 3D Minimum-Bend Orthogonal Graph Drawings (Problem46) Linear-Volume 3D Grid Drawings of Planar Graphs (Problem51) Queue-Number of Planar Graphs (Problem52) Smallest Universal Set of Points for Planar Graphs (Problem45) Thrackles (Problem30) graphs: Minimum-Turn Cycle Cover in Planar Grid Graphs (Problem53) Smallest Universal Set of Points for Planar Graphs (Problem45) Thrackles (Problem30) Traveling Salesman Problem in Solid Grid Graphs (Problem54) linear programming: Linear Programming: Strongly Polynomial? (Problem8) lower bounds: 3SUM Hard Problems (Problem11) Sorting X + Y (Pairwise Sums) (Problem41) meshing: Hexahedral Meshing (Problem27) Most Circular Partition of a Square (Problem59) minimum spanning tree: Bounded-Degree Minimum Euclidean Spanning Tree (Problem48) Euclidean Minimum Spanning Tree (Problem5) numerical computations: Sum of Square Roots (Problem33) optimization: Bounded-Degree Minimum Euclidean Spanning Tree (Problem48) Freeze-Tag: Optimal Strategies for Awakening a Swarm of Robots (Problem35) Minimum-Turn Cycle Cover in Planar Grid Graphs (Problem53) Packing Unit Squares in a Simple Polygon (Problem56) Pallet Loading (Problem55) Planar Euclidean Maximum TSP (Problem49) Traveling Salesman Problem in Solid Grid Graphs (Problem54) packing: Most Circular Partition of a Square (Problem59) Packing Unit Squares in a Simple Polygon (Problem56) Pallet Loading (Problem55) planar graphs: Bar-Magnet Polyhedra (Problem32) Pointed Spanning Trees in Triangulations (Problem50) point sets: k-sets (Problem7) Bounded-Degree Minimum Euclidean Spanning Tree (Problem48) Minimum-Turn Cycle Cover in Planar Grid Graphs (Problem53) Planar Euclidean Maximum TSP (Problem49) Simple Polygonalizations (Problem16) Smallest Universal Set of Points for Planar Graphs (Problem45) Surface Reconstruction (Problem26) Traveling Salesman Problem in Solid Grid Graphs (Problem54) polygons: Hinged Dissections (Problem47) Simple Polygonalizations (Problem16) Transforming Polygons via Vertex-Centroid Moves (Problem60) polyhedra: 3-Colorability of Arrangements of Great Circles (Problem44) Bar-Magnet Polyhedra (Problem32) Edge-Unfolding Convex Polyhedra (Problem9) General Unfoldings of Nonconvex Polyhedra (Problem43) Hamiltonian Tetrahedralizations (Problem29) Vertex-Unfolding Polyhedra (Problem42) reconstruction: Surface Reconstruction (Problem26) robotics: Freeze-Tag: Optimal Strategies for Awakening a Swarm of Robots (Problem35) scheduling: Freeze-Tag: Optimal Strategies for Awakening a Swarm of Robots (Problem35) shortest paths: Euclidean Minimum Spanning Tree (Problem5) Minimum Euclidean Matching in 2D (Problem6) Minimum-Link Path in 2D (Problem22) Shortest Paths among Obstacles in 2D (Problem21) simplification: Polygonal Curve Simplification (Problem24) Polyhedral Surface Approximation (Problem25) stabbing: Minimum Stabbing Spanning Tree (Problem20) traveling salesman: Minimum-Turn Cycle Cover in Planar Grid Graphs (Problem53) Planar Euclidean Maximum TSP (Problem49) Traveling Salesman Problem in Solid Grid Graphs (Problem54) triangulations: Compatible Triangulations (Problem38) Flip Graph Connectivity in 3D (Problem28) Hamiltonian Tetrahedralizations (Problem29) Minimum Weight Triangulation (Problem1) Pointed Spanning Trees in Triangulations (Problem50) Simple Linear-Time Polygon Triangulation (Problem10) The Number of Pointed Pseudotriangulations (Problem40) visibility: Trapping Light Rays with Segment Mirrors (Problem31) Vertex -Floodlights (Problem23) Visibility Graph Recognition (Problem17) Voronoi diagrams: Voronoi Diagram of Lines in 3D (Problem3) Voronoi Diagram of Moving Points (Problem2) Footnotes ... project 1 Partially supported by NSF grants to the three editors. Next: Numerical List of All The Open Problems Project - November 01, 2005
Steiner Trees
Open Problems with Steiner Trees, maintained by Joe Ganley.
Steiner Trees: Open Problems ganley.org - The Steiner Tree Page - Open Problems Open Problems Of course, there are probably about a zillion open problems related to Steiner trees, but here are a few I've thought about. Full trees. Hwang's theorem allows us to construct an optimal rectilinear Steiner tree of a full set in linear time. I know of no other metric or type of graph in which computing the optimal Steiner tree of a full set is polynomial-time solvable but computing a general Steiner tree is NP-hard. Note that there isn't even a sufficiently strong analogue of Hwang's theorem for rectilinear Steiner trees in three dimensions. Multidimensional rectilinear Steiner ratio. What is the rectilinear Steiner ratio in arbitrary dimension d? It is at least 2-1 d, as the d-dimensional analogue of the "cross" has this ratio. It is obviously at most 2. It is generally believed that the lower bound is correct, but this hasn't been proven. Even an upper bound lower than 2 would be interesting. Rectilinear Steiner arborescence. These are Steiner-like trees on points in the (first quadrant of the) plane, in which every segment in the tree is directed left to right or bottom to top. It is unknown whether computing an RSA is NP-complete. (A good reference to start with is Rao, Sadayappan, Hwang, and Shor .) SOLVED! J.-D. Cho published a paper in which a purported polynomial-time algorithm is presented for the problem. I was notified, however, by both Adil Erzin and Andrew Kahng, that it turns out the network flow problem to which Cho reduced the RSA problem is, itself, NP-complete. SOLVED! Shi and Su, in a paper in the 2000 Symposium on Discrete Algorithms, have proven that the RSA problem is (as suspected) NP-complete. It's a very pretty proof, too! Line separators. Smith proves that there exists a separator of length O(sqrt(n)) in the Hanan grid graph that cuts an optimal rectilinear Steiner tree in at most O(sqrt(n)) places. On the other hand, Deneen, Shute, and Thomborson prove that with high probability, a horizontal or vertical line bisecting the terminal set is crossed at most O(sqrt(n)) times by some optimal RST. I conjecture that there always exists a horizontal or vertical line that separates the terminal set into two subsets, each of size at most pn for p 1, such that the line is crossed by some optimal RST at most O(sqrt(n)) times. A proof of this would lead to a more implementable version of Smith's exact algorithm (or a non-randomized version of Deneen, Shute, and Thomborson's). SOLVED! Warren Smith has come up with a counterexample consisting of a spiral of points such that every line that separates the points into two O(n)-sized subsets intersects the optimal RST O(n) times, disproving the conjecture. He has also proven a number of related results, including a slightly restricted version of the original conjecture. Details will appear here once Warren has written them up. Bounding the number of rectilinear full sets. The best known bound on the worst-case number of full sets on n terminals is O(n1.384n) [ Fmeier and Kaufmann ]. However, empirically the number of full sets seems quite polynomial -- perhaps O(n log n) [ Salowe and Warme ]. These bounds might be narrowed from either side.
Open Problems from the Geometry Junkyard
Compiled by David Eppstein of the University of California at Irvine.
The Geometry Junkyard: Open Problems Open Problems Antipodes . Jim Propp asks whether the two farthest apart points, as measured by surface distance, on a symmetric convex body must be opposite each other on the body. Apparently this is open even for rectangular boxes. Bounded degree triangulation . Pankaj Agarwal and Sandeep Sen ask for triangulations of convex polytopes in which the vertex or edge degree is bounded by a constant or polylog. Centers of maximum matchings . Andy Fingerhut asks, given a maximum (not minimum) matching of six points in the Euclidean plane, whether there is a center point close to all matched edges (within distance a constant times the length of the edge). If so, it could be extended to more points via Helly's theorem. Apparently this is related to communication network design. I include a response I sent with a proof (of a constant worse than the one he wanted, but generalizing as well to bipartite matching). The chromatic number of the plane . Gordon Royle and Ilan Vardi summarize what's known about the famous open problem of how many colors are needed to color the plane so that no two points at a unit distance apart get the same color. See also another article from Dave Rusin's known math pages. Covering points by rectangles . Stan Shebs discusses the problem of finding a minimum number of copies of a given rectangle that will cover all points in some set, and mentions an application to a computer strategy game. This is NP-hard, but I don't know how easy it is to approximate; most related work I know of is on optimizing the rectangle size for a cover by a fixed number of rectangles. Cube triangulation . Can one divide a cube into congruent and disjoint tetrahedra? And without the congruence assumption, how many higher dimensional simplices are needed to triangulate a hypercube? For more on this last problem, see A lower bound for the simplexity of the cube via hyperbolic volumes , W. D. Smith, Eur. J. Comb.; Triangulating an n-dimensional cube , S. Finch, MathSoft; and Asymptotically efficient triangulations of the d-cube , Orden and Santos. Embedding the hyperbolic plane in higher dimensional Euclidean spaces. D. Rusin summarizes what's known; the existence of an isometric immersion into R4 is apparently open. Geombinatorics: Making Math Fun Again . A journal of open problems of combinatorial and discrete geometry and related areas. Geometric graph coloring problems from " Graph Coloring Problems ", a book by T. Jensen and B. Toft including a chapter on geometric and combinatorial graphs. Hermite's constants . Are certain values associated with dense lattice packings of spheres always rational? Part of Mathsoft's collection of mathematical constants . Integer distances . Robert Israel gives a nice proof (originally due to Erds) of the fact that, in any non-colinear planar point set in which all distances are integers, there are only finitely many points. Infinite sets of points with rational distances are known, from which arbitrarily large finite sets of points with integer distances can be constructed; however it is open whether there are even seven points at integer distances in general position (no three in a line and no four on a circle). Mirrored room illumination . A summary by Christine Piatko of the old open problem of, given a polygon in which all sides are perfect mirrors, and a point source of light, whether the entire polygon will be lit up. The answer is no if smooth curves are allowed. See also Eric Weisstein's page on the Illumination Problem . Open problems: Demaine - Mitchell - O'Rourke open problems project From Jeff Erickson, Duke U. From Jorge Urrutia, U. Ottawa . From the 2nd MSI Worksh. on Computational Geometry . From SCG '98 . Primes of a 14-omino . Michael Reid shows that a 3x6 rectangle with a 2x2 bite removed can tile a (much larger) rectangle. It is open whether it can do this using an odd number of copies. Prince Rupert's tetrahedra? One tetrahedron can be entirely contained in another, and yet have a larger sum of edge lengths. But how much larger? From Stan Wagon's PotW archive . Pushing disks together . If unit disks move so their pairwise distances all decrease, does the area of their union also decrease? A quasi-polynomial bound for the diameter of graphs of polyhedra , G. Kalai and D. Kleitman, Bull. AMS 26 (1992). A famous open conjecture in polyhedral combinatorics (with applications to e.g. the simplex method in linear programming) states that any two vertices of an n-face polytope are linked by a chain of O(n) edges. This paper gives the weaker bound O(nlog d). Rational triangles . This well known problem asks whether there exists a triangle with the side lengths, medians, altitudes, and area all rational numbers. Randall Rathbun provides some "near misses" -- triangles in which most but not all of these quantities are irrational. See also Dan Asimov's question in geometry.puzzles about integer right-angled tetrahedra. Squares on a Jordan curve . Various people discuss the open problem of whether any Jordan curve in the plane contains four points forming the vertices of a square, and the related but not open problem of how to place a square table level on a hilltop. This is also in the geometry.puzzles archive . Sums of square roots . A major bottleneck in proving NP-completeness for geometric problems is a mismatch between the real-number and Turing machine models of computation: one is good for geometric algorithms but bad for reductions, and the other vice versa. Specifically, it is not known on Turing machines how to quickly compare a sum of distances (square roots of integers) with an integer or other similar sums, so even (decision versions of) easy problems such as the minimum spanning tree are not known to be in NP. Joe O'Rourke discusses an approach to this problem based on bounding the smallest difference between two such sums, so that one could know how precise an approximation to compute. Tiling problems . Collected at a problem session at Smith College, 1993, by Marjorie Senechal. Tiling the unit square with rectangles. Will all the 1 k by 1 (k+1) rectangles, for k 0, fit together in a unit square? Note that the sum of the rectangle areas is 1. According to fourth-hand rumor, Marc Paulhus can fit them into a square of side 1.000000001, to appear in J. Comb. Th. Erich Friedman shows that the 5 6 by 5 6 square can always be tiled with 1 (k+1) by 1 (k+1) squares . Triangulations with many different areas . Eddie Grove asks for a function t(n) such that any n-vertex convex polygon has a triangulation with at least t(n) distinct triangle areas, and also discusses a special case in which the vertices are points in a lattice. Unfolding convex polyhedra . Catherine Schevon discusses whether it is always possible to cut a convex polyhedron's edges so its boundary unfolds into a simple planar polygon. Dave Rusin's known math pages include another article by J. O'Rourke on the same problem. Unsolved problems . Naoki Sato lists several conundrums from elementary geometry and number theory. From the Geometry Junkyard , computational and recreational geometry pointers. Send email if you know of an appropriate page not listed here. David Eppstein , Theory Group , ICS , UC Irvine . Semi-automatically filtered from a common source file.
Open Problems
Collected by Jeff Erickson. Mainly in geometry.
Open Problems Open Problems These are open problems that I've encountered in the course of my research . Not surprisingly, almost all the problems are geometric in nature. A name in brackets is the first person to describe the problem to me; this may not be original source of the problem. If there's no name, either I thought of the problem myself (although I was certainly not the first to do so), or I just forgot who told me. Problems in bold are described in more detail than the others, and are probably easier to understand without a lot of background knowledge. If you have any ideas about how to solve these problems, or if you have any interesting open problems you'd like me to add, please let me know . I'd love to hear them! 30 Jul 2003: Complete or partial solutions for several of these problems have been discovered in the two years since I last updated this site. Over the next few weeks, I'm planning to add pointers to these new results, as well as descriptions of several new open problems. (Search for "soon" on this page.) Stay tuned! Existence Problems: Does Object X exist? Unfolding convex polytopes [ Joe O'Rourke and Komei Fukuda ] Acute triangulation of the cube [Alper ngr] (soon) Combinatorial Problems : How complex is Object X? Degenerate facets of polytopes Faces of intricate polytopes Point-hyperplane incidences - update soon Union of intersecting unit balls - Solved! Line traversals of Delaunay triangulations [ Nina Amenta ] - Solved! Halving lines and k-sets Tangent pairs of (pseudo-)circles Medial surfaces of polyhedra and Voronoi diagrams of lines Forced convex subsets [The Erds-Szekeres "happy end" problem] Maximum genus of an n-vertex 3d polyhedron (soon) Minimum spread of a neighborly triangulation (soon) Visibility complex of disjoint unit spheres (soon) Algorithmic Problems : How fast can Problem X be solved? Minimum-area triangles Complex colinearities [ David Eppstein ] Extreme points A dynamic programming problem [ David Eppstein ] Shortest paths in line arrangements [ Marc van Kreveld ] - update soon Straight skeleton of a simple polygon [ Franz Aurenhammer ] Crashing motorcycles efficiently - update soon Klee's measure problem Generating random simple polygons Building convex polytopes [ Erik Demaine and others] Other People's Problems Some geometrical problems! from Olivier Devillers Lots of stuff from David Eppstein : Open problems from The Geometry Junkyard Open problems in mesh generation Some of David's own research projects Unsolved Mathematics Problems collected by Steven Finch Bounty problems collected by Peter Kwok , including several problems first posed by Paul Erds [According to Ron Graham, rewards for Erds problems may not be honored now that Erds has died.] Open Problems on Discrete and Computational Geometry from Jorge Urrutia Combinatorial Geometry Open Problems contributed by Pavel Valtr to the DIMACS collection of Open Problems for Undergraduates Open problems from the 1996 AMS summer research conference "Discrete and Computational Geometry: Ten Years Later", some of which have already been solved Issue 8 of the computational geometry newsletter CG Tribune includes several open problems from the 1997 Dagstuhl computational geometry workshop. Electronic Scottish Caf: Open problems from Ulam Quarterly : part 1 and part 2 Recommended Books Unsolved Problems in Geometry by Hallard Croft, Kenneth Falconer , and Richard Guy (Springer-Verlag, 1991) Old and New Unsolved Problems in Plane Geometry and Number Theory by Victor Klee and Stan Wagon (MAA, 1991). We should try to love the questions themselves, like locked rooms and like books that are written in a very foreign tongue. - Rainer Maria Rilke Problem solving is hunting. It is savage pleasure, and we are born to it. - Thomas Harris, The Silence of the Lambs If you keep proving stuff that others have done, getting confidence, increasing the complexities of your solutions - for the fun of it - then one day you'll turn around and discover that nobody actually did that one! And that's the way to become a computer scientist. - Richard Feynmann, Feynmann Lectures on Computation Open Problems - Jeff Erickson ( jeffe@cs.uiuc.edu ) 09 Apr 2001
Algebraic Geometry Notebooks for Non-Experts
By Aksel Sogstad. Short introductory sketch of some topics in the algebraic geometry of curves.
The Algebraic Geometry Notebooks For Non-Experts By Aksel Sogstad
Algebraic Curves
An overview.
8.1 Algebraic Curves Next: 8.2 Roulettes (Spirograph Curves) Up: 8 Special Plane Curves Previous: 8 Special Plane Curves 8.1 Algebraic Curves Curves that can be given in implicit form as f(x,y)=0, where f is a polynomial, are called algebraic. The degree of f is called the degree or order of the curve. Thus conics (Section 7 ) are algebraic curves of degree two. Curves of degree three already have a great variety of shapes, and only a few common ones will be given here. The simplest case is when the curve is the graph of a polynomial of degree three: y=ax +bx +cx+d, with a 0. This curve is a (general) cubic parabola (Figure 1 ). It is symmetric with respect to the point B where x=-b 3a. Figure 1: The general cubic parabola for a 0. For a 0, reflect in a horizontal line. The semicubic parabola (Figure 2 , left) has equation y =kx ; by proportional scaling one can take k=1. Figure 2: The semicubic parabola, the cissoid of Diocles, and the witch of Agnesi This curve should not be confused with the cissoid of Diocles (Figure 2 , middle), which has equation (a-x)y =x with a 0. The latter is asymptotic to the line x=a, while the semicubic parabola has no asymptotes. The cissoid's points are characterized by the equality OP=AB in Figure 2 , right. One can take a=1 by proportional scaling. More generally, any curve of degree three with equation (x-x )y =f(x), where f is a polynomial, is symmetric with respect to the x-axis and asymptotic to the line x=x . In addition to the cissoid, the following particular cases are important: The witch of Agnesi has equation xy =a (a-x), with a 0, and is characterized by the geometric property shown in Figure 2 , right. The same property provides the parametric representation x=a(1+sin 2 ), y=a tan . Once more, proportional scaling reduces to the case a=1. The folium of Descartes (Figure 3 , left) has equation (x-a)y =-x ( x+a), with a 0 (reducible to a=1 by proportional scaling). By rotating 135(right) we get the alternative and more familiar equation x +y =cxy, where c= a. The folium of Descartes is a rational curve, this is, it has a parametric representation by rational functions. In the tilted position such a representation is x=ct (1+t ), y=ct (1+t ) (so that t=y x). Figure 3: The folium of Descartes in two positions, and the strophoid. The strophoid has equation (x-a)y =-x (x+a), with a 0 (reducible to a=1 by proportional scaling). It satisfies the property AP=AP'=OA in Figure 3 , right; this means that POP' is a right angle. The strophoid has the polar representation r=-a cos 2 sec , and the rational parametric representation x=a(t -1) (t +1), y=at(t -1) (t +1) (so that t=y x). Among the important curves of degree four are the following: A Cassini's oval is characterized by the following condition: given two foci F and F', a distance 2a apart, a point P belongs to the curve if the product of the distances PF and PF' is a constant k . If the foci are on the x-axis and equidistant from the origin, the curve's equation is (x +y +a ) -4a x =k . Changes in a correspond to rescaling, while the value of k a controls the shape: the curve has one smooth piece, one piece with a self-intersection, or two pieces depending on whether k is greater than, equal to, or smaller than a (Figure 4 ). The case k=a is also known as the lemniscate (of Jakob Bernoulli); the equation reduces to (x +y ) =a (x -y ), and upon a 45 rotation to (x +y ) =2a xy. Each Cassini's oval is the section of a torus of revolution by a plane parallel to the axis of revolution. Figure 4: Cassini's ovals for k=.5a, .9a, 1.1a and 1.5a (from the inside to the outside). The foci (dots) are at x=a and x=-a. The black curve, k=a, is also called Bernoulli's lemniscate. A conchoid of Nichomedes is the set of points such that the signed distance AP in Figure 5 , left, equals a fixed real number k (the line L and the origin O being fixed). If L is the line x=a, the conchoid has polar equation r=a sec +k. Once more, a is a scaling parameter, and the value of k a controls the shape: when k -a the curve is smooth, when k=-a there is a cusp, and when k -a there is a self-intersection. The curves for k and -k can also be considered two leaves of the same conchoid, with cartesian equation (x-a) (x +y )=k x . Figure 5: Defining property of the conchoid of Nichomedes (left), and curves for k=.5a, k=a, and k=1.5a (right). A limaon of Pascal is the set of points such that the distance AP in Figure 6 , left, equals a fixed positive number k measured on either side (the circle C and the origin O being fixed). If C has diameter a and center at (0,a), the limaon has polar equation r=a cos + k, and cartesian equation (x +y -ax) =k (x +y ). The value of k a controls the shape, and there are two particularly interesting cases. For k=a we get a cardioid (see also Figure 8.2.2 , right). For a=k we get a curve that can be used to trisect an arbitrary angle : if we draw a line L through the center of the circle C and making an angle with the positive x-axis, and if we call P the intersection of L with the limaon a=2k, the line from O to P makes an angle with L. Figure 6: Defining property of the limaon of Pascal (left), and curves for k=1.5a, k=a, and k=.5a (right). The middle curve is the cardioid, the one on the right a trisectrix. Hypocycloids and epicycloids with rational ratio (see next section) are also algebraic curves, generally of higher degree. Next: 8.2 Roulettes (Spirograph Curves) Up: 8 Special Plane Curves Previous: 8 Special Plane Curves The Geometry Center Home Page Silvio Levy Wed Oct 4 16:41:25 PDT 1995 This document is excerpted from the 30th Edition of the CRC Standard Mathematical Tables and Formulas ( CRC Press ). Unauthorized duplication is forbidden.
The Brill-Noether Project
Aims to answer the basic questions this theory B(n,d,k) for vector bundles on algebraic curves. Welcomes contributions.
bnt.html THE BRILL-NOETHER PROJECT To celebrate New Year 2003 I propose the following project: to answer the basic questions in Brill-Noether theory for vector bundles on algebraic curves within 10 years, that is by 1 January 2013. By basic questions, I mean: for a general curve C and for all triples of integers (n,k,d) with n greater than or equal to 2, k greater than or equal to1, to obtain the following information concerning the Brill-Noether locus B(n,d,k), consisting of all stable bundles of rank n, degree d, with at least k independent global sections: (1) Is B(n,d,k) non-empty? (2) Is B(n,d,k) connected, and, if not, what are its connected components? (3) Is B(n,d,k) irreducible, and, if not, what are its irreducible components? (4) What is the dimension (of each component) of B(n,d,k)? (5) What is the singular set of B(n,d,k)? For many (n,d,k), complete or partial information is known and I will try to post the latest information on this page. The project is an open one; all contributions are welcome (and will be needed) if we are to answer these questions. There are of course many further questions, for example to discuss the detailed geometry of the Brill-Noether loci, their classes in the Chow ring and or cohomology ring of the moduli space, and the links with the moduli spaces of coherent systems, but I have chosen to concentrate on these basic ones to focus attention. The intention is that this page should be linked to other pages containing information on these problems. In the first instance, I would like to compile files of people who are interested in Brill-Noether theory and of papers, both with links where these are available. Please let me know by email if you would like to be included in the file of people and give me your web address so I can put in a link to your own webpage. (If you don't have a webpage or prefer not to be linked, I can include your email address.) For the time being, pending reconstruction of this page, see Vincent Mercat's presentation. Peter Newstead newstead@liv.ac.uk
Introduction to Algebraic Geometry
Illustrated webnotes by Donu Arapura.
