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# Polytopes

## PAGES WEB

<http://www.ac-noumea.nc/maths/amc/polyhedr/>
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> <http://xavier.hubaut.info/coursmath/pol/polyd.htm>
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>
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>
by Klaus Ã. Mogensen
<http://hjem.get2net.dk/Klaudius/Dice.htm>
I've noticed that the characteristic polynomials of symmetric figures are more factorable than asymmetric figures. (adjacency matrix )
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>
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> <http://membres.lycos.fr/villemingerard/Geometri/Tectoedr.htm#tecto>
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/>

<http://mathworld.wolfram.com/topics/Flexagons.html>
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/> <http://www.ifor.math.ethz.ch/~fukuda/fukuda.html> <http://www.CS.McGill.CA/~avis/>

## EXEMPLES - EXAMPLES

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

Geometry arising from the simultaneous comparison of multiple DNA or protein sequences.
<http://merlin.bcm.tmc.edu:8001/bcdusa/Curric/MulAli/align3d.html> <http://frey.newcastle.edu.au/~andrew/rsch_pix.html> <http://www.math.toronto.edu/gif/polytope.gif>

## LOGICIELS - SOFTWARES

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>
PolyDraw, PolyNet , Stellate Software from `Fortran Friends'
<http://www.argonet.co.uk/users/fortran/Poly/index.htm>
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

<http://www.math.tu-berlin.de/diskregeom/polymake/doc/> <http://www.ifor.math.ethz.ch/~fukuda/cdd_home/cdd.html>
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> <http://cgm.cs.mcgill.ca/~avis/C/>
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

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

## EDUCATION

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

## LIVRES - BOOKS

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

## SUJETS - SUBJECTS

Les polyÃ¨dres flexibles  et la conjecture du soufflet (PDF)
Thierry Lambre Univ. Bl. Pascal Les CÃ©zeaux Clermt-F.
<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

<http://www.ac-poitiers.fr/math/prof/resso/ima/sar1/index.htm#sommaire>
E.M. Feichtner, K. Fukuda and P. Parrilo
<http://www.math.ethz.ch/~feichtne/ATDM/semi04winter/semi04winter.html>

## JEUX - GAMES

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

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

## GROUPES DE DISCUSSION - NEWS

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>
some quite interesting maps among the other Archimedean polytopes
<http://www.ics.uci.edu/~eppstein/junkyard/archimedean.html>

## FIRMES - FIRMS

to cut out and build yourself
<http://www.seedtech.co.uk/stardust/> <http://www.polydron.com/>

<liens_math.html>
From the Geometry Junkyard, computational and recreational geometry pointers.

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