Accueil
Plan du site
Bloc notes
Jeux
Graphes
Combinatoire
Arithmétique
Algèbre
Analyse
Probabilités
Automates
Programmation
Divers
Liens
Home

UniversitySurf.net
Votre portail e-Learning

CultureMATH
ENSup. et Minist. EN

OE
I S

Encyclopédie des suites d'entiers

Séminaire MaMuX
Mathématiques, musique et relations avec d'autres disciplines

IHÉS

Institut des Hautes Études Scientifiques

L'INVERSEUR
de Plouffe

Polytopes

PAGES WEB

une balade dans le monde des polyÃ¨dres

Entre autres, les patrons des parallÃ©lÃ©pipÃ¨des rectangles.

<http://www.ac-noumea.nc/maths/amc/polyhedr/>

Beating Kelvin's Partition of Space

In 1887, Lord Kelvin asked how to partition space into cells of volume 1 such that the total area of the interfaces between the cells is a minimum. The best partition Kelvin could come up with was made of slightly curved 14-sided polyhedra. Two of Kelvin's tetrakaidecahedra are pictured here:

<http://www.susqu.edu/facstaff/b/brakke/kelvin/kelvin.html>

PolyÃ¨dres

Xavier Hubaut UniversitÃ© Libre de Bruxelles

<http://xavier.hubaut.info/coursmath/pol/polyd.htm>

Soap Film Cones

If one magnifies a point of a soap film, one gets a tangent cone. Recall that a cone is composed of rays emanating from a point.

<http://www.susqu.edu/facstaff/b/brakke/cones/cones.htm>

Virtual Polyhedra

By George W. Hart

<http://www.georgehart.com/virtual-polyhedra/vp.html>

Peek N-dimensional polytope visualization by cross-section and projection

<http://www.cs.utah.edu/~gk/peek/peek.html>

Properties of Dice

by Klaus Ã. Mogensen

<http://hjem.get2net.dk/Klaudius/Dice.htm>

The 34 Convex Heptahedra and their Characteristic Polynomials

I've noticed that the characteristic polynomials of symmetric figures are more factorable than asymmetric figures. (adjacency matrix )

<http://www.mathpuzzle.com/charpoly.htm>

The Szilassi and CzÃ¡szÃ¡r Polyhedra

<http://web.ukonline.co.uk/polyhedra/toroids/toroids1.html>

Polyhedra - UKonline

<http://web.ukonline.co.uk/polyhedra/>

Les cinq polyÃ¨dres de Platon

<http://www.saliege.com/dynamique/projet/ballade/platon1.htm>

Beating Kelvin's Partition of Space

Ken Brakke

In 1887, Lord Kelvin asked how to partition space into cells of volume 1 such that the total area of the interfaces between the cells is a minimum. For over a century, nobody could improve on Kelvin's partition. Then in 1993, Denis Weaire and Robert Phelan came up with a partition of space into two kinds of cells (of equal volume, of course) that beat Kelvin's partition by 0.3% in area.

<http://www.susqu.edu/facstaff/b/brakke/kelvin/kelvin.html>

PolyÃ¨dres

Jorge Rezende - Mathematical Physics Group (GFMUL) University or Lisbon

On the Puzzles with polyhedra and numbers, Polyhedron puzzles and groups, Photos of polyhedron puzzles

(Patrons de polytopes)

<http://gfm.cii.fc.ul.pt/Members/JR.en.html>

TectoÃ¨dre

GÃ©rard Villemin

<http://membres.lycos.fr/villemingerard/Geometri/Tectoedr.htm#tecto>

Mathematical models of surfaces

At the end of the 19th century, the interest in geometry gave place to the construction of geometrical models of surfaces. These models were mostly designed by mathematicians and made of different materials such as plaster, wood or string...

Drs Irene Polo Blanco, together with Lotte van der Zalm, have made an inventory of this collection and have put the information on this website.

<http://www.math.rug.nl/models/>

PAGES PERSONNELLES - HOME PAGES

Flexagons - MathWorld

<http://mathworld.wolfram.com/topics/Flexagons.html>

Bob Connelly

The Bellows Conjecture - Consider a polyhedral surface in three-space that has the property that it can change its shape while keeping all its polygonal faces congruent. Adjacent faces are allowed to rotate along common edges. Mathematically exact flexible surfaces were found by Connelly in 1978. But the question remained as to whether the volume bounded by such surfaces was necessarily constant during the flex.