Algebraic Geometry Introduction to Algebraic Geometry (Math 665, spring 06) Donu Arapura Blow up Examples Geometry of 2x2 nilpotent matrices Blowing up a cusp Lines in projective 3 space Computer Examples Nilpotent matrices revisited Singular Locus Pluecker Equation Fano Variety Dual Curve Books Examples In order to get a feeling for what algebraic geometry is, let's to go through some simple examples. Geometry of 2x2 nilpotent matrices Consider the space of 2x2 matrices over a field k with trace 0. These can be parameterized by 3-tuples (x,y,z) where x, y and z appear in the 11, 12, and 21 positions; therefore this can be identified with three dimensional affine space A3. In these coordinates the determinant det(x,y,z) = -x2 -yz. The determinant is an regular map A3 A1. The fibers Xt=det-1(t) = {(x,y,z) | det(x,y,z) = t} are algebraic subsets of A3. These are in fact varieties since the polynomials -x2 -yz-t are irreducible for each t in k. Let's study the geometry of these sets. Suppose k=C, then I claim that Xt is isomorphic to X1 whenever t is nonzero. To see this, choose a such that a2 = t, then the map (x,y,z) (ax,ay,az) defines an isomorphism between X1 and Xt; this can be checked by comparing coordinate rings. ( This works even if C is replaced by an algebraically closed field. But it fails in general. For example when k = R, X1 is connected in it's usual topology while X-1 isn't.) I claim that X1 is a homogeneous space which implies that any point looks like any other point. To see this, observe that the algebraic group SL2(C) acts on X1 by matrix conjugation, and that this is a transitive action. This last statement can be checked by brute force: a matrix A in SL2(C) sends (1,0,0) to (1+2a21a12, -2a11a21, 2a12a22)... X1 X0 is the space 2x2 matrices with zero trace and zero determinant. It follows by the Cayley-Hamilton theorem that this is precisely the set of nilpotent matrices of the form N2=0. There is a subtle point here. If I is the ideal generated by trace and det of a generic 2x2 matrix, and J is the ideal generated by the entries of its square, then the I and J have the same radical but they not the same (see below ). Thus they define different schemes with the same reduced structure. The zero matrix is a singular point of, while the other points are nonsingular. This can be checked by setting the partial derivatives of the equation -x2 -yz to zero. It follows that X0 is not homogeneous, and therefore X0 is not isomorphic to X1. Alternatively, these cannot be isomorphic since all the points of X1 are nonsingular. X0 What we've been doing so far is affine geometry. We get a little more insight into the structure of these sets by doing projective geometry. A matrix in X-1 has 1 and -1 as its eigenvalues. The eigenvectors span two distinct lines in C2. Conversely, a pair of distinct lines determines an element of X-1. Thus we have a bijection, and in fact isomorphism, between the X-1 and the product of two copies of the projective line minus the diagonal. Thus X-1 is a so called doubly ruled surface. These rulings, which are fibers of the projections onto the factors, are embedded as lines in A3. In fact, after a linear change of coordinates the embedding of X-1 into A3 extends to the Segre embedding of P1xP1 to P3. -1 Double ruling (red and black lines) It follows from above that X-1 is birational to the P1xP1 and therefore to A2. However, it is not isomorphic to A2. This is because the coordinate ring A(A2) = k[x1,x2] is a unique factorization domain, while A(X-1)= k[x,y,z] (1-x2-yz) isn't (the image of yz can be factored in two different ways, as (1-x)(1+x) and the obvious way). Blowing up a cusp The second example, which is the one indicated in the picture at the top of this page, is the blow up of the affine plane . This consist of a quasiprojective variety Bl and a morphism p:Bl A2, where Bl consists of pairs (x,L) where x is point in A2 and L point in P1 containing x, and p(x,L) = x. The morphism p induces isomorphism p -1 A2-{(0,0)} A2-{(0,0)}. Therefore p is a birational equivalence . However, p is not an isomorphism, since the preimage E = p -1 (0,0) is P1 rather than a point. Bl can be described as a union of two affine varieties, Bl1={(x,y,t) | y=xt} and Bl2={(x,y,u) | x=yu} glued via (x,y,t) (x,y,1 t). Bl1 is blue surface depicted above. The curve C = {(x,y) | y2=x3} in A2 is singular at (0,0). Its preimage under p, called its total transform, is the union of E with a curve C2 (the red curve). C2, which can be described as the closure of the preimage of C-{(0,0)}, is called the strict transform. This curve lies in Bl1 and is given parametrically by x=s2, y=s3, t=s. The map A1 C2 given by s (x,y,t) is an isomorphism, since t s is the inverse, therefore C2 is smooth. Since C2 and C are isomorphic away from (0,0), they are birational. The map on coordinate rings A(C) = k[x,y] (y2-x3) A(A1) = k[s] is given by x s2, y s3, so A(C) can be identified with the subring of k[s] generated by s2 and s3. C2 C is an example of resolution of singularities. Exercise: Carry out a similar analysis for the node y2=x2(x+1) Lines in projective 3 space A line in the projective plane is given by a nonzero linear equation in 3 variables, and two equations determine the same line if they are multiples of each other. Thus the set of lines in P2 is parameterized by another P2 called the dual plane. Now let's look at the the set of lines in P3. we claim that the set of these lines is parameterized by a projective variety called the Grassmanian G(2,4). We want to know how big it is; in other words, we want to compute its dimension of . A line in P3 is determined by a pair of distinct points, thus we arrive at an upper bound 6=2x3. Under the identification P3 =k4-{0} scalars, a line corresponds to a two dimensional subspace of V= k4. A pair of distinct points on the projective corresponds to a pair of linear independent vectors v1, v2 i.e. a basis of the subspace. Let M be the 2x4 matrix with rows given by the v's. M and M' determine the same subspace if and only if M' = AM for A GL2(k). Thus we could hope to identify G(2,4) with quotient of the affine space of 2x4 matrices by GL2(k). In particular, we should obtain the dimension as the difference 4=2x4-4. Since there is no a priori reason for the quotient variety to exist, we sketch an alternative construction. We form the wedge product of the vectors pl(M)= v1 v2 in the 2nd exterior power 2V. Note that pl(AM) = det(A)pl(M), thus the class of [pl(M)] in the projective space P(V) depends only on the point of the Grassmanian. This gives an embedding of G(2,4) to P(V). The image is the set of points satisfying the Pluecker condition pl pl =0. After identifying 2V with k6, the Plueker condition becomes a polynomial condition (see below ). Thus G(2,4) becomes a hypersurface in P5. In particular, its dimension is 4 as claimed. Let U G(2,4) be the open set of lines meeting A3 and with nonconstant projection to the x-axis. The lines in U admit a unique parameterization of the form y =ax + b, z=cx+d. Thus U is isomorphic to A4. Thus G(2,4) is birational to A4. Computer Examples There are a few of software packages that can help with working out examples. For general purpose algebraic manipulation, graphics etc. there's Maple or Mathematica . (I tend to use Maple since it's is available on most of our machines including all our Suns.) Some examples of the use of Maple for algebro-geometric calculations can be found in the books of Cox et. al below. Here, I'll just give the code for generating the graph of the blow up at the top of this page. with(plots): Bl := plot3d([x,x*t, t], x=-1..1, t=-1..1, style=WIREFRAME, color =blue): A2 := plot3d([x,y,0], x=-1..1, y=-1..1, color=yellow, style=PATCHNOGRID): E := spacecurve([0,0,t], t=-1..1, color=black, thickness=2): C := spacecurve([s^2, s^3, 0], s=-1..1, color=black, thickness=1): C2 := spacecurve([s^2,s^3,s], s=-1..1, color=red, thickness=2): display({Bl,A2, E, C, C2}); **** For doing calculations in algebraic geometry and commutative algebra, Grayson and Stillman's Macaulay2 program is more powerful than Maple or Mathematica. Documentation is available on the web . (There are a couple of other programs, CoCoA and Singular, with similar capabilities, but I'm less familiar with these.) To get a sense for what it can do, let's consider some simple examples. Assuming things have been set up properly, you can start the program by typing M2 in a terminal window of one of our Suns or whatever machine you're using. If you plan to do anything serious, you'll need to learn how to run it under emacs. Nilpotent matrices revisited As a first example, let's check a special case of the Cayley-Hamilton theorem. R = QQ[x_1..x_4] This sets up a polynomial ring over Q (= QQ in Macaulay) in 4 variables. Next define the ideal I generated by det and trace of the "universal" 2x2 matrix M by typing M = matrix{{x_1,x_2},{x_3,x_4}} D = det M T = trace M I = ideal {D, T} The algebraic set V(I) in A4 (= space of 2x2 matrices) is the set of matrices with det= trace = 0 Let Nilp be the set of matrices whose square vanishes. The entries of M2 generate an ideal J which can be constructed by J = ideal M^2 Then Nilp = V(J). By the Cayley-Hamilton theorem, V(I) coincides with Nilp as well. Let's check this directly. In order verify V(I) = V(J) (over any algebraically closed field of char. 0), it's enough, by the Nullstellensatz , to check the equality of the radicals of I and J: radical I == radical J Exercise: check whether I = J? If not, then what goes wrong? Singular Locus Next, let's check that the hypersurface f = z2 -(y-1)(y2-x)= 0 in A3 has exactly one singular point at (1,1,0) over an algebraically closed field of large positive characteristic (Macaulay2 computes more efficiently over finite fields). We need to clear the previous use of x before defining the new objects erase symbol x R = ZZ 31991[x,y,z] f = z^2 -(y-1)*(y^2-x) The graph (which is a bit misleading) suggests that the hypersurface might be reducible. We can check that it's irreducible over the prime field Z 31991 by trying to factor it: factor f Next define the ideal Jac generated by f and its partials. Jac = ideal { f, diff(x,f), diff(y,f), diff(z,f)} The singular locus of f is precisely V(Jac). It is enough by the Nullstellensatz to check the radical of Jac is the maximal ideal (x-1,y-1,z) radical Jac Pluecker Equation For the next example, we want to find the explicit Pluecker equations for G(2,4). Let's identify A8 with space of 2x4 matrices. Consider the morphism pl:A8 A6 given by sending a matrix to the vector of its 2x2 minors (in some order). Let X be the image of F, and let's find I(X) . Let R and S be the coordinate rings of these affine spaces. Rather than clearing all the previous variables on by one, we can restart the program: restart R = QQ[x_1..x_8] S = QQ[y_1..y_6] Let p be homomorphism of coordinate rings S R determined by pl. M = genericMatrix(R,x_1,2,4) M2 = exteriorPower(2,M) p = map(R,S, M2) Then I(X) = ker(p) ker p Fano Variety For this example, we show that the Fermat cubic surface in P3 has only finitely many lines on it. We do this by computing the dimension of the so called Fano variety of lines lying on it. To simplify our analysis, we work on the affine surface x3 + y3 + z3 + 1=0. Also we will be content to restrict our attention to the lines in U G(2,4) described above. We leave it as an exercise to finish the analysis. Substituting y= ax+b, z=cx+d into the above equation and expanding yields conditions on a,b,c,d which defines the Fano variety. Now, we compute the dimension of the coordinate ring F of the Fano variety. restart R = QQ[a..d] I = ideal{a^3+c^3+1, a^2*b+c^2*d, a*b^2+c*d^2, b^3+d^3+1} F= R I dim F should yield 0. Exercises: 1. Complete the analysis. 2. Any nonsingular cubic contains 27 lines. Find them in this example. Dual Curve For the last example, we compute the equation of the dual curve of the Fermat quartic curve C: x4 + y4 +z4 = 0. This is the set of tangent lines to this curve in the dual projective plane (P2)*. We can compute the dual curve by projecting the incidence variety {([x,y,z],[a,b,c]) P2x(P2)* | [x,y,z] C, a=4x3,b=4y3,c=4z4} to (P2)*, and then using elimination theory to find the relation among a,b,c. In algebraic terms this amount to intersecting the ideal I defining the incidence variety in k[a,b,c,x,y,z] with k[a,b,c]. We do this in 2 ways. In the first approach, we have to get our hands a bit dirty, but we do get a glimpse of what is really being computed behind the scenes. The idea (see Eisenbud pp 360-361 ) is to compute a Groebner basis with respect to an ordering called an elimination order with respect to x,y,z, and then discard the elements of this basis involving x,y, or z. It will turn out that the first element is the one we want, at least in Macaulay2. restart K= ZZ 31991 R = K[x,y,z,a,b,c, MonomialOrder = Eliminate 3] I = ideal {x^4+y^4+z^4, a-4*x^3, b-4*y^3, c-4*z^3} G = gens gb I G_{0} This should have degree 12. The second approach is simply to express the intersection of I with k[a,b,c] as the kernel of natural map f: k[a,b,c] k[x,y,z,a,b,c] I. Since Macaulay doesn't allow variables to live in different rings, so we use A, B, C for the initial ring and map these to a, b, c in the quotient by f. Q = R I S = K[A,B,C] f = map(Q,S,{a,b,c}) ker f When we're finished, type quit Books Harris was the text for this class the last time I taught it. I haven't decided about next semester. This and additional references are M. Atiyah, I. MacDonald, Commutative Algebra D. Cox, J. Little, D. O'Shea, Ideals, Varieties and Algorithms D. Cox, J. Little, D. O'Shea , Using Algebraic Geometry D. Eisenbud, Commutative Algebra D. Eisebud, J. Harris, Geometry of Schemes J. Harris, Algebraic geometry: a first course R. Hartshorne, Algebraic geometry* D. Mumford, Red book of varieties and schemes* I. Shafarevich, Basic Algebraic Geometry The starred references would be a little heavy going as an introduction. Back to beginning. Last revision Sep 27, 05
AGATHOS - Algebraic Geometry: A Total Hypertext Online System
An online system for learning algebraic geometry.
AGATHOS: Introduction AGATHOS Algebraic Geometry: A Total Hypertext Online System AGATHOS is a hypertext-based, online system for learning algebraic geometry. I developed AGATHOS while teaching a year-long graduate class at the University of Maryland in College Park. Consequently, it makes some nontrivial assumptions about the backgrounds of its readers. The most important assumption is that the reader has completed a year-long graduate course in abstract algebra. You can approach the subject in a variety of ways: As a linear text following the traditional order of presentation (definition, lemma, theorem, proof). Starting with a list of interesting examples of algebraic varieties. Starting with a list of important theorems in algebraic geometry. Starting with a list of key concepts in algebraic geometry. Starting with a list of exercises that can be solved using the ideas found elsewhere in these notes. agathos 1. adj. good. (From the Greek, .) 2. n. An online system for learning algebraic geometry. Warning: The course in question occured during Fall Semester 1998. (You may find the selected quotations from that semester somewhat entertaining.) The front end looks really impressive, but a great many links are still empty. Not long after that, I started working on bioinformatics instead of algebraic geometry (I know; the connection between the two seems a bit tenuous), and have had little time to fill in the blanks. Setup When I first put these pages together back in 1998 and 1999, the HTML 4.0 specification was new, and none of the available browsers supported the named entities needed to include mathematical symbols. So, I resorted to a work-around that used the Symbol font, pointing out that I knew this would eventually have to be changed. In recent months, I have received several comments from readers about symbols not being displayed correctly, so I decided it was time to update the display instructions. Sadly, the situtation is worse in 2005 than it was in 1999. Mozilla Firefox and the latest versions of Netscape (at least 7.0 and higher) have decided that it is "wrong" to use the Symbol font in the way that people have been using it for years. Instead, they have embraced the named entitites in the proposed HTML 4.0.1 standard. Current versions of Microsoft Internet Explorer (at least through version 6.0) fail to recognize the majority of the named entities. Even worse, instead of displaying the named entity (as they would if it were really unknown, like unknownEntity), which would allow most people to puzzle out the meaning, they replace many of them with useless blank boxes. And so, providers of mathematical content on the web have to make a choice. Do they continue to use the Symbol font (via FONT FACE="Symbol" ) and support Internet Explorer, or do they switch to the HTML 4.0 named entities and support Netscape and Firefox? For these pages, I have decided that Firefox is the future, so mathematical symbols and Greek characters in the documents on this site are provided using the HTML4.0 character entities. As a result, most of the pages are difficult or impossible to interpret using Internet Explorer. For readers unwilling to install the free version of Firefox as an alternate browser, the chapters are collected together as PDF files. In this way, at least, the situation has improved since 1999. Back then, I made TeX dvi files available (since most mathematicians had access to something that understood DVI files, even if their browser didn't quite know what to do with them). Improvements to pdftex and pdflatex have now made it easier to convert TeX source code into PDF outputs that look good. The disadvantage from the point of view of a truly hyperlinked text is that I haven't taken the time to make all teh hyperlinks work correctly in the PDF files, so that aspect of the text only works in the HTML version. Comments on this web site should be addressed to the author: Kevin R. Coombes Department of Biostatistics and Applied Mathematics University of Texas M.D.Anderson Cancer Center 1515 Holcombe Blvd., Box 447 Houston, TX 77030
Noncommutative Geometry and Algebra
People, groups and publications. Compiled by Paul Smith, University of Washington.
Paul Smith's Research Noncommutative geometry and algebra My main interest is the non-commutative world in all its aspects: geometric, algebraic, topological, physical, et cetera. Here are some homepages I like to visit. Some of these people are also interested in the non-commutative world: Jacques Alev John Baez Allen Bell Dave Benson David Ben-Zvi George Bergman Tom Bridgeland Ken Brown Tomasz Brzezinski Daniel Chan Alistair Craw Bill Crawley-Boevey Robbert Dijkgraaf Igor Dolgachev Paul Garrett Ken Goodearl Iain Gordon Mark Haiman Tim Hodges Martin Holland Colin Ingalls Srikanth Iyengar Peter Jorgensen Lars Kadison Sheldon Katz Dennis Keeler Bernhard Keller Alistair King Steffen Knig Henning Krause Rajesh Kulkarni Lieven Le Bruyn Tom Lenagan Edward Letzter Thierry Levasseur Martin Lorenz Kirill MacKenzie Shahn Majid Jim Milne Izuru Mori Dave Morrison John Murray Ian Musson Hiraku Nakajima Dmitri Nikshych Adam Nyman Sasha Polishchuk Claudio Procesi Miles Reid Markus Reineke Jeremy Rickard Claus Ringel John Roe Dan Rogalski Jonathan Rosenberg Dimitry Rumyin Aidan Schofield Karen Smith Darin Stephenson Balazs Szendroi Michaela Vancliff Michel Van den Bergh Chuck Weibel Alan Weinstein Amnon Yekutieli James Zhang Birge Zimmermann-Huisgen More pages. Antwerp Ring Theory Seminar University of Leeds Algebra Group Limburgs Universitair Centrum Algebra Research Group Trondheim Algebra Group University of Wisconsin-Milwaukee Algebra Group String Theory seminar at Duke University Representation theory of Finite dimensional algebras Official String Theory Web site Groupoids Homepage Calabi-Yau Homepage Groups, Representations and Cohomology Preprint Archive Hopf topology archive SGA by the Bourbakistas Lieven LeBruyn's noncommutative algebra geometry forum etc. Conferences: Generalized McKay Correspondences and Representation Theory , March 20-24, 2006. Higher dimensional geometry , April 1-6, 2006. Noncommutative algebra and algebraic geometry , May 7-13, 2006. PIMS UNAM Summer School , July 1-6, 2006. Here are some articles that might be of interest: Noncommutative curves and noncommutative surfaces by Toby Stafford and Michel Van den Bergh A Mad Day's Work: From Grothendieck to Connes and Kontsevich: evolution of the notions of space and symmetry by Pierre Cartier Noncommutative Geometry for Pedestrians by J. Madore Deformation quantization of algebraic varieties by Maxim Kontsevich Two Lectures on D-Geometry and Noncommutative Geometry by Michael Douglas A Point's Point of View of Stringy Geometry by Paul Aspinwall D-branes, Discrete Torsion and the McKay Correspondence by Paul Aspinwall and M. Ronen Plesser A short course in geometric motivic integration by Manuel Blickle Clay INstitute Lectures by Greg Moore Stacks for Everybody by Barbara Fantechi Algebraic Stacks by Tomas Gomez What is a stack? by Bill Fulton Notes on Stacks by Herb Clemens, Aaron Bertram, et al. Topological Stacks by Angelo Vistoli Topics in algebraic geometry-Algebraic stacks by A. Kresch A straight way to stacks by M. Romagny (Pre-)sheaves of Ring Spectra over the Moduli Stack of Formal Group Laws by Paul Goerss Discrete Torsion and Gerbes II by Eric Sharpe Algebraic orbifold quantum products by Dan Abramovich, Tom Graber, and Angelo Vistoli Gerbes over Orbifolds and Twisted K-theory by Ernesto Lupercio and Bernardo Uribe Orbifolds as Groupoids: An introduction by Ieke Moerdijk Differential Geometry of Gerbes by Lawrence Breen and William Messing An overview of the search for minimal models of algebraic threefolds by Sinan Sertoz Weak Hopf algebras and quantum groupoids by Peter Schauenburg Quantum Groupoids by Ping Xu Finite Quantum Groupoids and their applications by Dmitri Nikshych and Leonid Vainerman Ravenel's appendix on Hopf algebroids Toric Mori theory and Fano manifolds by J.A. Wisniewski Functoriality and Morita equivalence of operator algebras and Poisson manifolds associated to groupoids by N.P. Landsman Course on Deformation Theory and moduli spaces by Ravi Vakil A brief survey of non-commutative algebraic geometry by Snigdhayan Mahanta Non-commutative geometry and topology from Connes's perspective. Noncommutative Geometry, Year 2000 by Alain Connes A short survey of Noncommutative Geometry by Alain Connes Quantum Spaces and Their Noncommutative Topology by Joachim Cuntz Applications of Non-commutative geometry to topology J. Rosenberg's minicourse at the Clay Institute Symposium on Non-commutative Geometry Review of Noncommutative Geometry by Alain Connes by Vaughan Jones and Henri Moscovici Review of Noncommutative Geometry by Alain Connes by Andrew Lesniewski Review of Noncommutative Geometry by Alain Connes by John Roe My research efforts over the past decade concern non-commutative algebraic geometry. The field is emerging slowly, with several different perspectives. There is no good introduction to the subject yet, but the following may give some idea. Blowing up non-commutative smooth surfaces by Michel Van den Bergh; Noncommutative Geometry @n by Lieven LeBruyn A course dvi , ps , pdf I taught in spring 1999. This version was posted on June 14, 2000. Some lectures by Bodo Pareigis. A book by Bodo Pareigis. Intersection theory on non-commutative surfaces by Peter Jorgensen; Non-commutative projective schemes dvi ps pdf . These lectures, given in 1994, are rather dated now. I report on work of Artin and Zhang setting out the basic properties of cohomology for non-commutative projective schemes. It cries out for some examples. Some physics Spacetime 101 Cosmology FAQ Spacetime and the Philosophical Challenge of Quantum Gravity by J.Butterfield and C.J.Isham Superstrings 101 Black Holes FAQ Physics Lecture notes UseNet Relativity FAQ General Relativity 101 by John Baez A book: Physics in Noncommutative World: I Field Theories Ed by Miao Li and Yong-Shi Wu. More articles Derived Categories, Derived Equivalences and Representation Theory by Thorsten Holm Algebra at the turn of the Century by Claus Ringel
Algebraic Curves
Lecture Notes from H.W. Lenstra's course Math 274, Berkeley 1995.
Algebraic Curves Algebraic Curves Lectures in ps: Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Lecture 9 Lecture 10 Lecture 11 Lecture 12 Lecture 13 Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20 Lecture 21 Lecture 22 Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture 29 Lecture 30 Lecture 31 Lecture 32 Lecture 33 Lecture 34 Lecture 35 Lecture 36 Lecture 37 Lecture 38 Lecture 39 Lecture 40 Lecture 41 Complete Notes (in dvi) Lenstra's Scribe Notes, Fall 95 Robert F. Coleman Last modified: Tue May 9 17:39:36 PDT 2000
Differential Algebraic Geometry - A Scheme-theoretic Approach
Slides in multimedia format from a lecture by Henri Gillet at MSRI.
MSRI - Henri Gillet
The Hodge Conjecture
A description of the conjecture by Deligne (PDF) and details of the prize offered for its resolution.
Clay Mathematics Institute Clay Mathematics Institute Dedicated to increasing and disseminating mathematical knowledge HOME | ABOUT CMI | PROGRAMS | NEWS EVENTS | AWARDS | SCHOLARS | PUBLICATIONS Hodge Conjecture In the twentieth century mathematicians discovered powerful ways to investigate the shapes of complicated objects. The basic idea is to ask to what extent we can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. This technique turned out to be so useful that it got generalized in many different ways, eventually leading to powerful tools that enabled mathematicians to make great progress in cataloging the variety of objects they encountered in their investigations. Unfortunately, the geometric origins of the procedure became obscured in this generalization. In some sense it was necessary to add pieces that did not have any geometric interpretation. The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles. The Millennium Problems Official Problem Description Pierre Deligne Lecture by Dan Freed (video) Return to top Contact | Search | Terms of Use | Clay Mathematics Institute
The Cubic Surface Homepage
Images, software, links. At the University of Mainz, Germany.
