Polyhedra and Polytopes
This page includes pointers on geometric properties of polygons, polyhedra, and higher dimensional polytopes (particularly convex polytopes).

  • Bob Allanson's Polyhedra Page. Nice animated-GIF line art of the Platonic solids, Archimedean solids, and Archimedean duals.

  • Maurice Starck's Polyhedra Page. Several unusual all space fillers.

  • Almost research-related maths pictures. A. Kepert approximates superellipsoids by polyhedra.

  • Archimedean polyhedra, Miroslav Vicher. 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, , and Jean-Louis Loday's paper on associahedron coordinates.

    Symmetric view of associahedron

  • David Bailey's world of tesselations. Primarily consists of Escher-like drawings but also includes an interesting section about Kepler's work on polyhedra.

  • 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.

  • 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.

  • 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 by Paul Flavin.

  • Flexible polyhedra (Becky Alexander and Joseph O'Rourke) From Dave Rusin's known math pages.

  • Geodesic math. Apparently this means links to pages about polyhedra by Geodesic Designs Inc.

  • Geometria Java-based software for constructing and measuring polyhedra by transforming and slicing predefined starting blocks by Geometria.

  • 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.

  • 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.
  • 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.

  • Hyperspace star polytope slicer, Java animation by Mark Newbold.

    sliced hyperspace star polytope

  • Icosamonohedra, icosahedra made from congruent but not necessarily equilateral triangles. by Becky Alexander and Joseph O'Rourke

  • 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.

  • Sándor 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.

  • 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.

  • Martin's pretty polyhedra. Simulation of particles repelling each other on the sphere produces nice triangulations of its surface.

  • Mathematica Graphics Gallery: Polyhedra

  • Minesweeper on Archimedean polyhedra, Robert Webb.

  • Netlib polyhedra. by Jack Dongarra and Eric Grosse. 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.

    Donoso'O'Rourke polyhedron

  • Parallelopoids by Paul Bourke.. All space fillers
    Donoso'O'Rourke polyhedron

  • Pappus on the Archimedean solids. Translation of an excerpt of a fourth century geometry text.

  • 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 spheresby Dr. Karl- Dietrich Neubert. Java animation, with a discussion of platonic solid classification, Euler's formula, and sphere symmetries.

  • Poly by Pedagoguery Software Inc. 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, polyhedra, polytopes, R. Towle.

  • 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.

  • Polyhedron, polyhedra, polytopes, ... - Numericana.

  • Puzzles with polyhedra and numbers, J. Rezende. Some questions about labeling edges of platonic solids with numbers, and their connections with group theory.

  • 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).

    {3,3,5/2} cross-section

  • Rhombic spirallohedra, concave rhombus-faced polyhedra that tile space, R. Towle.

  • Ruler and Compass  by Literka.   Mathematical web site including special sections on the geometry of polyhedrons and geometry of polytopes.

  • SMAPO library of polytopes encoding the solutions to optimization problems such as the TSP.

  • Sperical polyhedra by Muarice Starck The "campanus" sphere and some geodesic domes.

  • Soap bubble 120-cell from the Geometry Center archives.

    Geometry Ctr. soap bubble 120-cell

  • Stella by Rob Webb. 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.

  • 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. by Melinda Green.Topics include deltahedra, infinite polyhedra, and flexible 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. 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.

    Szilassi polyhedron

  • 3D-Geometrie. T. E. Dorozinski provides a gallery of images of 3d polyhedra, 2d and 3d tilings, and subdivisions of curved surfaces.

  • 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.

    Ray-traced icosahedron

  • 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 Polytopeby Peter Serocka. Geometry arising from the simultaneous comparison of multiple DNA or protein sequences.

  • 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 by Steve Waterman. convex hulls as determined by employing incremental, root of an integer, radial sweeps upon the ccp.

  • Why "snub cube"? John Conway provides a lesson on polyhedron nomenclature and etymology. From the geometry.research archives.