<http://math.cornell.edu/~connelly/>

Komei Fukuda

<http://www.ifor.math.ethz.ch/~fukuda/fukuda.html>

David Avis

<http://www.CS.McGill.CA/~avis/>

DEMOS

Factoris

<http://wims.unice.fr/~wims/en_tool~algebra~factor.en.phtml>

Platonic Solids

<http://mathworld.wolfram.com/topics/PlatonicSolids.html>

EXEMPLES - EXAMPLES

The Csazar polyhedron

is the solution to a very interesting problem. The question is how many polyhedra exist such that every pair of vertices is joined by an edge. The first clear example is the well known tetrahedron (triangular pyramid). Some simple combinatorics specify how many vertices, edges, faces, and holes such polyhedra must have.

<http://www-personal.umich.edu/~clahey/Csaszar/>

the Szilassi polyhedron

In 1977, Hungarian mathematician Lajos Szilassi found a way to construct a toroidal heptahedron. Each face of his polyhedron is a hexagon (although none of them is a regular hexagon).

<http://www.qnet.com/~crux/szilassi.html>

Szilassi Polyhedron

<http://mathworld.wolfram.com/SzilassiPolyhedron.html>

Steffen's Simplest Known Flexible Polyhedron

Klaus Steffen took the construction of flexible polyhedron one step further by constructing a symmetric flexible Connelly sphere with only 9 verticies.

Bricard's flexible octahedra The flexible octahedron in figure 3 cannot be made with surfaces

<http://www.geocities.com/jshum_1999/polyhedra/steffen.htm>

flexor

Robert Connelly was looking for an answer to the following question: is convexity necessary for rigidity?

diagrams for the construction of two flexor models (left model : Robert Connelly, right : Klaus Steffen)

<http://www.icm.edu.pl/~delta/delta7/wielosci/connelly.htm>

IMAGES

Visualization of the Carrillo-Lipman Polytope

Geometry arising from the simultaneous comparison of multiple DNA or protein sequences.

<http://merlin.bcm.tmc.edu:8001/bcdusa/Curric/MulAli/align3d.html>

Some almost research-related maths pictures

<http://frey.newcastle.edu.au/~andrew/rsch_pix.html>

polytope

<http://www.math.toronto.edu/gif/polytope.gif>

LOGICIELS - SOFTWARES

Stella (Great and Small)

The site is here to tell you about my programs Great Stella and Small Stella, but I hope it will be of interest to anyone with an interest in polyhedra. There are lots of photos of polyhedra I've made using these programs, along with tips for building them.

< http://www.software3d.com/Stella.html>

Polyhedra examples

PolyDraw, PolyNet , Stellate Software from `Fortran Friends'

<http://www.argonet.co.uk/users/fortran/Poly/index.htm>

The Surface Evolver

is an interactive program for the modelling of liquid surfaces shaped by various forces and constraints. The program is available free of charge.

<http://www.susqu.edu/facstaff/b/brakke/evolver/evolver.html>

OUTILS - TOOLS

Polymake

polymake-1.4.tar.gz

<http://www.math.tu-berlin.de/diskregeom/polymake/doc/>

cdd and cddplus

<http://www.ifor.math.ethz.ch/~fukuda/cdd_home/cdd.html>

lrslib

is a self-contained ANSI C implementation of the reverse search algorithm for vertex enumeration/convex hull problems.

<http://cgm.cs.mcgill.ca/~avis/C/lrs.html>

~avis/C directory

<http://cgm.cs.mcgill.ca/~avis/C/>

Porta

is a collection of routines for analyzing polytopes and polyhedra. The polyhedra are either given as the convex hull of a set of points plus (possibly) the convex cone of a set of vectors, or as a system of linear equations and inequalities.