The Cubic Surface Homepage Have you ever played with cubic surfaces ? NEW!!! We have got our own domain now: WWW.CUBICSURFACE.NET . Please check out the new site ! This old one will no longer be supported, but some interesting pages, like singularities and Cubic Surface Movies are not ported to the new site yet... Do you know, what a double six is? ||||||||||||||||||||||||||||||||||||||||||||| ||||||||||||||||||||||||||| |||||| |||||| Oliver Labs , The Cubic Surface Homepage , Algebraic Geometry Group , University of Mainz, Germany A new project: Interactive Plane Geometry
Projections of Complex Algebraic Curves to Real 3-space
Graphics.
Projections of Complex Algebraic Curves to Real 3-space Projections of Complex Algebraic Curves to Real 3-space These projections were made using Chris Weigle's "Isosurf" program . The curve y5=x2-1 near the origin first projection second projection Return to my home
Algebraic Geometry
Links and list of algebraic geometers.
Algebraic Geometry Algebraic Geometry Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate. --David Mumford Sources for preprints and other texts Preprints at arXiv.org The canonical place to get and post the latest research in math. EMANI Electronic Mathematical Archiving Network Initiative Excellent free electronic source. Grothendieck-Circle.org the most complete source for material by and about Alexandre Grothendieck. JSTOR The scholarly journal archive (large collection of important sources, but limited access without subscription) Other General Sites European Algebraic Geometry: AGE and EUROPROJ Tata Institute of Fundamental Research IAS Park City Summer Mathematics Institute Algebraic Curves Projections of complex plane curves to real three-space . A Visual Dictionary of Special Plane Curves Overview of Algebraic Plane Curves (Geometry Center) Famous Curves Index (University of St. Andrews) Algebraic Surfaces Flchen mit vielen gewhnlichen Doppelpunkten , or Surfaces with many ordinary nodes. Mathematical Surfaces (Bergen) Algebraic Geometers and the Like General sources International Mathematical Union Electronic World Directory of Mathematicians AGE and EUROPROJ's list. AMS Combined Membership List. Individual Webpages I have removed my links to individual webpages, since they are easily located by any search engine and are hard to keep current. Related Sites Number Theory Web K-Theory Archives C. Weibel's algebraic K-theory book Return to Tyler Jarvis' home page Please send comments and suggestions to jarvis AT math DOT byu DOT edu Last modified: Fri Apr 15 12:55:41 MDT 2005
Activities For Pi Mathematics
Includes projects, history and videos to help learn about this unique constant.
Activities For Pi Mathematics Activities For Mathematics Introduction is the focus of this project. It includes a variety of activities to help you learn about . You will discover the approximate value of . You will use measurement and report data. In the History section you will be able to explore the history of using the World Wide Web. " Calculations Over Time" will link you to Internet sites on mathematicians who have calculated . In the Information Video section there are Fun Things About , Kathleen Smith's award- winning hyperstack, "A Slice of Pi", and a video called "A Pizza Project on ". You can send us information about your favorite pizza in the Project section. You will have the opportunity to participate with other students throughout the world on this collaborative project. Results of this project will be available on Day, the fourteenth of March ( 3 14). In the Application section you will apply formulas and do some practical problem solving. Visit and calculate the circumferences of the planets; then check your calculations using Internet resources. Materials: measuring tape string ruler or meter stick Objects to be measured: juice can soup can coffee can oatmeal box top garbage can top fruit can Directions: Use the measuring tape or the string and ruler meter stick to measure the circumference of the tops of the objects. Then measure the length of the diameter . List these measurements in a table like the one below: How does the measurement of the circumference compare to the measurement of the diameter? Is it twice as large? Is it three times as large or more than three times as large? This comparison is the ratio of the circumference to the diameter of the circle. This ratio is called . In the column marked COMPARISON, list the answer for the circumference divided by the diameter. The values we use for are 3.l4, 3.l4l6 or 3 l 7. If you would like to see the value of to 1000 places, click here . Some other values of that have been calculated can be found at History . Now take your values of and round them to the nearest hundredth . Thoughts to Consider: Compare your value to 3.14. Was your calculation greater or less than the 3.l4 value of ? Are the values of consistent? Why or why not? What reasons do you think would account for these differences? Activities History Information Video Project Application Home Page Teacher Resources
Pi Mathematics
An online collaborative project, using the concept of PI as the focus. This multidisciplinary project which includes math, history, English and thinking skills is designed for fifth through eighth graders.
Pi Mathematics Mathematics Mathematics, a collaborative project, using the concept of as the focus, is a home page on the WWW. This multidisciplinary project which includes math, history, English and thinking skills is designed for fifth through eighth graders. It will allow students to discover the approximate value of , an irrational number, using measurement and reporting the data, applying formulas, problem solving and participating in a collaborative project utilizing Internet resources. Students will also explore the history of , using Internet references. Our pages are: Our Homepage Our Activities Georgette Moore Yankee Ridge Elementary School Betty A. Ganas St. Pius X School
Pi Day
Activities for a Pi Day, March 14 (3.14) celebration including pi math problems, history, songs, stories, and greetings.
Math Forum: T2T FAQ: Pi Day Teacher2Teacher: FAQ Pi Day T2T || FAQ || Ask T2T || Teachers' Lounge || Browse || Search || Thanks || About T2T [ PoWs ] [ Discussions ] [ Dr. Math ] [ Literature ] [ Web Resources ] What are some activities I can use for a Pi Day celebration on March 14 (3.14)? How many celebrations are there in your math class? Each year on March 14th many classrooms break from their usual routines to observe the festivities of "Pi [ ] Day" because the digits in this date correspond with the first three digits of (3.14). Teachers use this date to engage students in activities related to the history and concept of . Students are often familiar with before this day and the projects and activities associated with Pi Day are meant to enrich and deepen the students' understanding of the concept. Activities may include investigations of the value of by approximating the ratio of the circumference to the diameter of a circle. Students may also share their own Pi Projects with their classmates. Some teachers choose to end their Pi Day celebration by eating pie! If you are looking for some ways to celebrate math in your math class, or if you would like to suggest some school-wide thematic activities, you are certain to find ideas on this page to help you design a Pi Day that students will enjoy. If honoring is not enough of a reason to make this day special, keep in mind that March 14 is also Albert Einstein's birthday. Problems of the Week: ElemPoW: Circles Around - posted January 8, 2001 [print version] Use this exploration problem to discover the relationship between circumference and diameter. MidPoW: Accuracy Please - posted February 7, 2000 [print version] Considering precision, accuracy, and appropriateness in finding the volume of a cone. MidPoW: Pi Day, Let's Eat Cake! - posted March 13, 2000 [print version] In honor of Pi Day we will be comparing the volumes of slices taken from round and rectangular cakes. MidPoW: Pizza Pie - posted March 8, 1999 [print version] Use the given information to decide how many slices of the small pizza would be equal to or greater than 3 slices from the large pizza. AlgPoW: Dot.Com Derby - posted January 8, 2001 [print version] Three "dots" run a race around a circular track in an unusual manner. You must determine the winners in two categories. AlgPoW: Round the World on a Wire - posted October 11, 1999 [print version] A wire is wrapped around the equator of the earth. Then extra wire is spliced in. Can you walk under the slack that results? [top] T2T discussion about Pi Day: Pi Day I am looking for some activities to use for upcoming "Pi Day" (March 14, i.e., 3.14) celebration. I am interested in ways to introduce Pi and its relationships with circles, circumference area. ...view discussion Pow-teach discussion about Pi: Teaching Pi I'm in the process of mentoring the midPOW for 2 26 and an understanding of pi is important to understanding how to solve the problem. ...view discussion [top] Ask Dr. Math resources: Pi = 3.14159... - Math Forum, Ask Dr. Math FAQ What is Pi? Who first used Pi? How do you find it? How many digits is it? more Facts about Pi What are some interesting facts about Pi? [top] Literature connections: Students' mathematical understanding can be enhanced through literature selected for its mathematical connections. The following books are available through Amazon.com: History of Pi by Petr Beckmann The Joy of Pi by David Blatner Pi in the Sky: Counting and Thinking by John D. Barrow Sir Cumference and the Dragon of Pi : A Math Adventure by Cindy Neuschwander (Illustrator), Wayne Geehan (Illustrator) [top] Resources on the Web: 123 Greetings Send a Pi Day greeting card ...more Activities Information Supporting Pi Day Celebration - Carolyn M. Morehouse A lesson plan for Pi Day activities, Pi Songs, and a Pi Trivia Quiz are among the resources Carolyn has provided. American Pi (song) Lyrics are by Lawrence Mark Lesser and he suggests singing it to the tune of Don McLean's "American Pie." Derivation of Pi - Jon Basden Students use real-world objects to understand the concept of a constant such as pi. Determination of Pi - Alexander Bogomolny This lesson presents an interesting alternative to the customary game plan for the determination of pi. Bogomolny's lesson presents pi in the framework of similarity, which makes the existence of pi self-evident. Approximation of pi does not require several objects. Pi can be approximated by taking measurements of a single circular object with different tools and different techniques. The Digit Connection - Lizzie A tribute to the number Pi through music, poetry, and fiction. (Meet Sailor Pi!) Also includes a brief history of Pi with some ways to calculate it, a quote collection, and ways to celebrate the number - even if it's not Pi ...more Discovering the Value of Pi by Jacobo Bulaevsky The purpose of this lesson is for students to discover a very special attribute of circles. Included is a lesson description for the teacher. Free Pi Day Card by Webmania Designs Jeff's Pi Memorization Tips Memorize 100 digits of pi using Jeff's tips. Math Library: Arithmetic Early: Number Sense About Numbers: Pi Pi resources that have been collected, organized, cataloged and annotated from diverse sources. Pi Day by Mr. Herte Suggested activities and links to other Pi Day ideas by Mr. Herte, a teacher in Glen Cove, New York. Pi Day: Making a Pi Necklace - Diana Funke Diana Funke teaches mathematics at Davisville Middle School in North Kingstown, RI. Her students make a Pi necklace for Pi Day to reinforce the idea that some numbers never repeat or end. They assign a color to each digit (including 0) and then ...more Pi Day Songs by LaVern and Mark C. Christianson Pi Mathematics - Ganas, Moore; National Center for Supercomputing Applications (NCSA) Whether it is considered to be of historical significance, mathematical importance, or a personal goal, Pi has universal appeal. This site is an adventure in exploring the concept of Pi. ...more The Pi Pages - Centre for Experimental and Constructive Mathematics (CECM) Links to many sources of information: the story of the history of the computation of Pi, current records of computation, and ...more The Pi Search Page - Dave Andersen Search the first 50 million digits of pi for any numeric string up to 120 digits. The page also documents a few observations about pi's numeric strings, including self-locating strings, palindromes, and month-day combinations. ...more Pi Through the Ages - MacTutor Math History Archives A history of Pi: the Rhind papyrus (Egypt), Ptolemy, Tsu Ch'ung Chi, al'Khwarizmi, Al'Kashi, Viet, Romanus, Van Ceulen, Gregory, Shanks, Lambert, Euler, Buffon, and many more, with other Web sites and 30 references (books articles). ...more The Ridiculously Enhanced Pi Page - The Exploratorium, San Francisco, CA Pi Day celebrations, Pi music, links, beads, and pie. A certified 98% content-free page. ...more The Ridiculously Enhanced Pi Page (2) - The Exploratorium, San Francisco, CA In celebration of Pi Day (March 14), a site that is certified 98 percent content-free. Includes Pi Remote and the 100-digit ...more The Story of Pi Video segments available for viewing on the Web or the videotape and workbook is available at cost from Project MATHEMATICS! Submit your own question to T2T. || Let us know what you think of this page. [ Privacy Policy ] [ Terms of Use ] Math Forum Home || The Math Library || Quick Reference || Math Forum Search Teacher2Teacher - T2T 1997-2005 The Math Forum http: mathforum.org t2t
E-Z Geometry
For high school teachers and students. Products include an interactive textbook, class video clips, projects, glossary, and resource links.
e-zgeometry.com Geometry Projects, Geometry Links, Glencoe Geometry Textbook Notes, Geometry Glossary, High School Geometry Project Ideas, Interactive Geometry Experiences, Geometer's Sketchpad Applets, Geometry Video Footage and much more
Polyhedra: Learning by Building
Teaching project that let's students learn about polyhedra by using kite materials to create life size forms. Exercises include measuring surface area, volumes, and changing the form to simpler shapes.
Polyhedra, learning by building The "Polyhedra: Learning by Building" project. Eva looking through an octahedron with two clear faces Building polyhedra with large triangles lets people handle and see geometric shapes from the inside as well as the outside. These new experiences have caused excitement and curiosity giving better understanding by children mathematics lessons. The originator of the triangles, artist Eva Knoll, regards mathematics as 'so beautiful it is worth making art out of it', thus drawing people into mathematics. However art is so creative we can use it to encourage creative original thinkers in mathematics, science, technology and design. Simon Morgan, a mathematics PhD graduate from Rice University, Houston TX, Eva Knoll, now a graduate student in the School of Education, University of Exeter, U.K. and Jackie Sack, Mathematics Chair at Lanier Middle school, Houston TX, have collaborated with the support of Rice University School Mathematics Project to develop activities for geometry teaching in schools using large brightly colored triangles. The triangles are made from kite materials for their size, strength, appearance, precision, and light weight. In the spring semester of 1999 five activities were tried out with three classes. We started with with an origami based exercise of folding a triangular grid on a paper disc and cutting out a snowflake shaped net that closed up into an icosahedron. Then the lessons in the table were carried out giving hands on experience at building and observing patterns in these shapes. Finally a giant eighty sided 'Endopentakis-icosi-dodecahedron' was made one afternoon for the school artfest. The children, teachers and parents were all really excited by these activities and the children learned things well that are usually hard to teach. Since then more lessons have been developed through a wide range of grade levels, fifth through high school, and with varying students from gifted and telented to low performing. To join the project as a teacher or parent email morgan@math.umn.edu (Minneapolis St Paul USA) or e.s.p.knoll@ex.ac.uk (Exeter, England) Gallery of photographs and images. Report with lesson outlines published in Bridges 2000 (download pdf) (download postscript file). Three middle and high school geometry lessons (Lanier, spring 1999) Lesson Shapes built How learning took place Lesson 1 Filling a tetrahedron Double edge length tetrahedra and shapes to fill it By comparing a small and a large tetrahedron, see how length, surface area and volume scale. A surprise happens when we try to fill the large tetrahedron with smaller shapes. Lesson 2 Stella Octanguli Tetrahedra, octahedra, double edge length tetrahedra and stella octanguli Construcing two stella octanguli in different ways shows how they relate to simpler shapes. There is also is an interesting observation about space filling that can be made. Lesson 3 Colored Icosahedra Icosahedra Coloring the icosahedron with 5 colors Co-operative geometric problem solving in going from a colored plan of a net on paper, to assembling a net and closing it up to make the icosahedron
An Introduction to Projective Geometry (for Computer Vision)
A monograph aiming to provide a readable introduction to the field of projective geometry and a handy reference for some of the more important equations. HTML, PS or PDF versions.
An Introduction to Projective Geometry (for computer vision) Next: Introduction An Introduction to Projective Geometry (for computer vision) Stan Birchfield Printable version: [PDF -- 247KB] [ps.gz -- 71 KB] ** Erratum ** In Section 2.1.3, "The unit sphere," it is stated that the projective plane is topologically equivalent to a sphere. In fact, it is only locally topologically equivalent to a sphere, as pointed out by John D. McCarthy . Introduction The Projective Plane Four models Homogeneous coordinates Ray space The unit sphere Augmented affine plane Duality Pencil of lines The cross ratio Conics Absolute points Collineations Laguerre formula Projective Space Representing lines: The Plcker relations Intersections and unions of points, lines, and planes Projective Geometry Applied to Computer Vision Image formation Essential and fundamental matrices Alternate derivation: algebraic Alternate derivation: from the epipolar line Summary Vanishing points Demonstration of Cross Ratio in Acknowledgment Bibliography About this document ... Stan Birchfield April 23, A.D. 1998
Linear Systems of Conics
Graphics and formulae for systems of curves defined by linear combinations of quadratic equations.
Adam Coffman --- Conics Linear Systems of Conics Conics. A "conic curve" (or a "conic section," or just a "conic") in the two-dimensional plane is the solution set of an equation of the form: Ax2 + By2 + Cxy + Dx + Ey + F = 0 for real constant coefficients A, B, C, D, E, F, which are not all zero. There are different types of conic curves in the real plane. The following chart gives a representative for each type of quadratic equation, and a name for the corresponding conics. For our purposes, they are divided into five cases: Real Nondegenerate Imaginary Nondegenerate Ellipse x2+y2-1=0 Imaginary Ellipse x2+y2+1=0 Hyperbola x2-y2-1=0 Parabola x2-y=0 Real Degenerate Imaginary Degenerate Intersecting Pair of Lines x2-y2=0 Imaginary Pair of Lines, Intersecting at a Real Point x2+y2=0 Parallel Lines x2-1=0 Imaginary Parallel Lines x2+1=0 Pair of Lines, one at Infinity x=0 Square Degenerate Coincident Lines x2=0 Coincident Lines at Infinity 1=0 If you know the coefficients of a quadratic equation, you can find its type in the above chart, by using the table from a Geometry Center web page . We can abbreviate an equation as E1(x,y) = 0, and multiplying all the coefficients of E1 by the same nonzero constant, c, gives (c . E1)(x,y) = cAx2 + cBy2 + cCxy + cDx + cEy + cF. Then E1(x,y) = 0 if and only if (c . E1)(x,y) = 0, so E1 = 0 and c . E1 = 0 have the same solution set, and they define the same conic curve. Linear systems. A "linear system of conics" (also called a "pencil of conics") is the set of linear combinations of polynomials E1 and E2: t1 . E1 + t2 . E2. We assume E1 and E2 are linearly independent: one is not a constant multiple of the other. Plugging specific numbers into these coefficients, (t1,t2), gives a polynomial in (x,y), so each ordered pair (t1,t2) corresponds to a conic curve (t1 . E1 + t2 . E2)(x,y) = 0. Multiplying both t1 and t2 by the same nonzero constant c results in the same curve: ((c . t1) . E1 + (c . t2) . E2)(x,y) = c . (t1 . E1 + t2 . E2)(x,y) = 0. So, a linear system is really just a one-parameter family of different conic curves, and the non-zero pairs (t1,t2) could be considered as homogeneous coordinates [t1 : t2] for a projective line. (See my Steiner surfaces page for some details on homogeneous coordinates.) Theorem: In a real linear system of conics, there is at least one conic which contains a real line (so that conic is either "real degenerate" or "square degenerate"). Theorem: If there is one nondegenerate conic in a linear system, then there are at most three degenerate conics. However, there exist linear systems where all the conics are degenerate. Intersections. A point (x,y) is on both conics if E1(x,y) = 0 and E2(x,y) = 0. Any point on this intersection will also be a point on every conic in the linear system: for any t1, t2, (t1 . E1 + t2 . E2)(x,y) = t1 . 0 + t2 . 0 = 0. In fact, if any two conics in a linear system meet at a point, then that point is on every member of the linear system. Such points of intersection are called base points. If two conics in a linear system are disjoint, then every pair of conics in the linear system will also be disjoint, and there are no base points. Theorem: If a real linear system of conics has finitely many base points, then there can be 0, 1, 2, 3, or 4 of them, but no more than four. However, there exist linear systems with infinitely many base points. Theorem: Every point in the plane which is not a base point lies on exactly one conic in the linear system. The Classification The quadratic equation of a conic could also be written as a homogeneous expression in three variables, Ax2 + By2 + Cxy + Dxz + Eyz + Fz2 = 0, which gives the original x,y equation when z = 1. This corresponds to the geometric definition of a conic section as the intersection of a cone (defined by the homogeneous equation in xyz-space) with a plane (defined by z = 1). The following examples list all the types of linear systems of conics, up to "real projective equivalence" (real linear transformations of the homogeneous coordinates [x : y : z]). It turns out there are only finitely many types of linear systems; roughly speaking, they can be categorized by the number of base points and degenerate conics, but there are the usual issues of "multiplicity," and whether the base points have real coordinates. The numbering system for types I, Ia, ..., V follows H. Levy's book. The Roman numerals I, ..., VIII represent eight complex projective equivalence classes, some of which are further subdivided into real projective equivalence classes, for a total of 13 types. Click on the picture to see an animation. Type I t1 . (x2-y2) + t2 . (x2-1). These conics meet at four real base points. There are three degenerate conics: [t1:t2]=[1:0] The intersecting lines x2-y2=0, [t1:t2]=[0:1] The vertical parallel lines x2-1=0, [t1:t2]=[-1:1] The horizontal parallel lines y2-1=0. To see how I rendered the conic sections in the picture to the left, Click Here to see a "side view," showing the cones in xyz-space, whose intersection with the plane z=1 (the top of the box) is shown in the two-dimensional figure. Type Ia t1 . (x2+y2+1) + t2 . (x). The type Ia linear system has no real base points. In this representative, most of the conics you see in the picture are circles, but there are also "imaginary ellipses" with no visible points: for some values of [t1:t2], the polynomial equation (t1 . E1 + t2 . E2)(x,y) = 0 has no real solutions. There are three degenerate conics: [t1:t2]=[0:1] The vertical line x=0, [t1:t2]=[1:2] The point (-1,0) is the only real solution of x2+2x+1+y2=0, [t1:t2]=[1:-2] The point (1,0) is the only real solution of x2-2x+1+y2=0. In complex homogeneous coordinates, the linear system t1 . (x2+y2+z2) + t2 . (xz) = 0 has four base points: [0:i:1], [0:-i:1], [1:i:0], [1:-i:0]. Type Ib t1 . (x2+y2-1) + t2 . (x). The type Ib linear system has two real base points. Since all the conics in the system must meet both real base points, there are no "imaginary ellipses." There is one degenerate conic: [t1:t2]=[0:1] The vertical line x=0. In the complex linear system, where x, y, t1, and t2 are allowed to be complex, there are two more degenerate conics, at [t1:t2]=[1:2i] and [1:-2i]. In complex homogeneous coordinates, there are two more base points, [1:i:0], [1:-i:0]. Type II t1 . y(x+y-1) + t2 . x(x+2y-1). The type II linear system has three real base points, and at one of these points, all the nondegenerate conics have a common tangent line. There are two degenerate conics: [t1:t2]=[1:0] The pair of lines y(x+y-1)=0. [t1:t2]=[0:1] The pair of lines x(x+2y-1)=0. Type IIa t1 . (y2+(x-y-1)2) + t2 . x(x-y-1). The type IIa linear system has one real base point, at which all the nondegenerate conics have a common tangent line. There are two degenerate conics: [t1:t2]=[1:0] The pair of imaginary lines y2+(x-y-1)2=0, meeting at the real base point (1,0). [t1:t2]=[0:1] The pair of lines x(x-y-1)=0. In complex coordinates, there are two more base points, (0,(-1+i) 2), (0,(-1-i) 2). Type III t1 . (x2-1 4) + t2 . y2. The type III linear system has two real base points. At each of the two base points, all the nondegenerate conics have a common tangent line. There are two degenerate conics: [t1:t2]=[1:0] The vertical parallel lines x2-1 4=0, [t1:t2]=[0:1] The horizontal coincident lines y2=0. Type IIIa t1 . (x2+y2) + t2 . (y+1)2. The type IIIa linear system has no real base points. Like Type Ia, there are some imaginary ellipses. There are two degenerate conics: [t1:t2]=[1:0] The point (0,0) is the only real solution of x2+y2=0, [t1:t2]=[0:1] The horizontal coincident lines (y+1)2=0. In complex coordinates, there are two base points, (i,-1), (-i,-1). Type IV t1 . (x2+x+y2) + t2 . (xy). The type IV linear system has two real base points. At one of the base points, any two conics meet each other transversely, and at the other, they meet at an intersection with multiplicity 3. There is only one degenerate conic: [t1:t2]=[0:1] The pair of lines xy=0. Type V t1 . (x+y2) + t2 . (x2). The type V linear system has one real base point, at which any two conics meet each other at an intersection with multiplicity 4. There is only one degenerate conic: [t1:t2]=[0:1] The vertical coincident lines x2=0. Type VI t1 . (x2-y2) + t2 . (2xy). The type VI linear system has only real degenerate conics, which are pairs of lines that meet at the one real base point. In the complex linear system, where x, y, t1, and t2 are allowed to be complex, there are two square degenerate conics, at [t1:t2]=[1:i] and [1:-i]. Type VIa t1 . (x2+y2) + t2 . (2xy). The type VIa linear system has real and imaginary degenerate conics, which are pairs of lines that meet at the one real base point. There are also two square degenerate conics, at [t1:t2]=[1:1] and [1:-1] Type VII t1 . (x2-y2) + t2 . (x+y)(y-1). The type VII linear system has only real degenerate conics, which are pairs of lines, one of which is a line of base points. There is also one real base point not on that line. There are no square degenerate conics. Type VIII t1 . (x2) + t2 . (xy). The type VIII linear system has base points which form exactly one line. There are real degenerate conics, and one square degenerate conic at [t1:t2]=[1:0]. This case is related to the previous one, by moving the isolated base point so that it is on the line of base points. References MR 0167882 29 5147 H. Levy, Projective and related geometries. The Macmillan Co., New York; second printing 1967. MR 1384308 (97a:14013) A. Degtyarev, Quadratic transformations RP2 - RP2, in Topology of real algebraic varieties and related topics, 61--71, AMS Transl. Ser. 2, 173, 1996. MR 1673756 (99m:14012) R. Miranda, Linear systems of plane curves , Notices Amer. Math. Soc. (2) 46 (1999), 192-201. Links Conics - The Geometry Center Famous Curves - St. Andrews Back to the graphics gallery .
MathWorld
Index to articles on Projective Geometry.
Projective Geometry -- From MathWorld INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News (RSS) New in MathWorld MathWorld Classroom Interactive Entries Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Geometry Projective Geometry General Projective Geometry (48) Map Projections (61) Radon Transforms@
Projective Geometry for Multiple View Analysis
A new framework for studying the problems of camera calibration, motion determination, and 3D structure reconstruction in the most general case of totally uncalibrated views.
Projective geometry for multiple view analysis Projective geometry for multiple view analysis I proposed a new framework for studying the problems of camera calibration (how to find out the optical and geometrical characteristics of the camera), motion determination (how to find out the relative position and orientation of two cameras), and 3D structure reconstruction (how to find how the location of objects or features in the world) in the most general case of totally uncalibrated views (nothing is known about the cameras who took the images). Depending on the amount of information and the constraints available, the problem can be analyzed at a projective, affine, or Euclidean level. This is the idea of stratification, studied systematically through the Canonical representations . Let me describe some of the individual findings which have been obtained within that powerful general framework. Given a single image, and a point on this image, the only thing which can be infered geometrically is a ray of projection, but not the position along this ray. The relationship between the point and the ray is best described by projective geometry, a generalization of the familiar Euclidean geometry, which can deal with projections and objects at infinity, but lacks the notions of angles or distances. Since it is in impossible to geometrically infer 3D information from a single image, let now consider a pair of images, for instance the one formed by each of your eyes. If we pick two points at random in each image, they are unlikely to be corresponding points, ie, the images of a single 3D point of the world. So do corresponding points have any special properties ? It turns out that there is a remarkably simple answer to this question: their coordinates - in any arbitrary frame attached to each image - have to satisfy a bilinear relationship whose coefficients are derived from a particular 3 x 3 matrix, that I called the Fundamental matrix , because it encodes the only geometrical information which can be generally computed from pairs of images. This information is projective in nature. The seven parameters of this matrix can also be interpreted in terms of epipolar geometry. Its computation requires a minimum of eight point correspondences to obtain a unique solution, and particular numerical methods were investigated to improve the robustness of the computation. The homography matrix describes completely the relationship between corresponding planar points. We investigated the deep relationship between the Fundamental matrix and that matrix, and also with the so-called critical surfaces, which have consequences for computation stability. Those observations can be generalized when more than two views are available, and we obtain more geometric and algebraic constraints, but still have access only to the projective geometry of the scene. So how is it possible to recover Euclidean information about the world, which matches our experience ? Previous methods relied on knowledge of the world (surveyed landmarks, artificial calibration objects whose 3D coordinates are known), or on controled motions, which is often impractical. In contrast, we rely on the fact that over multiple views some of the internal characteristics of the cameras remain constant. For instance, if all the internal parameters remains the same, we need only three views to recover them. This is the idea of Self-Calibration . The calibration object is replaced by an entity which has the advantage of being always available for free: the absolute conic. However, this entity is quite abstract (being imaginary and situated at infinity), and to exploit it, we must go through a system of polynomial Kruppa equations, which is quite delicate to solve robustly. These ideas, initiated in my PhD thesis, have since then received a great deal of attention and helped start a rich subfield of research which has led us within a decade to a more or less complete understanding of the geometry of multiple views. The book "The geometry of multiple images" , co-written with my advisor O. Faugeras, describes in a principled and comprehensive way all that. Interactive demos Learn more about epipolar geometry: with a 3D interactive visualization. Compute the fundamental matrix: with your own images ! Back to Tuan's research page
Wilson Stothers' Cabri and Conics Pages
A site devoted to (mainly projective geometry) illustrated by Cabri II and CabriJava.
Wilson Stothers' Cabri Pages Wilson Stothers' Geometry Pages what's new? original page plane conics the klein view affine geometry projective geometry inversive geometry hyperbolic geometry dynamic geometry odds and ends references Welcome to my new-look geometry pages. These pages began as an experiment in teaching projective conics using Cabri. They are designed for students of Glasgow University and the Open University, though they do not follow the Open University M203 course in detail. A guide to the differences appears on the M203 page . As time went on, I have expanded them to include other geometries, illustrating the klein view of geometry . I have also used CabriJava to provide interactive diagrams. You can find more about the interactive geometry packages Cabri, CabriJava and Cinderella, and others, in the dynamic geometry page. If you preferred the old page, you can still find it via the original page link. Please e-mail me with any comments or suggestions. my home page webcounter You can seach the site for any topic by entering keywords below This uses the Atomz search tool
Projective Geometry Resources
Part of the Math Forum.
The Math Forum - Math Library - Projective Geom. Browse and Search the Library Home : Math Topics : Geometry : Non-Euclidean Geom. : Projective Geom. Library Home || Search || Full Table of Contents || Suggest a Link || Library Help Selected Sites (see also All Sites in this category ) An Introduction to Projective Geometry (for computer vision) - Stan Birchfield The contents of this paper include: The Projective Plane; Projective Space; Projective Geometry Applied to Computer Vision; Demonstration of Cross Ratio in P^1; and a bibliography. (Euclidean geometry is a subset of projective geometry, and there are two geometries between them: similarity and affine.) Also at http: vision.stanford.edu ~birch projective . more Projective Geometry - Nick Thomas Basics, path curves, counter space, pivot transforms, and some people involved in the development of projective geometry, which is concerned with incidences: where elements such as lines planes and points either coincide or not. For example, Desargues Theorem says that if corresponding sides of two triangles meet in three points lying on a straight line, then corresponding vertices lie on three concurrent lines. The converse is also true: if corresponding vertices lie on concurrent lines, then corresponding sides meet in collinear points. This illustrates a fact about incidences and has nothing to say about measurements, which is characteristic of pure projective geometry. Projective geometry regards parallel lines as meeting in an ideal point at infinity. more PyGeo - Arthur Siegel A dynamic geometry toolset written in Python, with dependencies on Python's Numeric and VPython extensions. It defines a set of geometric primitives in 3d space and allows for the construction of geometric models that can be manipulated interactively, while defined geometric relationships remain invariant. It is particularly suitable for the visualization of concepts of Projective Geometry. PyGeo comes with complete source code. more All Sites - 21 items found, showing 1 to 21 Asteroids in projective geometry - Flynn, Heath A game similar to "Asteroids," but set in the projective plane. Written in Director 8 by high school student David Flynn with some technical help from Dr. Deej Heath, it is playable via Shockwave. ...more Donald Coxeter, Mathematician and Geometer - Great Canadian Scientists (GCS) Harold Scott MacDonald Coxeter, Professor Emeritus, Math Dept., Univ. of Toronto, is best known for his work in hyperdimensional geometries and regular polytopes - geometric shapes that extend into the 4th dimension and beyond. In 1926 he discovered a ...more The Educational Validity of Visual Geometry - Marius Blok; Hull, UK A 1998 doctoral thesis describing the design of a new branch of geometry and its teaching, based on philosophy and visual art, avoiding the use of formulae and computations. The site provides a table of contents, preface, and summary, as well as contact ...more The Fundamental Theorem of Projective Geometry - Maska Law; Dept. of Mathematics, University of Western Australia A brief definition of projective geometry, by the author of an honours dissertation covering ideas from the areas of projective geometry and group theory. ...more Gordon Royle's Homepage: Research - Gordon Royle Catalogues and tables of interesting combinatorial objects: small graphs, cubic graphs, symmetric cubic graphs, vertex-transitive graphs, Cayley graphs (by group), vertex-transitive cubic graphs, cubic cages and higher valency cages, planar graphs, cubic ...more Homogeneous Transformation Matrices - Daniel W. VanArsdale Explicit n-dimensional homogeneous matrices for projection, dilation, reflection, shear, strain, rotation and other familiar transformations. ...more Important Concepts from Projective Geometry - Paul Beardsley Contents include: Homogeneous Coordinates; Manipulating Points and Lines; Projective Transformations; Projective Invariants; and Ideal Points and Vanishing Points. ...more Infinity - Jason Howald An online course dedicated to the concept of infinity, with "lectures" and problems; the problems are rated by difficulty. Topics include cardinality, geometry (especially problems of perspective), induction, a brief comment on philosophy and mathematical ...more Interactive Course on Projective Geometry - Juan Carlos Alvarez Paiva; Polytechnic University, Brooklyn, NY An interactive, discovery-based course on projective geometry. Lecture notes, sample exams, applets, and a list of recommended reading are provided. ...more Investigations in Projective Geometry - Dwayne Alvis and Wanda Nabors How three-dimensional scenery is reproduced on a two-dimensional piece of paper. Construction exercises and a Geometer's Sketchpad file to download. ...more Mathematical Machines - Museum of Natural Science and Scientific Instruments of the University of Modena Models of machines based on scientific and technical literature dating back to classical Greece. Among other didactic goals, La collezione di Macchine Matematiche aims to foster reflection on the historical relationships between mathematics, society, ...more Mathematics in John Robinson's Symbolic Sculpture - University of Wales, Bangor Art and mathematics are combined in this site, which explores Borromean Rings, The Mbius Band, Bernard Morin and the Brehm Model, The Projective Plane, Fibre Bundles, Knots and Links, Torus Knots, and Fractals through the work of sculptor John Robinson. ...more OpsResearch - DRA Systems A collection of Java classes for developing operations research programs and other mathematical applications. The site includes documentation and tutorials, and software download is free. Also features a bookstore and related links. ...more Papers on the History of Mathematics - Gregory Cherlin, Rutgers University A random selection of term papers submitted in undergraduate, one-semester courses on the history of mathematics. Topics include zero, Fermat, Euler, Newton, Euclid's proof of the Pythagorean Theorem, Fibonacci, Vieta, conic sections, mathematics at Princeton ...more Projective Conics (Gallery of Interactive Geometry) - Mathew Frank; The Geometry Center This discussion of Pascal's theorem in terms of projective geometry includes an interactive application that lets you specify points on a conic and see how the theorem applies to them. Conics and Hexagons: Pascal's Theorem; Brianchon's Theorem; Some Stronger ...more Projective geometry and transformations - Albert Goodman; Deakin University (Rusden), Clayton, Victoria, Australia Applications of projective geometry and transformations for computer graphics. 2-D projection and coordinate systems; and transformations. From a course in Computer Graphics and Imaging. ...more Sliders - Cut the Knot!, Alexander Bogomolny, with Don Greenwell A variation of the Fifteen puzzle invented by Sam Loyd in the early 1870's: puzzles on graphs, including a proof independent of Wilson's theorem that all Lucky 7 permutations are possible; and variations of Sliders, a Fifteen-like puzzle that can be played ...more Steiner Surfaces - Adam Coffman Equations and graphics of Steiner surfaces: algebraic varieties with quadratic rational parametrizations. The classification includes quartic models of the projective plane, ruled cubics, and quadric surfaces. Steiner surface patches can be used in computer ...more Understanding Projective Geometry (Question Corner and Discussion Area) - University of Toronto Mathematics Network An in-depth explanation of projective geometry in response to a high school student's question. ...more Vectors in Projective Geometry (Question Corner and Discussion Area) - University of Toronto Mathematics Network A short, basic description of the use of vectors in projective geometry. ...more Wilson Stothers' Geometry Pages - Wilson Stothers, University of Glasgow A guide to the various geometry topics on Stothers' pages, including Euclidean, affine, projective, inversive, and hyperbolic geometries, and the Klein View of geometry. These pages began as an experiment in teaching projective conics using Cabri to provide ...more Page: 1 Search for these keywords: Click only once for faster results: all keywords, in any order at least one, that exact phrase parts of words whole words Choose a Math Topic all math topics algebra analysis arithmetic early math calculus (single variable) calculus (multivariable) communicating math differential equations discrete math dynamical systems general geometry history and biography logic foundations number theory numerical analysis operations research pre-calculus probability statistics topology applications connections projective geometry all math topics Choose a Math Education Topic all math education topics teaching issues strategies assessment testing general programs approaches materials-reviews recommendations activities psychological affective issues special contexts specific math concepts techniques teaching styles practices technology in math ed writing communication in math professional ed career development continuing ed degrees higher ed job placement job market pre-service staff prof development math ed research reform curriculum materials development pedagogical research psychological research reform social issues public policy community outreach educational systems equity public understanding of math all math education topics Choose a Resource Type all resource types educational materials net-based resources organizations publications recreations reference sources software all resource types Choose a Level all levels elementary early elem. (prek-2) late elem. (3-5) middle school (6-8) high school (9-12) college early college late college research all levels Power Search [ Privacy Policy ] [ Terms of Use ] Home || The Math Library || Quick Reference || Search || Help 1994-2005 The Math Forum http: mathforum.org
Lecture Notes on Projective Geometry
By Balzs Csiks. Budapest Semesters in Mathematics. In DVI format.
Notes on Projective Geometry by B. Csiks Projective Geometry Budapest Semesters in Mathematics Lecture Notes by Balzs Csiks FAQ: How to read these files? Answer: The files below are dvi files produced by the TeX system. These files can be viewed by xdvi under linux and by windvi or yap under windows. You can download these viewers and much more from the Comprehensive TeX Archive Network . CONTENTS Part 1. Introduction. The invention of perspective drawing. The real projective space, points at infinity. Topological structure of the real projective straight line and plane. Part 2. Linear spaces and the associated projective spaces Groups, rings, division rings and fields. Vector spaces and their subspaces. Basis, coordinates, dimension. The projective space associated to a linear space. Projective coordinate systems. The theorems of Desargues and Pappus. Part 3. Examples Projective spaces over finite fields. Complex projective spaces and the Hopf fibration. Digression: Stereographic projection and inversion. The stereographic image of the Hopf fibration. Quaternions. Part 4. The axiomatic treatment of projective spaces The incidence axioms of an n-dimensional projective space. The duality principle, the dual space. Desargues' theorem and the incidence axioms. Collineations. Part 5. Desarguesian projective spaces Construction of the division ring F. Construction of a collineation between P and FP2. Part 6. The Fundamental Theorem of projective geometry The Projective General Linear group. Collineations induced by automorphisms of F. The Fundamental Theorem of projective geometry. Part 7. Cross-ratio preserving transformations between lines Cross-ratio. Characterizations of cross-ratio preserving transformations between straight lines. Cross-ratio preserving transformations between coplanar lines. Cross-ratio preserving transformations of a line, involutions.
Projective Geometry
Rudolf Steiner's approach.
PROJECTIVE GEOMETRY Projective geometry is a beautiful subject which has some remarkable applications beyond those in standard textbooks. These were pointed to by Rudolf Steiner who sought an exact way of working scientifically with aspects of reality which cannot be described in terms of ordinary physical measurements. His colleague George Adams worked out much of this and pointed the way to some remarkable research done by Lawrence Edwards in recent years. Steiner's spiritual research showed that there is another kind of space in which more subtle aspects of reality such as life processes take place. Adams took his descriptions of how this space is experienced and found a way of specifying it geometrically, which is dealt with in the Counter Space Page . A brief introduction to the basics of the subject is given in the Basics Page . The work of Lawrence Edwards is introduced in the Path Curves Page , and some explorations of his work on further aspects is described in the Pivot Transforms Page . This is mostly pictorial, with reference to documentation. YOU ARE INVITED TO EXPLORE! Nick Thomas References and favourites are listed on the People page. Feedback welcome! Please include the word "counterspace" in the text and mail to nctcs At worldemail.com, replacing At with @ of course. Public encryption key available at PGP , key ID is 0xCD18C497 locations of recent changes
Important Concepts from Projective Geometry
Webtext by Paul Beardsley.
Important concepts from projective geometry Next: Homogeneous Coordinates Important concepts from projective geometry Paul Beardsley We review some important concepts from projective geometry. For a full introduction, two texts which are written specifically from the perspective of computer vision are [Kanatani 1992] and the appendix of [Mundy 1992]. More general textbooks are [Springer 1964] which is an introduction and [Semple 1952] which is a comprehensive reference. Homogeneous Coordinates Manipulating Points and Lines Projective Transformations Projective Invariants Ideal Points and Vanishing Points About this document ... Bob Fisher Fri Nov 7 12:08:26 GMT 1997
Reviews of Numerical Recipes in C
First and second editions, collected at Lysator.