<http://www.iwr.uni-heidelberg.de/iwr/comopt/software/PORTA/>

Polylib - A library of polyhedral functions

I R I S A

Polylib is a free library written in C for the manipulation of polyhedra. The library is operating on objects like vectors, matrices, lattices, polyhedra, Z-polyhedra, unions of polyhedra and a lot of other intermediary structures. It provides functions for all the important operations on these structures.

documents (Chernikova's algorithm, Ehrhart's polynomials)

<http://www.irisa.fr/cosi/polylib/user/>

JAVA

Les cinq polyÃ¨dres de Platon

<http://www.saliege.com/dynamique/projet/ballade/platon1.htm>

Cones

<http://www.susqu.edu/facstaff/b/brakke/wulff/circle.html>

HyperGami and JavaGami

are software environments for designing and building paper sculpture using polyhedra and custom variants of polyhedra.

<http://www.cs.colorado.edu/~ctg/projects/hypergami/>

EDUCATION

http://www.geocities.com/RainForest/Vines/2977/gauss/formulae/gon17.html

<http://www.geocities.com/RainForest/Vines/2977/gauss/formulae/gon17.html>

LIVRES - BOOKS

The fifty-nine icosahedra

Reference: The fifty-nine icosahedra, by H. S. M. Coxeter, P. Du Val, H. T. Flather, and J. F. Petrie.

<http://www.scg.uwaterloo.ca/~hqle/stellate/ico/>

SUJETS - SUBJECTS

Les polyÃ¨dres flexibles et la conjecture du soufflet (PDF)

<http://www.apmep.asso.fr/IMG/pdf/Asm18_Thierry_Lambre.pdf>

DOCUMENTS - PAPERS

Matrices hermitiennes et convexitÃ© (M. Audin). Polytopes convexes entiers (M. Brion). Construction de courbes rÃ©elles (J.-J. Risler). Volumes mixtes des corps convexes (B. Teissier)

<http://math.polytechnique.fr/xups/vol95.html>

COURS - COURSES

J.-M. Sarlat, Acad. Poitiers

<http://www.ac-poitiers.fr/math/prof/resso/ima/sar1/index.htm#sommaire>

Advanced Topics in Discrete Mathematics

E.M. Feichtner, K. Fukuda and P. Parrilo

Polytopes - Program

<http://www.math.ethz.ch/~feichtne/ATDM/semi04winter/semi04winter.html>

JEUX - GAMES

Rubik-style Polyhedral Puzzle

This page describes a Rubik-style puzzle I made out of cardboard. To be honest, it really doesn't work very smoothly!

<http://home.connexus.net.au/~robandfi/Puzzle.html>

QUESTIONS - FAQ

Frequently Asked Questions in Polyhedral Computation

Komei Fukuda

<http://www.ifor.math.ethz.ch/~fukuda/polyfaq/polyfaq.html>

GROUPES DE DISCUSSION - NEWS

Szilassi polyhedron toroidal polyhedron

In 1977, the Hungarian mathematician Lajos Szilassi found a way to construct a seven sided toroidal polyhedron.

<http://www.ics.uci.edu/~eppstein/junkyard/szilassi.html>

Archimedean Solids John Conway

some quite interesting maps among the other Archimedean polytopes

<http://www.ics.uci.edu/~eppstein/junkyard/archimedean.html>

FIRMES - FIRMS

Stardust Polyhedron Kits

to cut out and build yourself

<http://www.seedtech.co.uk/stardust/>

Welcome to Polydron's Web Site

<http://www.polydron.com/>

LIENS - LINKS

<liens_math.html>

Polyhedra and Polytopes

From the Geometry Junkyard, computational and recreational geometry pointers.

<http://www.ics.uci.edu/~eppstein/junkyard/polytope.html>

Polyhedral Links

<http://web.ukonline.co.uk/polyhedra/links.html>

Advertising : If you see a reference in one of the files that is not linked, and you know of a link address to the appropriate document, please send me mail, and I will include the link in the document. Thanks very much in advance.
Avertissement : Le classement par catégories est approximatif. Certains liens se retrouvent dans des rubriques différentes et sur plusieurs pages. Les commentaires sont généralement des courts extraits des pages référencées. Il est possible que certains liens nécessitent une mise à jour.
Tous commentaires ou remarques sont les bienvenus, vous pouvez les adresser à :
écrire

Les mises à jour demandées sont réalisées dès que possible et, sauf si c'est nécessaire, aucun message de réponse n'est expédié. Merci de m'écrire.

Copyright © 1999-2012 Jean-Paul Davalan - Reproduction interdite.