Reviews: Numerical Recipes Reviews of Numerical Recipes ``Numerical Recipes,'' with its versions for C and Pascal in addition to the original FORTRAN one, is one of the most frequently used compendiums on numerical programming methods. The flaws of the first edition were popularized by a long list of reviews that was published on USENET and some ftp servers. By 1995, the second edition of Numerical Recipes in C (Press et al., ISBN 0-521-43108-5, Cambridge University Press) has almost completely replaced the first edition, making it hard to judge whether the praise it has accumulated stems from different criteria of judgement or from improvements to the text. Bad reviews Favorable reviews Numerical Recipes on the World-Wide Web Users of the Numerical Recipes books might be interested in the Numerical Recipes WWW site . The subtree features an FAQ, ordering information, links to other information on numerical programming, and online copies (in PDF and PostScript) of the C and Fortran versions. The criticism about the first version is mentioned in the Rebuttal from the publisher as an ``urban legend'' on a line with the poodle in a microwave oven and the Mrs. Fields cookie recipe. Bad reviews of Numerical Recipes [First edition] [Editor's note: The reviews in this whole section are the feedback I (and other participants) received from a poster to USENET who asked for people's experiences with the `Numerical Recipes' books. Unfortunately, not only have all the names of the submitters been removed (which I agree with), but also that of the initial poster, making it impossible to give credits and date the survey or the books surveyed. If you can contribute details, please contact me . Thanks.] From an internal memo: "There is a series of books and associated software with the name 'Numerical Recipes' in the titles that provide descriptions of numerical algorithms and associated programs in popular programming languages... "The good news is that this series gives exceptionally broad coverage of computational topics that arise in scientific and engineering computing at a very reasonable price... The bad news is that the quality and reliability of the mathematical exposition and the codes it contains are spotty. It is not safe, we have found, to take discussions in the book as authoritative or to use the codes with confidence in the validity of the results. "The authors are identified on the book jacket as 'leading scientists' and [we] have no reason to think that they are not. However, there is no claim that they have special competence in numerical analysis or mathematical software. At least in the parts of the book that [we] have studied closely, they do not demonstrate any such competence. "Published reviews of the book[s] have fallen into two classes: Testimonials and reviews by scientists [including Kenneth Wilson, Nobel Laureate] and engineers tend to extol the broad scope and convenience of the products, without seriously evaluating the quality, while reviews by numerical analysts are very critical of the quality of the discussions and the codes... "Two reviews by numerical analysts are: (1) Lawrence F. Shampine, "Review of 'Numerical Recipes, The Art of Scientific Computing'," The American Mathematical Monthly 94, 9 (Nov 87) 889-892. (2) Richard J. Hanson "Cooking with 'Numerical Recipes' on a PC," SIAM (Society for Industrial and Applied Mathematics) News 28, 3 (May 90) 18. "Professor Shampine is a specialist in the numerical solution of ordinary differential equations (ODEs). He gives specific criticisms of chapter 15, which deals with ODEs, and says, in summary: 'This chapter describes numerical methods for ODE's from the viewpoint of 1970. If the authors had consulted an expert in the subject or read one of the good survey articles available, I think they would have assessed the methods differently and presented more modern versions of the methods.' "He also remarks that adaptive methods for numerical quadrature problems are not treated in NR although they are much in favor by numerical analysts. "Dr. Hanson is a former editor of the algorithms department of the Association for Computing Machinery Transactions on Mathematical Software (ACM TOMS). He ran tests of the nonlinear least-squares codes from NR and made comparisons with published results of better known codes LMDIR from MINPACK and NL2SOL... He found the NR codes sometimes required up to 20 times as many iterations as the comparison codes. He noted that the control of the Levenberg-Marquardt damping parameter \lambda was not sufficiently sophisticated, permitting overflow or underflow of \lambda to occur... the algorithm in NR is a very bare-bones implementation of the ideas presented in the referenced 1963 paper by Marquardt. Many significant enhancements of that idea have been given in the intervening 27 years. [We] would expect the codes LMDIR, NL2SOL, and their successors to be much more EFFICIENT AND RELIABLE [my emphasis]. "[Our] present attention to the NR products was initiated by calls for consultation ... Two involved the topics mentioned above... Other calls led us to scrutinize Sections 6.6, 'Spherical Harmonics' and 14.6, 'Robust Estimation'... "...The discussion, algorithms, and code given in section 6.6 is internally consistent and the choices of the recursions to use in computing the associated Legendre functions are ones recommended by specialists in the topic as being stable. No warning is given, however, regarding the fact that there are a number of alternative conventions in use regarding signs and normalization factors... [If one naively combined results from NR codes with results from other sources] one would probably obtain incorrect results. "In reading the section on robust estimation, [we were] skeptical of Figure 14.6.1(b) that shows a 'robust straight-line fit' looking substantially better than a 'least-squares fit'... "To check [our] doubts about this figure, [we] enlarged it and traced the points and the 'fitted' lines onto graph paper to obtain data with which [to] experiment... "We computed a least squares fit... The particular 'robust' method illustrated by figure 14.6.1(b) is not identified. However, since the only method for which NR attempts to give code in this area is L1 fitting, [we] computed an L1 fit to the data as an example of a 'robust' fit... [We] used a subroutine CL1, that was published in the algorithms department of ACM TOMS in 1980, to obtain an L1 fit in which [we] could have confidence. [We] also applied the NR code MEDFIT to the data and obtained a fit that agreed with the CL1 fit to about three decimal places. ..."As expected, the least-squares fit is not as far from the visual trend as in figure 14.6.1(b) and the L1 fit is not as close... It appears that the lines labelled 'fits' in the NR figure 14.6.1(b) are not the result of any computed fitting at all, but are just suggestive lines drawn by the authors to buttress their enthusiasm for 'robust' fitting. An uncritical reader would probably incorrectly assume that figure 14.6.1(b) illustrates the performance of actual algorithms. "The objective function in an L1 fitting problem is not differentiable at parameter values that cause the fitted line to interpolate one or more data points. The authors indicate some awareness of this fact but not of all its consequences for a solution algorithm. Typically, the solution to this problem will interpolate two or more data points, and in the authors' algorithm, it would be common for trial fits in the course of execution of the algorithm to interpolate at least one data point. ... suffice it to note that it is easy to produce data sets for which the MEDFIT ROFUNC code will fail. "One data set which causes looping is [x = 1, 2, 3; y = 1, 1, 1]. Another which causes looping in a different part of the code is [x = 2, 3, 4; y = 1, 3, 2]. A data set on which the code terminates, but with a significantly wrong result is [x = 3, 4, 5, 6, 7; y = 1, 3, 2, 4, 3]. Because of the faulty theoretical foundation, there is no reason to believe any particular result obtained by this code is correct, although by chance it will sometimes get a correct result... "Conclusions "The authors of 'Numerical Recipes' were not specialists in numerical analysis or mathematical software prior to publication of this book and its software, and this deficiency shows WHENEVER WE TAKE A CLOSE LOOK AT A TOPIC in the book [my emphasis]. The authors have attempted to cover a very extensive range of topics. They have basically found 'some' way to approach each topic rather than finding one of the best contemporary ways. In some cases they were apparently not aware of standard theory and algorithms, and consequently devised approaches of their own. The MEDFIT code of section 14.6 is a particularly unfortunate example of this latter situation. "One should independently check the validity of any information or codes obtained from 'Numerical Recipes'..." I have independently checked the codes for Bessel Functions and Modified Bessel Functions of the first kind and orders zero and one (J0, J1, I0, I1). The approximations given in NR are those to be found in the National Bureau of Standards "Handbook of Special Functions, Applied Mathematics Series 55," which were published by Cecil Hastings in 1959. Although the approximations are accurate, they are not very precise: don't trust them beyond 6 digits. Coding them in "double precision" won't help. Much work has been done in the approximation of special functions in the last 32 years. We haven't investigated the quality of every one of the NR algorithms and codes, nor the exposition in every chapter of NR (we have more productive things to do). But sampling randomly (based on calls for consultation) in four areas, and finding ALL FOUR faulty, we have very little confidence in the rest. I hope this helps some of you. Received in the mail, in response to USENET postings: "You can add the section on PDE's to the list of "bad". The discussion of relaxation solvers for elliptic PDE's starts off OK (in about 1950, but that is OK for a naive user if he is not in a hurry) but then fails to mention little details like boundary conditions! Their code has the implicit assumption that all elliptic problems have homogeneous Dirichlet boundary conditions! "Then they have their little coding quirks, like accessing their arrays the wrong way and putting unnecessary IF and MOD statements inside of inner loops.... "On the other hand, I did learn something from their discussion of the Conjugate Gradient technique for solving systems of linear equations. I did not like their implementation, but the discussion was OK." And: "Your posts in sci.physics about "Numerical Recipes" match my experience. I've found that "Numerical Recipes" provide just enough information for a person to get himself into trouble, because after reading NR, one thinks that one understands what's going on. The one saving grace of NR is that it usually provides references; after one has been burned enough times, one learns to go straight to the references :-). "Example: Section 9.5 claims that Laguerre's method, used for finding zeros of a polynomial, converges from any starting point. According to Ralston and Rabinowitz, however, this is only true if all the roots of the polynomial are real. For example, Laguerre's method runs into difficulty for the polynomial f(x) = x^n + 1 if the initial guess is 0, because f'(0) = f''(0) = 0." And: "As an aside, I have just received a preprint from Press describing what looks like chapter 18 for NR -- about the discrete wavelet transform. Now, I can tell you that this stuff is wrong, as the results which are in his figures are not reconstructable using his routines. Don't know why yet, but it just doesn't work. If anyone out there is using these routines -- toss them. If you have fixed these routines or have other discrete wavelet transform routines, I want to know about it. Thanks." And: " ... And so let me offer my personal caveat: SVDCMP does not always work. I found one example where the result is just wrong (fortunately it is easy to check, but one doesn't usually do so). I translated the NR fortran to c, and also tried the NR c code. Both wrong in the same way. I tried IMSL and Linpack in fortran, and tried translating Linpack to c; all three produced correct answers..." And: "The NR-recommended random number generators RAN1 and RAN2 should not be used for any serious application. If you use the top bit of RAN1 to create a discrete random walk (plus or minus 1 with equal probability) of length 10,000, the variance will be around 1500, far below the desired value of 10,000. "Both are low-modulus generators with a shuffling buffer, in one case with the bottom bits twiddled with another low-modulus generator. The moduli are just too low for serious work, and the resulting generators even out too well." And: "... It seems that everyone I talk to has a different part of the book that they don't like (The part I hate most is the section on simulated annealing and the travelling salesman's problem- there are far better approaches to the problem.)" And: "Yes, I was another numerical babe in the woods, told the NR was the ultimate word (obviously by professors and colleagues who had never used it!) and so I spent months trying to figure out why their QL decomposition routine didn't work. I thought it was me......" And: "I recently encountered a bug in Numerical Recipes in C (dunno what edition). Look at the code for ksprob(). When the routine fails to converge after so many iterations, they return 0.0. But a quick glance at the formula reveals that the sum will not converge for *small* lambda (and they must be small indeed), hence the correct value to return is 1.0, not 0.0. "Additionally, it would be nice to caution users that this formula is only an asymptotic approximation to the true function (which nobody, apparently, has figured out yet), and that the method is horribly unstable for small lambda." I don't know if the senders want to be publicly identified. If you're interested in contacting them, send me e-mail, and I'll ask them to contact you. Our experience, and that of many others, is that it is best to get numerical software from reliable sources. The easiest and cheapest is NETLIB, which includes the collected algorithms from ACM Transactions on Mathematical Software (which have all been refereed), and a great many other algorithms that have withstood the scrutiny of the peers of the authors, but in ways different from the formal journal refereeing process. If you don't know how to use NETLIB, the easiest way to get started is to send mail to netlib@ornl.gov, subject irrelevant, with the text consisting of the lines "send help" and "send index". If you're an X-windows user, you may want to consider having NETLIB send you 'xnetlib', an X-windows interface to the NETLIB collection. Best regards. nr-info.medfit: The problem in MEDFIT is more in the theory than in the code. It's hard to point to one place in the code and say AhA! The author of MEDFIT assumed that the minimum of the residual will occur at a place where the derivative is zero. But for L1 approximation, that's not true: the approximating function is only C0 continuous, so there are places where the derivative doesn't exist, namely at every data point. The criterion to use is spelled out in section 4-4 of "The Approximation of Functions -- Vol 1: Linear Theory" by John R. Rice, Addison-Wesley, 1964 (102ff): a minimum occurs where an arbitrary perturbation increases the residual. In the case of a continuous and continuously differentiable approximant, this is the same as saying the derivative is zero, but it's not the case with the approximant for L1 fitting. So the first defect in MEDFIT is in assuming that the minimum occurs at a zero of the derivative of the approximating function. The second defect arises in the transformation of the flawed theory into an algorithm. To understand this problem, consider first the problem of one parameter L1 fitting. In this case, what one wants is the median of the set. But if the size of the set is even, anywhere between the two media is an equally good solution. The algorithm in NR, however, considers only solutions that interpolate one or two data points. (In general, L1 fitting to N parameters will interpolate N data points). The result is that MEDFIT is always computing a derivative at a place where it doesn't exist. It turns out that what MEDFIT really needs is just a sign, and there's a 50-50 chance of getting that right, even if the derivative doesn't exist. But when there's just a little bit of data, and MEDFIT gets close to the solution, it will get alternating positive and negative derivatives, and just rattle back and forth between two points, one probably being a solution (but at which MEDFIT can't decide to quit). The third kind of defect arises in the transformation of the flawed algorithm into code. Others have already pointed out these defects, which can be spotted even if one has no idea what the code is intended to do. Incidentally, the Fortran and C versions could frequently have different behaviour because the tests used in the C version in place of the Fortran SIGN function do not do the same as what the Fortran-77 standard says the SIGN function does. Hope this helps. Favorable reviews of Numerical Recipes in C [From: Daniel Asimov] I would like to offer a contrasting opinion. I bought the original Numerical Recipes edition and found it to be excellent. It covers a wide range of topics and includes lucid explanations of why the algorithms are the way they are, as well as printed computer code for each algorithm. I, too, heard that the first edition contained some errors. But, after a lot of use, I encountered perhaps one error total. For a book that comprehensive, it is almost unimaginable for it to be errorfree. Now I have the second edition (C version), and it seems to me to be superb. All known errors in the first edition have been corrected, a number of topics have been revised or expanded, and a number of new topics (like wavelets) have been added. They will also sell a disk containing all the code (saving you the trouble of typing it in by hand from the book) for about $40. [From: mbk@jt3ws1.etd.ornl.gov (Matt Kennel)] It's by far the most useful single book for my research that I've ever owned. Its explanations are far better for getting scientists to understand the idea (like myself) than anything else I've read. Most other literature are oversimplified undergraduate textbooks (not sufficient for the broad range of problems encountered in research science) or highly inappropriate advanced research works focused either on a single topic and often directed at other people who are creating and proving numerical mathematics instead of just using it. Numerical Recipes is also good at exposing to the naive just what sorts of algorithms do exist "out there" that many people were previously unaware of. Its programs are explicitly not designed to be bullet-proof high performance black boxes. Though I have not personally encountered a problem with them. Still, I think that most people would be wiser LAPACK users after reading NR. Contributions to this collection are welcome. Send email to jutta@pobox.com .
Spline Bibliography
Begun by Larry Schumaker and kept up by Carl de Boor. The references are sorted alphabetically by author, with a searchable index.
spline bibliography spline bibliography This is the collection of references, presumably relevant in spline theory, begun by Larry Schumaker and presently kept up by Carl de Boor. It was last updated on 31dec04 For Europeans, there is a more convenient mirror site . SEARCH here for all the references fitting a given pattern. The search engine at the above-mentioned mirror site is more powerful, as is its original version at a mirror site in the US. On the other hand, the search here responds with an ascii file, ready to be incorporated into a \TeX\ file. email corrections and additions to: deboor@cs.wisc.edu or enter them via an electronic form The references are sorted alphabetically by author, with each reference carrying a unique label, fashioned from author name(s) and year of publication. A , B , C , D , E , F , G , H , I , J , K , L , M , N , O , P , Q , R , S , T , U , V , W , X , Y , Z . The references are in a format that (i) makes it easy to convert them via \TeX, into whatever format some journal or editor might desire (including BibTeX), yet (ii) makes it easy to type them in without the need for various code words. The \TeX\ file refmac.tex contains details and examples for the use of these references in a \TeX\ environment. The format is explained in detail at the end of the \TeX-file journal.tex which also contains the definition of the abbreviations of journal names used in the references. The abbreviations for standard proceedings used can be found in proceed.tex . updated{27jun05}
ArXiv Front: NA Numerical Analysis
Numerical analysis section of the mathematics e-print arXiv.
NA Numerical Analysis Thu 17 Nov 2005 Search Submit Retrieve Subscribe Journals Categories Preferences iFAQ NA Numerical Analysis Calendar Search Authors: AB CDE FGH IJK LMN OPQR ST U-Z New articles (last 12) 15 Nov math.NA 0511354 Two results on ill-posed problems. A. G. Ramm . NA ( FA ). 15 Nov math.NA 0511353 On Compactness of the Embedding. A. G. Ramm . NA . 7 Nov math.NA 0511103 Spherical Harmonics Expansion of the Vlasov-Poisson initial bounary value problem. Christian Dogbe (LMNO). NA . 31 Oct math.NA 0510638 Thermoacoustic tomography - implementation of exact backprojection formulas. Gaik Ambartsoumian (Texas AM University), Sarah K. Patch (University of Wisconsin - Milwaukee). 15 pages. NA . 27 Oct math.NA 0510573 Fast Monte-Carlo Low Rank Approximations for Matrices. Shmuel Friedland , Mostafa Kaveh , Amir Niknejad , Hossein Zare . NA . 26 Oct math.NA 0510516 On Numerical Algorithms for the Solution of a Beltrami Equation. Denis Gaydashev , Dmitry Khmelev . NA ( CV ). Cross-listings 17 Nov math.SP 0511408 On Temple--Kato like inequalities and applications. Luka Grubisic . SP ( NA ). 10 Nov math.DS 0511219 New Periodic Orbits for the n-Body Problem. Cristopher Moore , Michael Nauenberg . DS ( NA ). 8 Nov math.DS 0511178 Non-ergodicity of the Nose-Hoover Thermostatted Harmonic Oscillator. Frdric Legoll , Mitchell Luskin , Richard Moeckel . 18 pages. DS ( MP NA ). 28 Oct q-bio.GN 0510047 An Algorithm for Missing Value Estimation for DNA Microarray Data. Shmuel Friedland , Mostafa Kaveh , Amir Niknejad , Hossein Zare . ( NA ). Revisions 17 Nov math.NA 0505223 Metric based up-scaling. Houman Owhadi , Lei Zhang . NA ( AP PR ). 16 Nov math.NA 0502407 A Least Squares Functional for Solving Inverse Sturm-Liouville Problems. Norbert Roehrl . 8 pages. Inverse Problems 21 (2005) 2009-2017. NA . Recent Calendar 2005 75+43 November 3+3 October 6+3 September 7+4 August 14+6 July 3+3 June 7+5 May 8+5 April 6+1 March 7+6 February 9+5 January 5+2 2004 56+36 December 4+3 November 4+3 October 3+7 September 4+2 August 7+3 July 5+2 June 4+3 May 6+2 April 4+2 March 6+1 February 4+3 January 5+5 2003 46+36 December 4+2 November 2+5 October 2+3 September 6+3 August 0+1 July 2+3 June 5+6 May 9+4 April 3+1 March 2+2 February 4+3 January 7+3 2002 39+30 December 8+5 November 2+4 October 2+2 September 7+3 August 2+2 July 6 June 1+2 May 3+1 April 2+1 March 2+1 February 2+6 January 2+3 2001 50+28 December 4 November 4+4 October 5+1 September 2+2 August 3+1 July 2 June 5+4 May 7+5 April 6+2 March 2+2 February 3 January 7+7 2000 38+36 December 1+7 November 1+3 October 4+13 September 3 August 3+3 July 3 June 1+1 May 4+5 April 6+1 March 4 February 4+3 January 4 1999 46+29 December 0+4 November 4+10 October 2+5 September 9+1 August 5+1 July 6 June 5+2 May 5+1 April 4+1 February 1+2 January 5+2 1998 13+25 December 1+5 November 1 October 1+1 September 2+2 August 2+2 July 0+2 June 2+2 May 2+3 April 1+1 March 0+1 February 0+1 January 1+5 1997 0+3 July 0+1 June 0+1 March 0+1 1996 1+1 October 0+1 April 1 1994 1+1 September 1 April 0+1 1993 3+5 October 0+1 July 1+4 April 2 1992 2+1 January 2+1 Total: 370+274 articles (primary+secondary) Authors AB A Acharya, Amit Akram, Ghazala Alt, Alina Argentati, M. E. Adam, Gh. Aksenov, S. V. Altenbach, H. Argentini, Gianluca Adam, S. Alexandrov, Oleg Ambartsoumian, Gaik Arnold, Douglas N. Adukov, V. M. Allgower, E. L. Anitescu, Mihai Arponen, Teijo Adukov, Victor M. Almulla, N. Arabhi, Sundararajan Atfeh, Bilal B Babuska, Ivo Batrac, Andrew V. Bodon, E. Bradji, Abdallah Bailey, David H. Bayer, Christian Bodon, Elena Broadhurst, David J. Baillet, Laurent Becher, J. Boffi, Daniele Brown, David H. Bakirova, Margarita Bell, J. B. Boikov, I. Brown, Nathanial P. Bao, Weizhu Bell, John Bojanov, Borislav Bueler, Ed Barenblatt, G. I. Benichou, Bertrand Bollhoefer, Matthias Bueno-Orovio, Alfonso Barenblatt, Grigory I. Berngardt, Oleg I. Bornemann, Folkmar Bunn, Julian Barnes, David Bidegaray-Fesquet, Brigitte Bos, Len P. Bussolari, L. Barrera, D. Bjerregaard, Pablo Alberca Bowman, Michael Buyarov, V. Bates, D. J. Bobenko, A. I. Boyadjiev, T. L. CDE C Campos, Rafael G. Chebotarev, Alexander M. Ciraolo, Giulio Cortes, Jorge Candes, Emmanuel Chelyshkov, Vladimir Cohen, Albert Cotter, Colin Caprile, Bruno Chen, W. Coifman, Ronald R. Crainic, Marius Carlsson, Johan Cheung, Dennis Coley, A. A. Crainic, Nicolae Castro, D. Chorin, A. J. Constantin, Peter Crutchfield, W. Y. Celledoni, Elena Chorin, Alexandre J. Conti, C. Crutchfield, William Chan, Youn-Sha Chourkin, Andrei Contucci, P. Cucker, Felipe Charina, M. Ciesielski, Mariusz Cornelis, J. D da Mota, L. A. C. P. Deift, Percy De Wit, David Drake, James M. Daubechies, Ingrid D'Elia, J. Dogbe, Christian Duarte, L. G. S. de Boor, C. Del Popolo, A. Dolean, Victorita Duarte, S. E. S. Dedieu, Jean-Pierre Del Popolo, Antonino Dorn, Oliver Dumitriu, Ioana Degani, Ilan Demmel, James Dorodnitsyn, Vladimir Dumont, Y. de Groen, P. Dettmann, C. P. Dragomir, Sever Silvestru Dvornikov, Maxim Dehesa, J. S. E Elling, Volker Endresen, Lars Petter E., Weinan Eymard, Robert Elser, Veit Ergenc, T. FGH F Fairag, F. Fedoseyev, A. I. Fiziev, P. P. Fournier, Aime Fairag, Faisal Fenton, Flavio H. Foecke, Harold Freund, Roland W. Falk, Richard S. Fernandez, Noemi Lain Fornasier, M. Friedland, Shmuel Fannjiang, Albert C. Fischer, P. Fornasier, Massimo Friedman, M. J. Favini, A. Fischer, Paul G Gallouet, Thierry Genz, Alan Givelberg, Edward Groisman, Pablo Galstian, A. Gerdt, Vladimir P. Gjesdal, Thor Gunter, D. O. Gannon, Dennis Geroyannis, V. S. Glazunov, Nikolaj M. Gusynin, Valery P. Garbey, M. Giardina, C. Glowinski, Roland Gutman, S. Garbey, Marc Giberti, C. Gogilidze, Soso A. Gutman, Semion Garcia-Ripoll, Juan Jose Gil, Amparo Gonzalez, Candido Martin Gutowski, Marek W. Garikipati, K. Gilbert, Anna Graca, Mario M. Gu, Yun Gaydashev, Denis Gioumousis, Peter Grishin, Denis Gwin, Elyus Gelb, Anne Givelberg, E. Groby, Jean-Philippe H Hakopian, H. Han, YoungAe Henon, M. Holtz, Olga Hald, O. H. Hauser, Raphael Herbin, Raphaele Homeier, Herbert H. H. Hald, Ole H. Hein, Steffen Heyde, K. Horwitz, Alan Halim, A. A. He, J. Hinrichs, Aicke Humpherys, Jeffrey Han, Hai-Shan IJK I Ibanez, M. J. Iliev, A. Iliev, Anton I. Ingerman, Eugene A. Ida, Masato Iliev, A. I. Ingber, Lester Iserles, Arieh Iliescu, T. J Jacek, Leszczynski Jacquet, W. Jaksch, Dieter Jentschura, U. D. Jackson, Brad Jakobsen, Espen R. Jalnapurkar, Sameer M. Johnson, Charles R. K Kamdem, Jules Sadefo Katsoulakis, Markos A. Klimov, Yu. Kreuzer, Maximilian Kang, Sheon-Young Kaveh, Mostafa Knyazev, A. V. Kriecherbauer, Thomas Kansa, E. J. Kayumov, Alexander Koepf, Wolfram Kshevetskii, S. P. Kaper, H. G. Kevrekidis, Ioannis Koltracht, Israel Kumar, Manish Kaper, Hans G. Kevrekidis, Ioannis G. Kornyak, Vladimir V. Kuperberg, Greg Karasozen, B. Khmelev, Dmitry Korzh, A. Kupferman, R. Karczewska, Anna Kim, Tommy Kunhung Kosowski, Przemyslaw Kupferman, Raz Kast, A. Kirtman, Bernard Kozlov, Roman Kurganov, Alexander Kast, Anton LMN L Lackner, Klaus Leaf, G. K. Levy, D. Liu, Ying Ladipo, Kehinde O. Leble, S. B. Levy, Doron Li, Xing Lafon, Stephane Lee, Taeyoung Leykin, Anton Lombard, Bruno Langford, William F. Leimkuhler, Ben Lim, W. C. Lopez-Fernandez, Maria Lanzara, F. Leite, Ricardo S. Little, John B. Lubich, Christian Lasserre, J. B. Lemahieu, I. Litvinov, Grigori Luksan, L. Latche, Jean-Claude Leok, Melvin Litvinov, Grigori L. Lundh, Torbjorn Layton, William J. Leszczynski, Jacek Liu, Xu-Dong Luskin, Mitchell M Macia, Fabricio Maslova, Elena McLachlan, Robert I. Mohr, P. J. MacKenzie, T. Maslov, Viktor Melnik, R. V. N. Molari, L. Malajovich, Gregorio Mathar, Richard J. Meneses, Claudio Montana, J. L. Mariusz, Ciesielski Matthes, D. Menikoff, Ralph Moro, Esteban Marsden, Jerrold E. Maz'ya, V. Merlet, Benoit Moussa, Jonathan E. Martinez-Finkelshtein, A. McClamroch, N. Harris Mielke, Alexander Mumford, David Martin, J. San McLachlan, Robert N Nadiga, Balu T. Naumenko, K. Nie, Qing Nistor, Victor Nadler, Boaz Nefedov, I. M. Niesen, Jitse Noid, D. W. Nataf, Frederic Neuberger, B. Nigro, Noberto M. Nordborg, H. Natanzon, S. Neuberger, J. W. Niknejad, Amir Novak, Erich OPQR O Okunev, Pavel Owhadi, Houman P Pahlevani, Faranak Pardo, L. M. Perlmutter, Matthew Postnikov, Eugene B. Palencia, Cesar Parlett, Beresford N. Petit, J-M. Pouquet, Annick Palla, Gergely Patch, Sarah K. Piraux, Joel Priouret, Pierre Pan, Tsorng-Whay Paulino, Glaucio H. Pirumov, Ulyan G. Prostokishin, V. M. Paoli, L. Pekarsky, Sergey Plechac, Petr Puppo, G. Paoli, Laetitia Perez-Garcia, Victor M. Postnikov, E. B. Q Qian, Liwen Qi, Yingyong Quispel, G. R. W. Quispel, G. Reinout W. Qin, Mengzhao R Radwan, Samir F. Rawitscher, George Rodionov, Anatoli Rombouts, S. Ramm, A. G. Rein, Gerhard Roehrl, Norbert Rosenberg, Duane Ramm, Alexander G. Ren, Weiqing Roemer, Rudolf A. Rozmej, Piotr Rannacher, Rolf Roberts, A. J. Rojas, J. Maurice Russo, G. Rapin, Gerd Robidoux, Nicolas Romanyukha, N. Rusu, Dan D. Rasch, Christian Rodewis, Thomas Romberg, Justin ST S Sablonniere, Paul Schmidt, G. Siddiqi, Shahid S. Sondergaard, Niels Saito, Naoki Schorghofer, Norbert Sideris, Thomas C. Sopasakis, Alexandros Saldanha, Nicolau C. Schurz, Henri Silva, E. V. Correa Sorge, H. Sanchez-Lara, J. Schwab, Christoph Sivaloganathan, Sivabal Spedicato, E. Sangtrakulcharoen, Paungkaew Segura, Javier Skarke, Harald Spedicato, Emilio Sassi, Taoufik Semerdzhiev, Kh. I. Skea, J. E. F. Stinis, Panagiotis Savageau, M. A. Semerdzhiev, Khr. Smirnova, A. Storti, M. A. Sawant, Aarti Sethian, J. A. Smirnova, A. B. Storti, Mario A. Sbibih, D. Shabanov, Sergei V. Smoktunowicz, Alicja Strang, Gilbert Scargle, Jeffrey D. Shereshevski\ui, I. A. Sobolevskii, Andrei Strauss, Martin Schadle, Achim Shibberu, Yosi Soff, G. Strohmer, Thomas Schatzman, Michelle Shkoller, Steve Sokal, Alan D. Su, Hongling Schenk, Olaf Shutov, A. Sommese, A. J. Sun, Y. H. Scherzer, Otmar Shvets, Helen Sommese, Andrew J. Suris, Yu. B. Schiff, Jeremy T Tadmor, Eitan Tao, Terence Tenti, Guiseppe Truyen, B. Tanabe, S. Tausch, Johannes Todorov, M. D. Tsaban, Boaz Tan, Linda Taylor, Mark A. Tomei, Carlos Tsai, Tun Tao Tanner, Jared Teichmann, Josef Tonoyan, M. Tselyaev, V. I. Tannor, David Temme, Nico M. Traub, J. F. Tsogka, Chrysoula U-Z U Ujevic, Nenad Unguendoli, F. Uwano, Yoshio V Valvi, F. N. Veldhuizen, Todd L. Verschelde, Jan Voronov, Alexander L. Vanden-Eijnden, Eric Venakides, Stephanos Vertesi, P. Vrahatis, M. N. Vattay, Gabor Vernia, C. Viazminsky, C. P. W Wampler, C. W. Wei, G. W. Weniger, Ernst Joachim Wingate, Beth A. Wampler, Charles W. Weinstein, Alan West, Matthew Wozniakowski, Henryk Wang, Yusong Wells, G. N. Whitlow, Darryl Wrobel, Iwona Wasilkowski, Grzegorz W. Weniger, E. J. X Xia, Z. Xia, Zunquan Xiu, Dongbin Xu, Yuan Xia, Zun-Quan Xin, Jack Y Yazadjiev, S. S. Ye, Yinyu Z Zare, Hossein Zhao, Ailing Zou, Jing Zumbrun, Kevin Zeron, E. S. Zhou, Y. C. Zubelli, Jorge P. Zykov, P. S. Zhang, Lei Zhou, Yunkai zu Castell, Wolfgang Search Author Title ID Anywhere Cat MSC articles per page Show Help AC AG AP AT CA CO CT CV DG DS FA GM GN GR GT HO KT LO MG MP NA NT OA OC PR QA RA RT SG SP ST Authors: All AB CDE FGH IJK LMN OPQR ST U-Z Home Search Submit Retrieve Subscribe Journals Categories Preferences iFAQ - for help or comments about the Front - for help about submissions or downloading arXiv articles
Approximation People
A list of home pages kept at the University of Erlangen.
APPROXIMATION PEOPLE HOMEPAGE Approximation People A - E ** F - J ** K - O ** P - S ** T - Z A - E Peter Alfeld Brad Baxter Hubert Berens Ranko Bojanic Carl de Boor Peter Borwein Bruce Chalmers Wen Chen Ward Cheney Charles Chui Albert Cohen Stephan Dahlke Wolfgang Dahmen Oleg Davydov Ronald DeVore Kai Diethelm Sven Ehrich David Elliott Tamas Erdelyi F - J Greg Fasshauer Hans Georg Feichtinger Andrei M. Finkelshtein Martina Finzel Michael S. Floater Jeff Geronimo Tim Goodman Bin Han Douglas Hardin Chris Heil Klaus Hllig Don Hong Tom Hogan Yingkang Hu Bjorn Jawerth Rong-Qing Jia Kurt Jetter K - O Kirill Kopotun Tom Kunkle Michael Lachance Ming Jun Lai Seng Luan Lee Junjiang Lei Dany Leviatan William Light Tom Lyche Alphonse Magnus Detlef Mache Petra Mache Hrushikesh Narhar Mhaskar Knut Mrken Manfred Mller Fran Narcowich Mike Neamtu Edward Neuman Paul Nevai Erich Novak Konstantin Osipenko Konstantin Oskolkov Peter Oswald P - S Knut Petras Jorg Peters Pencho Petrushev Allan Pinkus Gerlind Plonka Jrgen Prestin Andrei Reztsov Lothar Reichel Sherman Riemenschneider Klaus Ritter Giuseppe Rodriguez Amos Ron Edward Saff Thomas Sauer Robert Schaback Hans Joachim Schmid Larry Schumaker Kathi Selig Robert Sharpley Zuowei Shen N. Sivakumar Joachim Stckler Gilbert Strang Wim Sweldens T - Z Vladimir Temlyakov Walter Van Assche Loc Vu-Quoc Grace Wahba Shayne Waldron Hans Wallin Guido Walz Joseph Ward Xiang Ming Yu top
Approximation and Imaging
Universit de Pau. Research topics, faculty, publications, meetings.
Welcome to the University of Pau - Math. Dpt - Approximation and Imaging HomePage Lab. de Math Appliques de l'Universit de Pau UMR CNRS 5142 Analyse Numrique (Approximation-Imagerie) Publication in interpolation, error rates... New collaboration A new work with UCLA, Los Angeles, started on nov. 10, 2004 Contract with Univ. Zaragoza in Spain A Research project with Spanish Math. Dpt from frontier regions (Aragon...) is in preparation THIRD INTERNATIONAL CONFERENCE Multivariate Approximation: Theory Applications at Cancun, MEXICO, April 24-29, 2003 Org. : C. Gout - C. Rabut - L. Traversoni Proceedings will appear in JULY 2005 Approximation Mathematical Imaging General information, Research Topics Publications Last Publications Future Meetings Next conferences in Approximation Imaging Faculty Home Pages Library Math. Library, Preprints Database (soon) Campus Life Services for the University AFA Association Franaise d'Approximation Math Department Telephone, mail, and email contact information Administration Administrative offices UPPA Sports From the Pyrnes to the Atlantic coast Approximation Theory Network Publications... Mathematical Database... Free search... SEARCH AMS MR Lookup SEARCH Science Database INGENTA Alphabetic index of the University of Pau Take an Online Tour of Pau! Mairie de Pau Section Paloise (rugby) Elan Barnais Pau-Orthez (basket ball) Contact the Webmaster - Copyrights C. Gout 1999 2000
A View from the Back of the Envelope
About approximation and the fun it enables. Includes examples, exercises, and links to help grasp concepts such as measurement, scale and magnitude. Sections include simplifying numbers, exponential notation, Fermi problems, using your body as a ruler, and other ways to visualize size and scale.
A View from the Back of the Envelope This page Introduction Table of Contents What's New About these pages These pages are about approximation, and some of the fun it enables. You can always get back to this page by clicking on Envelope above. one... ten... hundred Counting by powers of ten "What order of magnitude is ...?" game Counting to ten billion... on your fingers Lots of dots and A million dots on one page A dot for every second in the day (with a clock ). New! 2001 Apr Scaling the universe to your desktop Meters in your hands nano | micro | milli | meter | kilo | mega | giga | tera | peta Constructing your own desktops. How Big Are Things? New! 2001 Feb Simplify numbers How to simplify a number by rounding , sometimes to an order of magnitude , sliding the decimal point , and using a number you can remember . Another example of simplifying is A ball's volume and area are 1 2 of its box's . Exponential Notation is sliding the decimal point. An Exponential Notation Meta Page . Rounding to an order of magnitude and What is "order of magnitude"? Scientific Notation is exponential notation plus conventions. How to write and speak the exponential notation Fermis Fermi Questions Rough quantitative estimates about the world. Talamo's Fermi Problems site - "What order of magnitude is ...?" game - A Pinocchio estimation game New! 2001 Feb - a mayonnaise story - landmarks - bounding - honesty - Why be approximate? - On Being Approximate Scale Scale of some things `Powers of Ten' scales people , people seconds , volume , area , length , time , mass , energy , area volume-density , speed , volume rate , power ? Developing "deep" understanding If you can't explain it to a nine year old... ( Picturing altitude above maps - teleportation - probing near space with a flashlight ) Getting a feel for big numbers - For children - Atomic bonding Body Ruler Measuring length with your body. Measuring Angle Distance with your Thumb Resources Books some additional links. Cosmic View: The Universe in 40 Jumps A copy of the 1957 classic which inspired Powers of Ten. Standing on the back of the envelope, one can see much of the universe. I have found it a vantage point of great power and beauty, but unfortunately one much neglected. It is my hope, in collecting these resources, to increase its accessibility, and to draw greater attention to it. I welcome and appreciate your thoughts, your questions, and your contributions. - Mitchell N Charity mcharity@lcs.mit.edu Site alpha release 0.2 (1997.Aug.01) Rewrote this page 1998.Mar.28.
People in Approximation Theory
Amos Ron's list of links.
Amos Ron's list of people in Approximation Theory Amos Ron's list of people affiliated with Approximation Theory Please deposit your URL in the mailbox , if you wish me to include you in the list below This page is only the beginning! at a later stage, I will include photos taken at meetings, so that you may click on the face of your favorite person, in order to enter his her hompepage (no need to remember names any more !!!) Brad Baxter Faqir Bhatti Ranko Bojanic Carl de Boor Peter Borwein Abderrahman Bouhamidi Dietrich Braess Marcel de Bruin Martin Buhmann Paul Butzer Ward Cheney Charles Chui Steven Damelin Ron DeVore Stefano De Marchi Zeev Ditzian Sven Ehrich Thomas Erdelyi Michael Floater Jeff Geronimo Anatoly Golub Tim Goodman Christian Gout Bin Han Doug Hardin Chris Heil Margareta Heilmann Tom Hogan Don Hong Alan Horwitz Yingkang Hu Rong-Qing Jia Palle Jorgensen Kirill Kopotun Tom Kunkle Ming-Jun Lai Seng Luan Lee Daniel Lemire Xin Li Svenja Lowitzsch Tom Lyche Detlef Mache J-J. Martinez Knut Mrken Fran Narcowich Edward Neuman Paul Nevai Peter Oswald Jorg Peters A. Martinez Finkelshtein Allan Pinkus Gerlind Plonka Juergen Prestin Ewald Quak T.S.S.R.K. Rao Andrei REZTSOV Sherm Riemenschneider Giuseppe Rodriguez Ed Saff Thomas Sauer Robert Schaback Kathi Selig Zuowei Shen Sivakumar Ian Sloan Stamatis Koumandos Alexander Stepanets Grace Wahba Shayne Waldron Joe Ward Liu Yongping Xiang Ming Yu Zhiqiang Xu your name Check also the JAT list and tamu cat page
NZ Approximation Theory Group
Members, meetings, visitors and links.
NZ Approximation Theory Group New Zealand Approximation Theory Group (aka Approximation down under) Welcome to the homepage of the New Zealand Aotearoa Approximation Theory Group. The core of the group is Rick Beatson (Canterbury) and Shayne Waldron (Auckland), both of whom work fulltime on Approximation Theory, and consists of the following people . Here is a list of the addresses and interests of the Approximation Theorists working in Australasia . Conferences Surface Approximation and Visualisation University of Canterbury 15-18 February 1999 Surface Approximation and Visualisation II Westport 19-22 February 2002 Visitors Professor Len Bos of the University of Calgary visits Norm Levenberg at the University of Auckland every year. Here is a list of visitors to and from Australasia . One of the aims of the group is to support postdocs in Approximation Theory. You are most welcome to come and visit us ( visitor information ). Approximation Theory Network The AT-NET (Approximation Theory Network) is intended as a ready means for the distribution of information to the Approximation Theory community. The more people know about it and sign on, the better for the community. Here are the AT-NET bulletins . There is also NA-NET (Numerical Analysis Network) and OP-SF-NET (Orthogonal Polyomials and Special Functions Network). Bibliographic data bases The University of Auckland is a mirror site for the highly useful Spline Bibliography currently maintained by Carl de Boor , and provides a convenient electronic form for submitting references to it. It can be searched electronically in Madison or using the JAT version of find_out (faster). Journals The main journals in Approximation Theory are Approximation Theory and its Applications Constructive Approximation East Journal on Approximations Journal of Approximation Theory Journal of Computational Analysis and Applications History of Approximation Theory Surveys in Approximation Theory Use Paul Nevai and Giuseppe Rodriguez 's most excellent CA + EJA + JAT search to search the tables of contents of all three journals (simultaneously!). Approximation Theorists Here are searchable lists of Approximation Theorist's addresses and e-mails The address lists of persons whose last names start with the letter... A B C D E F G H I J K L M N O P Q R S T U V W X Y Z The e-mail lists of persons whose last names start with the letter... A B C D E F G H I J K L M N O P Q R S T U V W X Y Z (a database of htmls is on the way, some htmls ). Also, a list of those in Australasia . Other WWW sites See Tom Hogan's extensive list of Approximation Theory web sites, or visit Amos Ron in Madison. The French connection , the Wavelet IDR Center . Edited by Shayne ( waldron@math.auckland.ac.nz ). Last Modified on .
History of Approximation Theory
Articles and web resources developed by Allan Pinkus and Carl de Boor.
History of Approximation Theory Primary site - Technion Wisconsin Auckland Giessen (AT-NET) FoCM Focus Group History of Approximation Theory (HAT) --- Approximation People --- This is our short list of the main past contributors to approximation theory. --- Articles on HAT --- Articles devoted to the history of approximation theory. --- Historical Papers --- Some seminal papers in approximation theory. --- Links --- Links to other relevant sites. This page is presently being developed by Allan Pinkus and Carl de Boor. You are most welcome to join us or to comment, help, suggest, support, or criticize. We can be reached at either: pinkus@tx.technion.ac.il or deboor@cs.wisc.edu . Our thanks to the following people for their help . FastCounter by bCentral
A Short Course on Approximation Theory
Lecture notes for a graduate course by Neal Carothers, Bowling Green State University. (DVI,PS)
Approximation Theory A Short Course on Approximation Theory Please read the disclaimer and description before downloading these notes. This is a set of lecture notes for a short course on Approximation Theory that I offered to graduate students at Bowling Green State University , Bowling Green, Ohio, in the Summer of 1998 (and, in somewhat different form, in 1994 and 1990). A complementary textbook for the course was T. J. Rivlin's An Introduction to the Approximation of Functions, Dover, 1981. The course was intended as a survey of elementary techniques in Approximation Theory for novices and non-experts. Experts in the field seeking new, original, or research topics should look elsewhere. This is strictly for beginners! Prerequisites for a thorough understanding of the course include: A first course in advanced calculus or real analysis (pointwise and uniform convergence, compactness, etc.). A rudimentary knowledge of normed spaces and completeness. A course in linear algebra. There is enough material here for roughly 25 hour-and-a-half lectures; probably not quite enough for a full semester course. On the other hand, it is sufficient background to facilitate reading E. W. Cheney's Introduction to Approximation Theory, Chelsea, 1982, or G. G. Lorentz's Approximation of Functions, Chelsea, 1986, two excellent sources for further study. The notes are written in Plain TeX (plus AMSFonts) and are available here in dvi and Postscript format. The printed version is 159 pages. Available as a dvi file: Macintosh stuffed dvi file, 203K UNIX gnu-zipped dvi file, 203K Available as a Postscript file: UNIX gnu-zipped ps file, 325K Table of Contents Preface Preliminaries Exercises on Normed Spaces Approximation by Algebraic Polynomials Exercises on Approximation by Polynomials Approximation by Trigonometric Polynomials Exercises on Trigonometric Polynomials Characterization of Best Approximation Exercises on Chebyshev Polynomials Simple Application of Chebyshev Polynomials Lagrange Interpolation Exercises on Interpolation Approximation on Finite Sets Introduction to Fourier Series Exercises on Fourier Series Jackson's Theorems Orthogonal Polynomials Exercises on Orthogonal Polynomials Gaussian Quadrature The Mntz Theorems The Stone-Weierstrass Theorem Short List of References Neal Carothers - carother@bgnet.bgsu.edu
Approximation Theory Network (AT-NET)
Publishes a Bulletin and lists journals, theses, people, events and other links.
AT-NET's Homepage marginwidth=20 marginheight=20 name="rechts" scrolling=yes marginwidth=20 marginheight=20
Approximation Theory
Notes for courses given by Carl de Boor, Madison. (PS)
MA887 - Approximation Theory ------------------------------------- last change: 9may03 MA887 - Approximation Theory This page contains information for the Spring 2003 version of this math course. B B Check out the Announcement for the course which will take place MWF 13:16-14:10 at B231 Van Vleck. As promised in the Announcement, there will be some HOMEWORK ASSIGNMENTS , and the same page now also provides answers to all the assignments. The instructor is very willing to answer email, at the address deboor@cs.wisc.edu All email sent to the class list is archived . Old notes, for 1995 , 1998 , and for 2000 , are still available. However, I will be making changes here and there in these notes as I teach, and, for that reason, will maintain Notes (now up to page 87), in postscript and in pdf . I am assuming that you will have no difficulty printing out the latest pages from the postscript or pdf file. To help you with this, I am putting, for each page, the date of its most recent revision into its lower left corner. For an earlier version of this course, Amos Ron , a co-instructor then for the second part, prepared the following notes . I will use these notes for the last part of the present course but will not be able to resist the temptation to make changes; so, have a look at the emended(?) notes . In addition, that page (as of 23feb98) also contains an alternative way of looking at approximation order by convolution kernels . There is a 100-page survey, meant for a general audience, of NONLINEAR APPROXIMATION by Ron DeVore, that has appeared in Acta Numerica; 7; 1998.
Approximation Theory
Lecture notes for Part III of the Mathematical Tripos by A. Shadrin, DAMTP, University of Cambridge. (PS)
Approximation Theory Part III Mathematical Tripos: Part III Approximation Theory A. Shadrin, DAMTP, University of Cambridge Lent Term 2005, MWF, MR 15, 9:00 Lecture notes for the course The left icon produces a pdf- or a postscript-file. The course will mainly follow the same scheme as the year before, and the "postscript" icons corresponds to the lectures from the previous year which should give a general idea of the material to be covered. The updated lectures are within the "pdf" icons, but be aware that I use to make small changes until the last minute before the actual lecture. Syllabus Lecture 1 Lecture 7 Lecture 13 Lecture 19 Lecture 2 Lecture 8 Lecture 14 Lecture 20 Lecture 3 Lecture 9 Lecture 15 Lecture 21 Lecture 4 Lecture 10 Lecture 16 Lecture 22 Lecture 5 Lecture 11 Lecture 17 Lecture 23 Lecture 6 Lecture 12 Lecture 18 Lecture 24 Examples 1 Examples 2 Examples 3 Examples 4
Bibliography in the Approximation of Curves and Surfaces
Mainly in English, some French.
Conference Approximation et Representation de Courbes et Surfaces ACHIESER N.I., Theory of Approximation, Frederick Ungar Publishing, 1956. AHLBERG J.H., E.N. NILSON and J.L. WALSH, The Theory of Splines and their Applications, Academic Press, 1967. ATTEIA M., Hilbertian Kernels and Spline Functions, North holland, 1992. BARNHILL R. and W. BOEHM, Eds, Surfaces in Computer Aided Design. Proceedings of a conference held in Oberwolfach (Germany), April 25-30, 1982. North Holland, Amsterdam, 1983. BARNHILL R., W. BOEHM and J. HOSCHEK, Eds, Curves and Surfaces in CAGD'89, North Holland, 1990. Proceedings of a Conference held April 16-22 1989, in Oberwolfach, Germany. Reprinted from CAGD, Vol. 7, Numbers 1-4, 1990. BARNHILL Robert E., Ed., Geometry Processing for Design and Manufacturing, SIAM, 1992. BARSKY Brian A., Computer Graphics and Geometric Modeling using Beta-splines, Springer Verlag, 1988. BARTELS Richard H., John C. BEATTY and Brian A. BARSKY, An Introduction to Splines for Use in Computer Graphics and Geometric Modeling, Morgan Kaufmann Publishers, 1987. BERGER M., Geometrie 1, Geometrie 2, F. Nathan, 1990. BEZHAEV A. Yu. and V. A. VASILENKO, Variational Spline Theory, NCC Publisher, Novosibirsk, Series: Numerical Analysis, Special issue of NCC Bulletin 3, 1993. BREZINSKI C., History of Continued Fraction and Pade Approximants. Springer Verlag. BOHMER Klaus, Spline-Funktionen, Teubner, 1974. BOHMER Klaus, Gunter MEINARDUS and Walter SCHEMPP, Eds, Spline-Funktionen, B.I.-Wissenschaftsverlag, 1974. Proceedings of a conference held April 29-May 2, 1973 in Oberfolfach, Germany. BOJANOV B.D., H.A. HAKOPIAN and A.A. SAHAKIAN, Spline Functions and Multivariate Interpolations, Kluwer Academic Pub., Mathematics and its Applications, Vol. 248, 1993. BOWYER Adrian, Computer-Aided Surface Geometry and Design, The Mathematics of Surfaces IV, Clarendon Press, Oxford, 1994, Proceedings of the fourth conference of the mathematics of surfaces held Sept. 10-12, 1990, at the University of Bath, Great Britain. BREZINSKI Claude, History of Continued Fractions and Pade Approximants, Springer Verlag, Springer Series in Comput. Math., 12, 1991. BUTZER P.L. and J. KOREVAAR, Eds, Uber Approximationstheorie, Birkhauser verlag, 1964. Proceedings of a conference held August 4-10, 1963, in Oberwolfach, Germany. BUTZER P.L.and B. Sz.-NAGY, Eds, Abstract Spaces and Approximation, Birkhauser Verlag, 1969. Proceedings of a conference held July 18-27, 1968, in Oberwolfach, Germany. BUTZER P.L., B. Sz.-NAGY and E. GORLICH, Eds, Functional Analysis and Approximation, Birkhauser Verlag, 1981. Proceedings of a conference held August 9-16, 1980, in Oberwolfach, Germany. CHENEY E.W., Ed., Approximation Theory III, Academic Press, 1980. CHOI Byoung K., Surface Modeling for CAD CAM, Elsevier, 1991. CHUI C.K., L.L. SCHUMAKER and J.D. WARD, Eds, Approximation Theory IV, Academic Press, 1983, Proceedings of a symposium held Jan. 10-14, 1983, at Texas AM University, College Station, USA. CHUI C.K., L.L. SCHUMAKER and J.D. WARD, Eds, Approximation Theory V, Academic Press, 1986. Proceedings of the fifth international symposium on approximation theory held January 13-17, 1986, at Texas AM University, College Station, USA. CHUI C.K., L.L. SCHUMAKER and F.I. UTRERAS, Eds, Topics in Multivariate Approximation, Academic Press, 1987. Proceedings of an international workshop held at the University of Chile in Santiago, December 15-19, 1986. CHUI Charles K., Multivariate Splines, Regional conference series in applied mathematics, 54, SIAM, 1988. CHUI C.K., W. SCHEMPP and K. ZELLER, Eds, Multivariate Approximation Theory IV, Birkhauser Verlag, 1989. Proceedings of a conference held February 12-18, 1989, at Oberwolfach, Germany. CHUI C. K., Ed., Approximation Theory and Functional Analysis, Academic Press, 1991, Proceedings of a conference held Feb. 24-25, 1990, at Texas AM University, College Station, USA. CHUI Charles K., An Introduction to Wavelets, 1, Academic Press, 1992. CHUI Charles K., Wavelets: A Tutorial in Theory and Applications, 2, Academic Press, 1992. CIESIELSKI Zbigniew, Ed., Approximation and Function Spaces, North Holland, 1981. Proceedings of a conference held August 27-31, 1979, in Gdansk, Poland. COLLATZ L., H. WERNER and G. MEINARDUS, Eds, Numerische Methoden der Approximationstheorie, 3, Birkhauser Verlag, 1976. Proceedings of a conference held Mai 25-31, 1975, in Oberwolfach, Germany. COLLATZ L., H. WERNER and G. MEINARDUS, Eds, Numerische Methoden der Approximationstheorie, 4, Birkhauser Verlag, 1978. Proceedings of a conference held November 13-19, 1977, in Oberwolfach, Germany. COLLATZ L., H. WERNER and G. MEINARDUS, Eds, Numerische Methoden der Approximationstheorie, 5, Birkhauser Verlag, 1980. Proceedings of a conference held November 18-24, 1979, in Oberwolfach, Germany. CONTE A., V. DEMICHELIS, F. FONTANELLA, I. GALLIGANI, Eds, Computational Geometry, World Scientific, 1993, Proceedings of a workshop held June 23-26, 1992, in Torino, Italy. COX M.G. and J.C. MASON, Eds, Algorithms for Approximation III, In "Numerical Algorithms, Vol 5, 1-4, Proceedings of the NATO workshop held July 27-31, 1992, at Oxford, England. DAHMEN W., M. GASCA and C.A. MICCHELLI, Eds, Computation of Curves and Surfaces, Kluwer Academic Publishers, Netherlands, 1990. Proceedings of a Nato Adv. Inst. held 1989, in Tenerife, Canary Islands. de BOOR Carl, A Practical Guide to Splines, Springer Verlag, 1978. de BOOR C., K. HOLLIG, S. RIEMENSCHNEIDER, Box Splines, Springer Verlag, Applied Math. Sci. 98, 1993. de CASTELJAU Paul de Faget, Formes a Poles, Hermes, Mathematiques et CAO, volume 2, 1985. de CASTELJAU Paul de Faget, Shape Mathematics and CAD, Hermes Publishing, 1986. CHOI Byoung K., Surface Modeling for CAD CAM, Elsevier, Advances in Ind. Eng. 11, 1991. DAHMEN Wolfgang, Mariano GASCA and Charles A. MICCHELLI, Eds, Computation of Curves and Surfaces, Kluwer Academic Pub., Series C: Math. and Phys. Sci., Vol. 307, 1990, Proc. of a NATO advanced Study Institute held July 10-21, 1989, at Puerto de la Cruz, Tenerife, Spain. DELTHEIL R. and D. CAIRE, Geometrie et Complements, Ed. J. Gabay, 1989. DELVOS F.-J. and W. SCHEMPP, Boolean Methods in Interpolation and Approximation, Longman Scientific and Technical, 1989. DEVORE Ronald A. and Karl SCHERER, Eds, Quantitative Approximation, Academic Press, 1980. Proceedings of a symposium on "quantitative approximation" held August 20-24, 1979, in Bonn, Germany. EARNSHAW R.A., Ed., Fundamental Algorithms for Computer Graphics, Springer Verlag, 1985. Proceedings of the NATO advanced study institute on "Fundamental algorithms for computer graphics" held March 30-April 12, 1985, Ilkley, Yorkshire, England. EARNSHAW R.A., Ed., Theoretical Foundations of Computer Graphics and CAD, Springer Verlag, 1988. Proceedings of the NATO advanced study institute on "Theoretical foundations of computer graphics and CAD" held July 4-17, 1987, at ll Ciocco, Italy. FARIN Gerald E., Ed., Geometric Modeling: Algorithms and New Trends, SIAM, 1987. Proceedings of a SIAM conference on "Geometric Modeling and Robotics" held July 15-19, 1985, in Albany, New-York, USA. FARIN Gerald, Curves and Surfaces for Computer Aided Geometric Design, a Practical Guide, Academic Press, 1988. FARIN Gerald, Ed., NURBS for Curve and Surface Design, SIAM, 1991. FIOROT J. C. et P. JEANNIN, Courbes et Surfaces Rationnelles. Applications a la CAO. RMA 12, Masson, 1989. Rational Curves and Surfaces; Applications to CAD, Wiley and Sons, 1992. FIOROT J.C. et P. JEANNIN, Courbes Splines Rationnelles; Applications a la CAO. RMA 24, Masson, 1992. GASSER Th. and M. ROSENBLATT, Eds, Smoothing Techniques for Curve Estimation, Springer Verlag, 1979. Proceedings of a workshop held April 2-4, 1979, in Heidelberg, Germany. GRUSA Karl-Ulrich, Zweidimensionale, interpolierende Lg-Splines und ihre Anwendungen, Lecture Notes in Mathematics, 916, Springer Verlag, 1982. HAGEN Hans, Ed., Curve and Surface Design, SIAM, 1992. HAGEN Hans, Ed., Topics in Surface Modeling, SIAM, 1992. HAUSSMANN W. and K. JETTER, Eds, Multivariate Approximation and Interpolation, Birkhauser Verlag, 1990. Proceedings of an international workshop held August 14-18, 1989, at the University of Duisburg (Germany). HOFFMANN C.M., Geometric and Solid Modeling; an Introduction. Morgan Kaufmann, 1989. HOSCHEK Josef and Dieter LASSER, Grundlagen der geometrischen Datenverarbeitung, B.G. Teubner, 1989. Fundamentals of Computer Aided Geometric Design, A.K. Peters, 1993. JETTER Kurt and Florencio I. UTRERAS, Eds, Multivariate Approximation: From CAGD to Wavelets, World Scientific, 1993, Proceedings of an international workshop held Sept. 24-30, 1992, in Santiago, Chile. LANCASTER Peter and Kestutis SALKAUSKAS, Curve and Surface Fitting, an Introduction, Academic Press, 1986. LAURENT Pierre-Jean, Approximation et Optimisation, Hermann, 1972. LAURENT Pierre-Jean, Alain LE MEHAUTE and Larry L. SCHUMAKER, Eds, Curves and Surfaces, Academic Press, 1991. Proceedings of the AFA international conference on "Curves and Surfaces" held June 21-27, 1990, in Chamonix-Mont-Blanc, France. LAURENT Pierre-Jean, Alain LE MEHAUTE and Larry SCHUMAKER, Eds, Curves and Surfaces in Geometric Design, A. K. Peters, 1994, Proceedings of the second AFA international conference on "Curves and Surfaces" held June 10-16, 1993, in Chamonix-Mont-Blanc, France. LAURENT Pierre-Jean, Alain LE MEHAUTE and Larry SCHUMAKER, Eds, Wavelets, Images and Surface Fitting, A. K. Peters, 1994, Proceedings of the second AFA international conference on "Curves and Surfaces" held June 10-16, 1993, in Chamonix-Mont-Blanc, France. LAW Alan G. and Badri N. SAHNEY, Eds, Theory of Approximation with Applications, Academic Press, 1976. Proceedings of a conference held August 11-13, 1975, at the University of Calgary, Alberta, Canada. LEON J.C., Modelisation et Construction de Surfaces pour la CAO, Hermes, 1991. LORENTZ G.G., Approximation of Functions, Holt, Rinehart and Winston, 1966. LORENTZ G.G., Ed., Approximation Theory, Academic Press, 1973. Proceedings of a conference held January 22-24, 1973, in Austin, Texas (USA). LORENTZ G.G., C.K. CHUI and L.L. SCHUMAKER, Eds, Approximation II, Academic Press, 1976. LYCHE Tom and Larry L. SCHUMAKER, Eds, Mathematical Methods in Computer Aided Geometric Design, Academic Press, 1989. Proceedings of an international conference held June 16-22, 1988, at the University of Oslo (Norway). LYCHE Tom and Larry L. SCHUMAKER, Eds, Mathematical Methods in Computer Aided Geometric Design II, Academic Press, 1992. Proceedings of an international conference held June 20-25, 1991, at Biri (Norway). MASON J.C. and M.G. COX, Eds, Algorithms for Approximation, Clarendon Press, 1987. Proceedings of the IMA conference on "Algorithms for the approximation of functions and data" held July 1985, at the Royal Military College of Science, Shrivenham. MASON J.C. and M.G. COX, Eds, Algorithms for Approximation II, Chapman and Hall, 1990. Proceedings of the second international conference on "Algorithms for Approximation", held July 1988, at the Royal Military College of Science, Shrivenham. MEINARDUS Gunter, Approximation of Functions: Theory and Numerical Methods, Springer Verlag, 1967. MEINARDUS Gunter, Ed., Approximation in Theory und Praxis, B.I.-Wissenschaftsverlag, 1979. Proceedings of a symposium held January 31-February 2, 1979, in Siegen, Germany. MEIR A. and A. SHARMA, Eds, Spline Functions and Approximation Theory, Birkhauser Verlag, 1973. Proceedings of the symposium held May 29-June 1, 1972, at the University of Alberta, Edmunton, Canada. MEYER Yves, Les Ondelettes, Armand Colin, Collection "Acquis avances de l'informatique", 1992. MICCHELLI C.A., D.V. PAI and B.V. LIMAYE, Eds, Methods of Functional Analysis in Approximation Theory, ISNM 76, Birkhauser Verlag, 1986. Proceedings of the international conference held December 16-20, 1985, at the Indian Institut of Technology, Bombay, India. NEAMTU Marian, Multivariate Splines, Thesis, Dept of Applied Math., University of Twente, The Netherlands, Dec. 12, 1991. POWELL M.J.D., Approximation Theory and Methods, Cambridge University Press, 1981. RICE John R., The Approximation of Functions, Addison-Wesley Publishing Co., 1964. RISLER Jean-Jacques, Methodes Mathematiques pour la CAO, RMA 18, Masson, 1991. RIVLIN Theodore J., An Introduction to the Approximation of Functions, Blaisdell Publishing Company, 1969. SCHABACK R. and K. SCHERER, Eds, Approximation Theory, Springer Verlag, 1976. Proceedings of an international colloqium held June 8-11, 1976, at Bonn, Germany. SCHEMPP Walter and Karl ZELLER, Eds, Multivariate approximation theory, Birkhauser Verlag, 1979. Proceedings of a conference held February 4-10, 1979, in Oberwolfach, Germany. SCHONBERG I.J., Ed., Approximation with Special Emphasis on Spline Functions, Academic Press, 1969. Proceedings of a symposium held May 5-7, 1969, at the University of Wisconsin, Madison, USA. SCHOENBERG I.J., Cardinal Spline Interpolation, Regional conference series in applied mathematics, SIAM, SCHONHAGE Arnold, Approximationstheorie, Walter de Gruyter, 1971. SCHUMAKER Larry L., Spline Functions: Basic Theory, John Wiley, 1981. SEMPLE J.G. and G.T. KNEE BONE, Algebraic Projective Geometry, Oxford Univ. Press, 1979. SINGER Ivan, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer Verlag, 1970. SPATH Helmuth, Algorithmen fur elementare Ausgleichsmodelle, Oldenburg Verlag, 1973. SPATH Helmuth, Algorithmen fur multivariable Ausgleichsmodelle, Oldenburg, 1974. STRASSER W. and H.-P. SEIDEL, Eds, Theory and Practice of Geometric Modeling, Springer Verlag, 1989. Proceedings of the conference " Theory and practice of geometric modeling" held October 3-7, 1988, at the University of Tubingen, Germany. SU BU-QING and LIU DING-YUAN, Computational Geometry, Curve and Surface Modeling, Academic Press, 1989. TIMAN A.F., Theory of Approximation of Functions of a Real Variable, Pergamon Press, 1963. VARGA R.S., Functional Analysis and Approximation Theory in Numerical Analysis, Regional conference series in applied mathematics, 3, SIAM, 1971. WAHBA Grace, Spline Models for Observational Data, Regional conference series in applied mathematics, 59, SIAM, 1990. YAMAGUCHI Fujio, Curves and Surfaces in Computer Aided Geometric Design, Springer Verlag, 1988. ENST home page AFA home page June 1995. For any comment or suggestion mail to AFA
Association Franaise d'Approximation
The subgroup of SMAI concerned with approximation and representation of curves and surfaces. Features conferences, workshops and bibliography.
AFA home page Association Franaise d'Approximation AFA is the group of SMAI (Socit de Mathmatiques Appliques Industrielles) for academic and industrial people concerned with approximation and representation of curves and surfaces. It organizes the International Conferences CURVES SURFACES 2002 June 27- July 3 Fifth "CURVES SURFACES" at Saint-Malo (FRANCE) 1999 July 1 - 7 Fourth "CURVES SURFACES" at Saint-Malo (FRANCE) The proceedings are edited in two volumes. Contents can be obtained in dvi by clicking on the buttons below. "Curve and Surface Design" - Saint-Malo 1999 Pierre-Jean Laurent, Paul Sablonniere, and Larry L. Schumaker (eds) Vanderbilt University Press, 2000 "Curve and Surface Fitting" - Saint-Malo 1999 Albert Cohen, Christophe Rabut, and Larry L. Schumaker (eds) Vanderbilt University Press, 2000 1996 June 27 - July 3 Third "CURVES SURFACES" at Chamonix Mont-Blanc (FRANCE) June 1993 Second "Curves Surfaces " at Chamonix Mont-Blanc (FRANCE) The proceedings are edited by Pierre-Jean Laurent, Alain Le Mhaut and Larry L. Schumaker in two volumes "Curves and Surfaces in Geometric Design" and "Wavelets, Images and Surface Fitting" published both by A.K. Peters (1994) June 1990 First "Curves Surfaces " at Chamonix Mont-Blanc (FRANCE) The proceedings "Curves and Surfaces" are edited by Pierre-Jean Laurent, Alain Le Mhaut and Larry L. Schumaker published by Academic Press (1991) Yearly workshops at CIRM (Centre International de Rencontres Mathmatiques) November 1994 "New trends in Curves and Surfaces" April 1995 "Hilbertian Approximation" April 1996 "Geometric Modelling and Approximation" October 1998 "Modlisation et Approximation de Courbes et Surfaces" Some interesting pointers : A bibliography on Curves and Surfaces Approximation Enst june 1998. For any comment or suggestion mail to AFA
Numeritek
Home of Numerica, a library for solving hyperbolic partial differential equations.
Numeritek Website Numerical Methods Software Courses Conferences Publishing Consultancy
Beasy
Boundary Element software for various areas such as stress, durability analysis, heat transfer, fatigue, crack growth, acoustic design, corrosion, cathodic protection, mechanical design, contact and wear.
BEASY Software, Making Simulation Simpler "BEASY provides solutions and services for engineers in Industry" Integrity Assessment Corrosion Control Other Corrosion Modelling in the Oil and Gas Industry Interactive Workshop Wednesday 1st February 2006 Aberdeen, UK More Info Mechanical Design and Fracture Modelling Using BEASY Version 10 Interactive Workshop Tuesday 7th March 2006 Southampton, UK More Info Corrosion Modelling Optimisation 14th -17th March 2006 Ashurst Lodge, Southampton, UK More Info Fracture and Failure Modelling 22nd -24th November 2005 Ashurst Lodge, Southampton, UK More Info
Applied Research Associates NZ Ltd (ARANZ)
Developers of 3D digitizer (handheld laser scanner), point-cloud meshing and 3D data gridding software.
ARANZ: Applied Research Associates NZ Ltd 3D Scanning and Data Modelling [ Home | About | Contact ] ARANZ is an established developer of 3D scanning and modelling technologies. Click on the buttons below to find out about some of the products based on these technologies. ARANZ is an established developer of 3D scanning and modelling technology for applications as diverse as movie-making, geological modelling and medicine. ARANZ supplies both hardware and software solutions to industry. About - More details about the ARANZ Group of companies. Contact - How to contact ARANZ. FastSCAN is the industry's most portable and lightweight handheld laser scanner, and is distributed world-wide by Polhemus. www.fastscan3d.com In a collaborative effort between ARANZ and Sir Charles Gairdner Hospital's Radiation Oncology Department (Perth Australia), a system has been developed for producing immobilisation masks using FastSCAN. Information sheet Newspaper story Television News Story [2.2 MB Mpeg] In a collaboration with Hanger, the largest US prosthetics and orthotics provider, ARANZ developed a complete workflow solution for custom prosthetics and orthotics manufacturing: Hanger Insignia. The system includes a modified hand-held laser scanner and tailored software. Television News Story [3.1 MB WMV] www.hanger.com insignia The FastRBF product range offers developers, researchers, engineers and OEMs advanced techniques for modeling scattered 2D and 3D data. FastRBF enables data sets consisting of millions of points to be interpolated by Radial Basis Functions (RBFs). New filtering and approximation methods make FastRBF ideal for visualizing and processing non-uniformly sampled noisy data. The FastRBF family of products is marketed and sold through ARANZ's subsidiary FarField Technology Ltd. www.farfieldtechnology.com products toolbox SilverLining is a stand-alone graphical tool based on our FastRBF technology, for fitting smooth meshes to point-cloud data or incomplete mesh data. It produces watertight meshes for computer graphics and rapid prototyping. SilverLining is marketed and sold through ARANZ's subsidiary FarField Technology Ltd. www.farfieldtechnology.com products silverlining Leapfrog is a completely new way of processing, viewing and interpreting drill-hole data. With Leapfrog, you can immediately visualise mineralisation patterns in 3D through a single interactive processing environment. Leapfrog is developed and marketed by Zaparo Ltd, a joint venture between SRK Australia and ARANZ www.leapfrog3d.com Copyright ARANZ Ltd 2004. All rights reserved. Page last updated: January 2005.
(Norway) Norwegian University of Science and Technology
Department of Mathematical Sciences, Numerical Analysis Group. People, research projects, meetings.
Department of Mathematical Sciences, NTNU - Numerical Analysis Group Numerical Analysis Group The numerical analysis group is one of the groups at the Department of Mathematical Sciences , The Norwegian University of Science and Technology . We are located on campus, in the 13th floor of one of the two highrises. E. Celledoni, A. Kvrn, E. Rnquist, B. Owren S. P. Nrsett News Einar Rnquist spends the academic year 2005 2006 at MIT, Boston. Franz-Theo Suttmeier left the group November 1, 2005 to take up a position in Germany. We wish him the best of luck in his new job. Abel symposium May 2006. Mathematics and Computation, a Contemporary View. Research Exponential integrators BeMatA project Reduced Basis Methods BeMatA project Computational internal waves The SYNODE-project Computational Science and Engineering at NTNU SUP in CSE at NTNU Technical reports [ 2005 ] [ 2004 ] [ 2003 ] Academic staff Elena Celledoni Anne Kvrn Syvert P. Nrsett Brynjulf Owren Einar Rnquist PhD students Hvard Berland Bjarte Hgland Emil Alf Lvgren Brd Skaflestad Tormod Bjntegaard Vegard Kippe Niklas Svstrm Preprint server Also look at publications from the synode-project . Old projects and activities Inverse problems "Norsk Numerikk Mte" PhD defenses Runar Holdahl 02.09.2004 Hallgeir Melb 17.12.2001 Inge Morten Skaar 07.05.2001 Roman Kozlov 05.01.2001 Arne Marthinsen 13.05.1999 Editor: Instituttleder Contact address: Brynjulf Owren Updated: 2005-11-13
(Netherlands) Radboud University Nijmegen
Subfaculty of Mathematics, Department of Numerical Analysis.
Subfaculty of Mathematics 4 februari 2002 Bron: bp Department of Numerical Analysis Head Prof. dr. A.O.H. Axelsson Staff Research Programme Programme Numerical Analysis Zoeken in de openbare Research pagina's (m.b.v. de SURFnet Search Engine): Voorbeeld van zoeken met meer termen: reports AND 1998
(UK) University of Portsmouth
Numerical Analysis Sector. Approximation Theory; Integral and Integro-differential Equations. Members, publications, courses, events, resources.
Numerical Analysis at Portsmouth Numerical Analysis at Portsmouth Department of Mathematics 1st Floor Buckingham Building University of Portsmouth Portsmouth PO1 3HE, UK Telephone: +44 23 92 846369 Fax: +44 23 92 846364 Contact: Dr. Makroglou: athena.makroglou@port.ac.uk Recent scientific activities In October 2004, the SCIG (Scientific Computing Interest Group) of the Univ. of Portsmouth was established. Spring 2004-05: Graduate Student Seminars (jointly with SCIG). 2005-06: Computational Science and applications Interdisciplinary Research Seminars (jointly with SCIG). Organisation (A. M. and Prof. Yang Kuang, Arizona State University) of a mini-symposium titled "Differential and Integral Equations in Epidemiology and Medicine: Applications and Numerics" for the HERCMA 2005 Conference, 22-24 September, 2005, AUEB, Athens, Greece Organisation (A.M. and Prof. Claudia Klppelberg, Technical University of Munchen, Germany), of a mini-symposium titled "Actuarial Risk Models: Theory and Computations", for the HERCMA 2005 Conference, 22-24 September, 2005, AUEB, Athens, Greece . A short report which includes the titles of the talks of the mini-symposia and a couple of photos may be found here. HERCMA 2005 Conference short report (.pdf file). The Second International Workshop on Analysis and Numerical Approximation of Singular Problems Karlovassi, Samos, Greece, 6-8 September, 2006. Portsmouth contact: athena.makroglou@port.ac.uk What is Numerical Analysis Numerical Analysis is the area of Mathematics and Computer Science concerned with the solution of Mathematical problems using a Computer. Such problems include data fitting, evaluation of various types of integrals, solution of algebraic equations and systems of equations, solution of ordinary, partial, integral and integro-differential equations. Application areas related areas Numerical Analysis has applications in all areas of Science and Technology. A partial list of such areas includes Physics, Chemistry, Biology, Medicine, Engineering, Computer Graphics, Economics, Finance and Actuarial Sciences, Law, Psychology and Social Sciences. Numerical Analysis is closely related to Scientific Computing. (See SCIG (Scientific Computing Interest Group at Portsmouth) for information, members, communication letters). Numerical Analysis Research at the Dept. of Mathematics, Univ. of Portsmouth Two main Numerical Analysis research areas are represented at the Dept. of Mathematics of the University of Portsmouth currently: Approximation Theory (Dr. Graham Elliott) Integral and Integro-differential Equations (Dr. Athena Makroglou) Publication lists may be found in Some publications of Dr. Elliott and in Publications of Dr. A. Makroglou Current research students Maria Apostolou , Mphil., start date: October 2004. Title of thesis: Numerical algorithms and Mathematics for some of the early childhood diseases. Supervisor: Dr. A. Makroglou Iordanis Karaoustas, Ph.D., part time, start date: October 2005. Research area: Numerical Integral Equations. Supervisor: Dr. A. Makroglou. New applications information Applications of students for M.Phil and Ph.D. degrees are welcome for part or full time. Prospective students may contact Dr. A. Makroglou for preliminary discussions. Desirable qualification: M.Sc. in Numerical Analysis - Applied Mathematics. Application (and other) forms may be downloaded directly from the forms page which is part of the Academic Registry page for research degrees . The pdf file of the research degrees application form for 2006 is Research Degree Application Form 2006 entry . Numerical Analysis and Computing Teaching at the Dept. of Mathematics, Univ. of Portsmouth Classes with elements of Numerical Computational content offered to Mathematics students by members of the Mathematics Dept. include: Num 101 Numerical Methods CMP 108 Computer Packages in Mathematical Studies . Num 202 (Numerical Analysis). The Num 202 web page is at: Numerical Analysis (Num 202) Web page . Num 301 Finite Elements Nume 302 Function Approximation ( Nume 302, 2004-05 web page). Mth 305 (Optimisation). Optimisation (Math 305, 2004-05 web page). Mth 315 Partial Differential Equations Mth 317 Financial Derivative Pricing 2 and Financial Derivative Pricing 2 (Part B) (Math 317, part B, 2003-04 web page). CMP 305 C++ Programming . The CMP 305 2004-05 policy is at: CMP 305 2004-05 policy (pdf). 50 years of progress of Numerical Analysis, a Conference report In June 1998, the first 50 years of Numerical Analysis were celebrated in Manchester by a Conference titled `Numerical Analysis and Computers --- 50 Years of Progress'. It was organised by the Dept. of Mathematics of the University of Manchester. The report of the Conference in long and short format was written by one of the members of our Numerical Analysis sector at the Univ. of Portsmouth, Dr. A. Makroglou (a former graduate of the University of Manchester). The Conference talks went through some of the important developments of the subject during the past 50 years with extensions into the future. The two reports may be downloaded from the Conference page as (Short version - 9 pages, in compressed postscript form) (Longer version - 17 pages, in compressed postscript form) Numerical Analysis and Computing links Some links for general interest Mathematics periodical publications Mamthematics Today , by IMA (The Institute of Mathematics and its Applications) Plus , part of the Millennium Mathematics Project . The Mathematical Gazette , by The Mathematical Association, UK . Page created in July 2003 by A. Makroglou Mathematics Department Homepage UoP Faculty of Technology Homepage University of Portsmouth homepage Portsmouth Math-Net Homepage SCIG (Scientific Computing Interest Group at Portsmouth). Page maintained by Athena Makroglou Athena Makroglou |
(USA) Texas A and M University
Numerical Methods for PDE Group. Primarily concerned with the efficient numerical approximation of solutions of partial differential equations.
Texas AM Numerical Methods for PDE Group Home | People | Seminar | Projects | Former Visitors and Students Numerical Methods for PDE Group The numerical methods group is primarily concerned with the efficient numerical approximation of solutions of partial differential equations. The techniques and expertise include the development and analysis of iterative methods, stability and error analysis for finite element, finite difference and finite volume approximations, and large scale scientific computation with industrial application. The group consists of eight permanent faculties, graduate students and numerous visitors. As well as doing research into theoretical numerical analysis, the group works closely with the Institute for Scientific Computation in the development of large scale scientific simulations for serial and parallel computing architectures. The graduate program in numerical analysis includes courses in basic numerical analysis and analysis of iterative methods. More advanced courses concerning the theory of finite elements, domain decomposition, multigrid, and mixed finite elements are also offered on a regular basis. Numerical Analysis Qualifying Examination Texas Finite Element Rodeo Home | People | Seminar | Projects | Former Visitors and Students Last revised Sept. 29, 2003. mail to Webmaster
Computational Fluid Dynamics, Mathematical Modeling and Numerical Methods
The focus of this group is to find numerical solutions of differential equations coming from several areas of applications, as for example. Participants are from various universities in the US and Europe.
Research Group: Computational Fluid Dynamics, Mathematical Modeling Numerical Methods Computational Fluid Dynamics, Mathematical Modeling Numerical Methods GOALS The Computational Fluid Dynamics, Mathematical Modeling Numerical Methods research group has as its main goals: To integrate its participant members through joint research, departing from common interests in Computational Fluid Mechanics, Mathematical Modeling and Numerical Methods. To cooperate in the teaching load of graduate courses, supervision of dissertations and thesis, allowing for a solid background development for the undergraduate and graduate students. To estimulate the exchange of ideas with other research groups and institutions, from inside and outside of the University of Sao Paulo, and to collaborate for the diffusion of the scientific knowledge, to cooperate for the projects and other specialized services. PARTICIPANT MEMBERS Currently, this research group has six researchers, about one third of the Applied Mathematics Department . Prof.Dr. Saulo Maciel Rabello de Barros (MS-5) Dr. Rer. Nat., Universitt Bonn Prof.Dr. Alexandre Megiorin Roma (MS-3) Ph.D., New York University - Courant Institute of Mathematical Sciences Prof.Dr. Antonio Elias Fabris (MS-3) Ph.D., University of East Anglia - School of Information Systems Prof. Dra. Joyce da Silva Bevilacqua (MS-3) Doctor, Universidade de Sao Paulo - Institute of Astronomy and Geophysics Prof.Dr. Luis Carlos de Castro Santos (MS-3) Ph.D., Georgia Institute of Technology - School of Aerospace Engineering Prof.Dr. Nelson Mugayar Kuhl (MS-3) Ph.D., New York University - Courant Institute of Mathematical Sciences RESEARCH AREAS The focus of this group is the numerical solution of differential equations coming from several areas and applications, as for example: Computational Fluid Mechanics To development numerical techniques to solve problems in aerodynamics, biofluid dynamics, physics of plasms, and weather numerical prediction. Regarding to these fields, the group researchs multigrid methods, self-adaptive refinement techniques, parallel algorithms, and inverse methods for compressible and incompressible flows. Mathematical Modeling and Numerical Analysis To study algorithms to solve numerically ordinary and partial differential equations. To investigate the modeling of systems in biology and industry. To study numerical techniques for solving systems of simultaneous equations, eigenvalue problems and for parallelizing numerical algorithms. ACADEMIC ACTIVITIES Supervision of Students: Currently, the participant members of this group supervise 10 graduate students enrolled in the scientific initiation program, 5 MSc and 6 Doctorate program students. Mathematics Education: Investigation and developing new ways to teach mathematics, focusing its applications, employing modern technological resources such as audio and video, computer multimedia, and internet. Graduate Courses: Among others, the most important graduate courses offered by this group are: MAP5724 NUMERICAL RESOLUTION OF ELLIPTIC PARTIAL DIFFERENTIAL EQNS. Contents: Second-order elliptic partial differential equations. Discretization methods: finite differences and finite elements. Classical relaxation methods: Gauss-Seidel and SOR. Gradient conjugate method; pre-conditioning. Direct methods. Fast Poisson solvers. Introduction to multigrid methods. Bibliography: W. Hackbusch, Theorie und Numerik elliptischer Differentialgleichungen, Teubner, Stuttgart, 1986; W. Hackbusch, Multigrid methods and applications, Springer, 1985; K. Stuben, U. Trottenberg, Multigrid methods: fundamental algorithms, model problem analysis and applications. Springer, 1982. (Lecture Notes in Math. 960); J. Stoer, R. Bulirsch, Introduction to numerical analysis, Springer, 1980. MAP5725 NUMERICAL TREATMENT OF ORDINARY DIFFERENTIAL EQUATIONS Contents: 1. A brief introduction to the theory of ordinary differential equations: existence, uniqueness, continuity, differentiability, and periodicity of solutions. First order systems. 2. Discretization methods: consistency, stability, and convergence. Forward step methods. Single and multiple step methods. Consistency. 3. Stability criteria. 4. Fixed point theorem. 5. Predictor-corrector methods. 6. Numerical stability of ODE wth initial conditions. Stiff ODEs. 7. Variable time stepping. Local truncation error. 8. Comparison between several methods. Bibliography: I.Q. Barros, Mtodos numricos em lgebra linear, vol. I., IMECC--UNICAMP, 1970; L. Collatz, The numerical treatment of differential equations, Springer, 1960; P. Henrici, Discrete variational methods in ordinary differential equations, Wiley, 1962; J.D. Lambert, Computational methods in ordinary differential equations, Wiley, 1973; L.S. Pontriaguin, Ordinary differential equations, Addison--Wesley, 1962; H.J. Stetter, Analysis of discrete methods for ordinary differential equations, Springer, 1973. MAP5726 INTRODUCTION TO THE COMPUTATIONAL FLUID DYNAMICS I: INCOMPRESSIBLE FLUIDS Contents: 1. A brief introduction to tensor algebra and analysis 2. Equations of motion 3. The solution in closed form for some particular examples 4. Some basic concepts of numerical analysis for partial differential equations 5. Non-dimensional form, similarity solutions, and Prandtl boundary layer equations 6. Generalized coordinates 7. Equivalent forms for the Navier-Stokes equations (conservative form, vorticity-stream,...) 8. Selection among several numerical methods: projection methods, vortex methods, spectral methods, finite element methods, etc. Bibliography: Chorin, A.J.; Marsden, J.E., A mathematical introduction to fluid mechanics, Springer-Verlag; Gurtin, M.E.,An introduction to continuum mechanics, Academic Press; Serrin, J.,Mathmatical principles of classical fluid mechanics, Handbuch der Physik, VIII 1, Springer-Verlag; Strikwerda, J.C. Finite difference schemes and partial differential equations, Wadsworth and Brooks Cole Mathematics Series; Peyret, R.; Taylor, T.D., Computational methods for fluid flow, Springer-Verlag. MAP5729 INTRODUCTION TO NUMERICAL ANALYSIS Contents: 1. Resolution of linear systems: direct and iterative methods. 2. Resolution of non-linear equations: Richardson method, Newton. 3. Interpolation: polynomial (Lagrange and Hermite), splines. 4. Gaussian quadrature, Romberg method (based on extrapolation). 5.Numerical resolution of ordinary differential equations: initial value problems - single and multistep methods; contour problems; finite difference methods. Bibliography: J. Stoer, R. Bulirsch, Introduction to numerical analysis, Springer, Berlin, 1980; E. Isaacson, H.B. Keller, Analysis of numerical methods, Wiley, 1966. MAP5745 FLUID MECHANICS Contents: 1.Brief introduction to algebra and analysis. 2. Cinematics. 3. Momentum and mass. 4. Forces. 5. Constitutive hypothesis - inviscid fluids. 6. Change in the position of the observer - material invariance 7. Newtonian fluids - Navier--Stokes equations. 8. Finite elasticity. 9. Linear elasticity. Bibliography: M.E. Gurtin, An introduction to continuum mechanics, New York, Academic Press, 1981. 265p. (Mathematics in Science and Engineering Series) MAP5822 MULTIGRID METHODS Contents: 1. Introduction: basic concepts. 2. The model problem: Poisson equation on a rectangle. Convergence analysis. Computational optimality.Variational formulation. 3. Elliptic systems. 4. Multigrid methods for finite elements. 5. Higher order discretizations, residual correction. 6.Multigrid e local refinement. FAC and AFAC. 7. Multigrid methods for hyperbolic and parabolic problems. 8. Parallelism. 9.Other topics as time permits. Bibliography: A. Brandt, Multigrid techniques: 1984 guide with applications to fluid dynamics,GMD--Studie, 85 (St. Augustin), 1984; W. Hackbusch, Multigrid methods and applications, Berlin, Springer, 1985; S. McCormick, Multilevel adaptive methods for partial differential equations, SIAM, Philadelphia, 1989; K. Stber, U. Trottenberg, Multigrid methods: fundamental algorithms, model problem analysis and applications, in: Hackbuich and Trottenberg (eds.), Berlin, Springer, 1982, (Lecture Notes in Mathematics, 960).
(UK) University of Cambridge
Department of Applied Mathematics and Theoretical Physics, Numerical Analysis Group. People, reports, seminars, lecture notes, publications, web resources.
Cambridge Numerical Analysis
(USA) NIST
Information Technology Laboratory, Mathematical and Computational Sciences Division. Software, reports, meetings and links.
Mathematical Computational Sciences Division About MCSD The Mathematical and Computational Sciences Division, of NIST 's Information Technology Laboratory , provides technical leadership in modern analytical and computational methods for solving scientific problems of interest to American industry. What MCSD does MCSD's organization Mathematical Modeling Group Mathematical Software Group Optimization Computational Geometry Group Scientific Applications Visualization Group MCSD's Staff Reports Summary of Division Activities 2003-04 ... 2002-2003 ... 2001-2002 ... 2000-2001 ... 1999-2000 ... 1998-99 ... 1997-98 ... 1996-97 ... 1995-96 ... 1994-95 On-line Access to Selected Staff Publications Numerical Evaluation of Special Functions Search math.nist.gov Search ITL Webspace Search NIST Webspace Services MCSD Seminar Series Software Developed by Division Staff Guide to Available Mathematical Software Matrix Market Java Resources for Numerical Computing ACM Transactions on Mathematical Software SIAM Activity Group on Orthogonal Polynomials and Special Functions Current Projects AlgoCEM Time-Domain Algorithms for Computational Electromagnetics DLMF Digital Library of Mathematical Functions IMPI Interoperable Message Passing Interface OOMMF Object-Oriented MicroMagnetic computing Framework OOF Object-Oriented Finite Element Modeling of Material Microstructures PHAML Parallel Adaptive Multigrid Methods and Software SciVis Scientific Visualization Sparse BLAS Basic Linear Algebra Subprograms TNT Template Numerical Toolkit for C++ Past Projects JazzNet Personal Supercomputing on PC Clusters Reflectance Computer Graphic Rendering of Material Surfaces Multi-Laboratory Projects CTCMS NIST Center for Theoretical and Computational Materials Science CTCMS-ALCOM Liquid Crystal Polymer Working Group muMAG NIST Micromagnetic Modeling Activity Group MSORS Measurement Science for Optical Reflectance and Scattering Quantum Information Recent Events Topics in Operations Research , A Symposium in Honor of Dr. Christoph Witzgall; Gaithersburg, MD, May 13, 2004. Workshop on the Changing Face of Mathematical Software , Washington, DC, June 3-4, 2004. MCSD News Quantum Computing with Realistically Noisy Devices Pete Stewart Elected to National Academy of Engineering Lozier Presents Keynote Lecture on Mathematical Knowledge Management Past News Highlights ... Contact Dr. Ronald F. Boisvert NIST, 100 Bureau Drive, Stop 8910 Gaithersburg, MD 20899-8910 USA 301-975-3800 boisvert@nist.gov Location Directions. Opportunities Postdoctoral Research Program Student Employment Opportunities Privacy Policy | Disclaimer | FOIA NIST is an agency of the U.S. Commerce Department's Technology Administration. Date created: 1995-01-01, Last updated: 2005-07-22. Contact mcsdweb @ nist.gov
(UK) University of Brighton
Computational Mathematics Group. Interests include Acoustics, Bubble dynamics, Diffusion in gels and Plasma physics.
School of Computing, Mathematical and Information Sciences undergraduate postgraduate professional consultancy research contacts Information about... --- Faculty of MIS Brighton Business School School of Service Management CENTRIM --- B.U.R.K.S. (Resource Kit) CMIS Intranet --- Undergraduate Courses Postgraduate Courses Courses for Working Professionals Consultancy Research Contact Us Search for CMIS Staff --- University of Brighton Student Intranet Staff Central Computational Mathematics Group School of Computing, Mathematical and Information Sciences, University of Brighton PRINCIPAL STAFF: Dr. Roma Chakrabarti, Dr. Steve Ellacott , Dr. Paul Harris , Dr. David Henwood, Mr. Keith Parramore and Dr. William Wilkinson. For further information about staff, look at the CMIS Staff Search . CURRENT RESEARCH STUDENTS: David Chappell, Philip Newman The Computational Mathematics Group is involved with a number of on-going research projects. The details of some of these are listed below. The Boundary Element Method Previous research has covered a variety of application areas, notably bubble dynamics. However current activity focuses on acoustics. The group is currently working on the modelling of the transient sound field radiated by a loud-speaker cone as part of a long-standing collaboration between the University of Brighton and BW Loudspeakers Ltd . Prediction of the sound field enables the design process to be simplified. Calculated results are compared to actual measurements obtained using a laser interferometer and anechoic chamber at BW. The animation shows the calculated pressure field generated on the surface of the loudspeaker cone by an axisymmetric oscillation of the cone. The group is also collaborating with the University of Liverpool on precondtioners for iterative solution of boundary element equations. Paul Harris is on the steering scientific committee for two upcoming conferences. They are the 5th UK Boundary Integral Method conference in Liverpool, 12th-13th September 2005 and the 9th International Conference on Integral Methods in Science and Engineering , 23 - 27 July 2006 in Niagra Falls, Canada. Space Plasmas The group's research interest in space plasmas focuses on the various aspects of collisionless shock waves; quasi-perpendicular and quasi-parallel shock structure; shock thermalization processes; particle dynamics at shocks; upstream and downstream wave phenomena; particle distributions and wave-particle interactions and co-rotating interaction regions in the solar wind. The work involves theoretical and computer simulation studies supported by some analysis of spacecraft measurements. The supersonic solar wind is deflected past the Earth by a standing shock wave (pink in the picture). This picture is from the ESA NASA SOHO spacecraft portfolio and is courtesy of NASA. The plasma is heated as it passes through the shock. Phase space analysis allows the particle distribution to be determined. On the diagram, the dots show the calculated velocity distribution (white dots, one standard deviation; black dots, mode and two standard deviations), and the ellipses indicate a Gaussian approximation to the distribution. The diagram is taken from Ellacott and Wilkinson, 2003 (see publications below). Recent PhD H. Wang, Boundary integral modelling of transient wave propagation with application to acoustic radiation from loudspeakers, University of Brighton, 2004 Recent Publications Boundary Elements K. Chen and P. J. Harris, On efficient preconditioners for iterative solution of a Galerkin boundary element equation for the three dimensional exterior Helmholtz problem, The Journal of Computational and Applied Mathematics, 156, pp. 303-318, 2003. H. Wang, P. J. Harris, R. Chakrabarti and D. Henwood, Modelling transient acoustic radiation using the retarded potential integral equation method, Proc. Third UK Conference on Boundary Integral Methods (ed. S. Amini), University of Salford, pp. 57-66, 2003. P. J. Harris, An investigation into the use of the boundary integral method to model the motion of a single gas or vapour bubble in a liquid, Proc. Third UK Conference on Boundary Integral Methods (ed. S. Amini), University of Salford, pp. 37-46, 2003. K. Chen, S. Hawkins and P. J. Harris, Easily inverted approximation type preconditioners for matrices arising from wavelet discretisations of boundary integral equations, Proc. Third UK Conference on Boundary Integral Methods (ed. S. Amini), University of Salford, pp. 165-174, 2003. P. J. Harris and K. Chen, A discontinuous linear boundary element method for solving the three-dimensional exterior Helmholtz problem, Proc. Third UK Conference on Boundary Integral Methods (ed. S. Amini), University of Salford, pp. 57-66, 2003. P. J. Harris, An investigation into the use of the boundary integral method to model the motion of a single gas or vapour bubble in a liquid, Engineering Analysis with Boundary Elements, 28, Issue 4, pp. 325-332, 2004. P. J. Harris, W. K. Soh and H. Al-Awadi, An investigation into the effects of heat transfer on the motion of a spherical bubble ANZAM J 45, pp. 361-371, 2004. P. J. Harris, H. Wang, R. Chakrabarti and D. Henwood, A method for modelling non-harmonic periodic acoustic radiation from a loudspeaker, Proceedings of Integral Methods in Science and Engineering 2002, Ed. C. Constanda, Birkhauser, Boston, 2004. Space Plasma Physics P. J. Moran, A. R. Breen, C. A. Varley, P. J. S. Williams, W. P. Wilkinson and J. Markkanen Measurements of the direction of the solar wind using interplanetary scintillations, Annales Geophysicae, 16 (10), pp. 1259-1264, 1998. A. R. Breen, P. J. Moran, C. A. Varley, W. P. Wilkinson, P. J. S. Williams, W. A. Coles, A. Lecinski and J. Markkanen, Interplanetary scintillation observations of interaction regions in the solar wind, Annales Geophysicae, 16 (10), pp. 1265-1282, 1998. W. P. Wilkinson, Multiple encounters of a specularly reflected ion with planar quasi-perpendicular shocks, Planetary and Space Science 47(3-4), pp. 529-543, 1999. W. P. Wilkinson, Constraints from the theory of magnetohydrodynamics on the supersonic interaction of two coplanar plasmas Journal of Technical Physics 40 (1), pp. 49-52, 1999. W. P. Wilkinson What the Rankine-Hugoniot relations tell us about co-rotating interaction regions in the solar wind, and what they dont, Proceedings of the Les Woolliscroft Memorial Conference Sheffield Space Plasma Meeting: Multi-Point Measurements versus Theory, Sheffield (UK), 24-26 April 2001, European Space Agency Special Publication SP-492 (ed. B. Warmbein), pp. 45-52, July 2001. W. P. Wilkinson The Earth's quasi-parallel bow shock: review of observations and perspectives for Cluster, Planetary and Space Science 51 (11), pp. 629-647, 2003. (Invited review article for a special issue of the journal). S. W. Ellacott and W.P. Wilkinson, Heating of directly transmitted ions at low mach number perpendicular shocks: new insights from a statistical physics formulation, J. Geophys Res., vol.108, no. A11, 1409, 2003. S. W. Ellacott and W.P. Wilkinson, Irreversibility and entropy issues concerning the heating of directly transmitted ions at low mach number perpendicular collisionless shocks, to appear in J. Advances in Space Research. Click here for a preprint. Other S. W. Ellacott, The arithmetical structure of Celtic key patterns, Mathematics Today, vol.36 no.5 pp142-150, October 2000. Computational Mathematics Research Group, Brighton Research Last updated 9th March 2005 Webmaster Contact
(UK) University of Leicester
Centre for Mathematical Modelling. Specializing in interdisciplinary solutions based on advanced simulation and modelling tools. Members, research projects.
about the centre research groups directions msc phd home calendar news colloquium Newton Description Newton Access Info Newton Usage HEX blank A Newton\'s balls javascript simulation (repulsive \'soft-spheres\' connected by springs). Click to toggle motion on off. Past Colloquia Current Series
(UK) Rutherford Appleton Laboratory
Numerical Analysis Group. Large software base.
Welcome to CLRC's Computational Science and Engineering Department 15:10:30 GMT Thursday 17 November 2005 Search: Enter text and press return Home About us Contact us Support and services Research and development Advanced research computing Atomic and molecular physics Band theory Computational engineering Computational material science Molecular simulations Numerical analysis Software engineering Quantum chemistry Events calendar Newsroom Useful links Site map index Online resources CSE projects CSE people Software library Publications library Chemical databases Networks Training courses Numerical Analysis The Numerical Analysis Group is involved in research into techniques for the solution of large scale problems in science, engineering, operations research and economics. One of the main strengths of the group, and one for which it is internationally recognized, is in the solution of large sparse systems of linear equations, and of large-scale linear and nonlinear optimization problems. Members of the group are involved in many collaborative ventures overseas, including projects with Belgium, France, Norway, Sweden, Taiwan, and the USA. This work is described in our series of technical reports and in various publications. Information is also disseminated through invited and contributed presentations at international meetings and conferences and some of our talks are available from the Web. Furthermore, implementations of the underlying numerical algorithms are generally available in software developed both for HSL (formerly the Harwell Subroutine Library), for GALAHAD and for the public domain. The group is supported in this technology transfer by an agreement with Hyprotech UK Ltd. Some of the recent work in the Group is concerned with exploiting parallelism. Members of the Group are also involved in teaching and supervision of research students, locally, nationally, and internationally. Further details are given in the Groups' biennial progress report . For more information about the Numerical Analysis Group please contact Professor Iain Duff. back to top Quick links Software Support Technical Support Technical Reports Talks, Seminars and Courses Seminar Series Bath-RAL Numerical Analysis Day Biennial progress report External Links People
(Germany) Universitt Erlangen-Nrnberg
DIME Project: Data Local Iterative Methods for the Efficient Solution of Partial Differential Equations. People, publications, software.
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(Germany) Ruprecht-Karls-Universitt Heidelberg
Interdisciplinary Center for Scientific Computing. Objectives: mathematical modelling and computational simulation of complex systems in science and technology; development and use of computer methods and software for applications in industry and economy; visualization, computer graphics, image processing; education in scientific computing.
IWR - University of Heidelberg University IWR Contact Research Groups Sort in alphabetical order by: subjects | names Parallel Computing - Prof. P. Bastian Simulation and Optimization - Prof. H. G. Bock Theoretical Chemistry - Prof. L. S. Cederbaum Inorganic Chemistry, Molecular Modeling - Prof. P. Comba BiophotonicsandInformationProcessing - Prof. C. Cremer Statistics - Prof. R. Dahlhaus Intelligent Bioinformatics Systems - Prof. R. Eils Applied Physical Chemistry - Prof. M. Grunze Multiphase Flows and Combustion - Prof. E. Gutheil Multidimensional Image Processing - Prof. F. Hamprecht Computational Biophysics - Prof. D. W. Heermann Statistical Physics - Prof. H. Horner Applied Analysis - Prof. W. Jger Image Processing - Prof. B. Jhne VisualizationandNumericalGeometry - Dr. S. Krmker Bioinformatics Computational Biophysics - Dr. U. Kummer* Biophysics of Macromolecules - Prof. J. Langowski Computer Architecture Computer Engineering - Prof. V. Lindenstruth Parallel and Distributed Systems - Prof. T. Ludwig Massively Parallel Computer Systems - Prof. R. Mnner* Algorithmic Algebra - Prof. B. H. Matzat Theoretical Physics - Prof. H.-J. Pirner Environmental Physics - Prof. U. Platt Numerical Methods - Prof. R. Rannacher Discrete and Combinatorial Optimization - Prof. G. Reinelt Terrestrial Systems - Prof. K. Roth Phytosphere - Prof. U. Schurr* Soft Matter Theory and Biological Physics - Dr. U. Schwarz Computational Molecular Biophysics - Prof. J. Smith Theoretical Astrophysics - Prof. W. M. Tscharnuter and Prof. R. Wehrse Modeling Simulation of Reactive Flows - Prof. J. Warnatz Technical Simulation Group - Prof. G. Wittum Physical Chemistry - Prof. J. Wolfrum (* Guest Members of the IWR) R e s e a r c h G r o u p s G. Bock Last Update: 05.10.2005 webmaster[at]iwr.uni-heidelberg.de?subject=About page groups index.php
(Germany) University of Heidelberg
IWR - Technical Simulation Group. Concerned with the development of algorithms for the efficient simulation of problems from physics and engineering.
Simulation in Technology Center Lehrstuhl Technische Simulation Simulation in Technology SiT Home People Research Events Publications Lectures Courses Contact Local User Information The Simulation in Technology Center of Prof. Wittum is concerned with the development of algorithms and software for the efficient simulation of problems from physics and engineering. With the simulation system UG we have created a general platform for the numerical solution of partial differential equations in two and three space dimensions on serial and on parallel computers. UG supports distributed unstructured grids, adaptive grid refinement, derefinement coarsening, dynamic load balancing, mapping, load migration, robust parallel multigrid methods, various discretizations, parallel I O, and parallel visualization of 3D grids and fields. Last change: $Date: 2004 07 15 10:03:03 $
(Austria) Johannes Kepler University, Linz
Spezialforschungsbereich F013: Numerical and Symbolic Scientific Computing.
SpezialForschungsBereich SFB F013 Your Browser does not support frames
(Austria) Johannes Kepler University, Linz
Institute of Computational Mathematics (formerly Department of Computational Mathematics of the Institute of Analysis and Computational Mathematics).
Numerische Mathematik - NUMA - University Linz Viewing this page requires a browser capable of displaying frames. Follow this link to an HTML document. numerik.html
(Germany) Universitt Erlangen
System Simulation Group - IMMD X. Projects include: Cache-oriented multigrid project (DiME); Plasma Immersion Ion Implantation Simulation Project (PII); Simulation and Visualisation of large data sets (Adhoc3D).
Informatik 10 - System Simulation Department of ComputerScience10 Department of Computer Science 10 Contact People Research Publications Teaching HPC-Cluster Miscellaneous Intranet Computational Engineering Elite Program CE German Scientific Computing Diese Seite auf deutsch Dept. of Computer Science Computer Science 10 Department of Computer Science 10 System Simulation The Department of Computer Science 10 deals with the modelling, efficient simulation and optimization of complex systems in science and engineering. The main focus is on the design and the analysis of algorithms and tools for these purposes. Head of the department is Prof.Dr.UlrichRde LSS auf der Langen Nacht der Wissenschaften Contact Address Phone , Travel Information , Maps People Research Publications Technical Reports , Dissertations , Degree Dissertations, Master and Short Theses , Papers , Talks Teaching Courses , Bachelor and Master theses, Studien- and Diplomarbeiten , ComputationalEngineering , EliteprogramCE HPC-Cluster Miscellaneous Events Intranet News Weltrekord im Gleichungslsen von cf am 14. November 2005 Anmeldeschluss Elitestudiengang CE am 30.11.2005 von cf am 04. November 2005 The NuSiF Showroom von cf am 03. November 2005 Diese Seite auf deutsch Contact Last modified: 2005-11-04 10:47 cf
(UK) Manchester Centre for Computational Mathematics (MCCM)
Formed by the Numerical Analysis groups at the Department of Mathematics in the University of Manchester and at UMIST, with University College Chester. Activities, publications, resources.
Manchester Center for Computational Mathematics ___MCCM___ MANCHESTER CENTRE FOR COMPUTATIONAL MATHEMATICS About MCCM | People | Technical reports | Seminars | Conferences | Postgraduate Teaching | About MCCM MCCM is formed by the Numerical Analysis groups at the Department of Mathematics in the University of Manchester and at UMIST , with links elsewhere, especially at University College Chester. The activities of MCCM include running an M.Sc. in Applied Numerical Computing , organizing Numerical Analysis conferences and a seminar series , and publishing a series of technical reports . The research interests within MCCM include numerical linear algebra, numerical solution of (ordinary and partial) differential and functional differential e