Polyhedra and Polytopes

• Adventures among the toroids. Reference to a book on polyhedral tori by B. M. Stewart.

• Alice visits the fourth dimension. Stereoscopically animated cross sections of a hypercube, with German text.

• 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 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.

• 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.

• Les cinq polyédres de Platon

• 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 T-shirts from the makers of Mathematica.

• 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. See also Fearnley's gallery of basic polyhedra.

• Five Platonic solids and a soccerball.

• Flexible polyhedra. From Dave Rusin's known math pages.

• 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.

• Geometry, algebra, and the analysis of polygons. Notes by M. Brundage on a talk by B. Grünbaum 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.

• Glowing green rhombic triacontahedra in space. Rendered by Rob Wieringa for the May-June 1997 Internet Ray Tracing Competition.

• Daniel 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.

• HypArr, Unix software for modeling and visualizing convex polyhedra and plane arrangements.

• Hypergami polyhedral playground. Rotatable wireframe models of platonic solids and of the penguinhedron.

• Hyperspace. Kyoto University, Group for Hyperspace, English version. Graphic images of regular polytopes. See also their page of 4-polytope images (not linked to from their main page).

• The icosahedron, the great icosahedron, graph designs, and Hadamard matrices. Notes by M. Brundage from a talk by M. Rosenfeld.

• Johnson Solids, convex polyhedra with regular faces. From Eric Weisstein's treasure trove of mathematics.

• Kepler-Poinsot Solids, concave polyhedra with star-shaped faces. From Eric Weisstein's treasure trove of mathematics.

• Kepler's plague (vertex figures of regular polytopes and regular tilings).

• Links2go: 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 3.0 Graphics Gallery: Polyhedra

• Models of Platonic solids and related symmetric polyhedra.

• Netlib polyhedra. Coordinates for regular and Archimedean polyhedra, prisms, anti-prisms, and more.

• Occult correspondences of the Platonic solids. Some random thoughts from Anders Sandberg.

• Origami and three dimensional models. A geometry class project to construct polyhedra out of colored paper and straws.

• 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 software in Java.

• The Platonic solids. With Java viewers for interactive manipulation. Peter Alfeld, Utah.

• 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.

• Polygon symbology.

• 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.

• 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, Imperial College.

• The Polyhedra Page, Bruce Ross.

• A polyhedral analysis. Ken Gourlay looks at the Platonic solids and their stellations.

• Polyhedron challenge: cuboctahedron.

• 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.

• The Puzzling World of Polyhedral Dissections. Stewart T. Coffin's classic book on geometric puzzles, now available in full text on the internet!

• 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(n^{log d}).

• Realization Spaces of 4-polytopes are Universal, G. Ziegler and J. Richter-Gebert, Bull. AMS 32 (1995).

• 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.)

• Right Pentagonal Dodecahedron. Tessellating 3-space in hyperbolic geometry. Robert Grzeszczuk, U. Chicago.
  

• Rolling polyhedra. Dave Boll investigates Hamiltonian paths on (duals of) regular polyhedra.

• Polyhedron web scavenger hunt

• 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.

• Smoothing regular and stellated polyhedra by spline surfaces. J. Chen, Purdue.

• 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.

• Symmetries of torus-shaped polyhedra

• 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. Here's another picture with a Hungarian caption and some literature references. See also Dave Rusin's page on polyhedral tori with few vertices.
  

• 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

• Tom's Branch of Polytopia. An introduction to multi-dimensional regular solids.

• 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.

• 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 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.

• 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.

• 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.

• VRML models: Platonic solids. Four out of the five, anyway. And some Archimedean ones too. From the VRML mall.

• 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. Last update: 02 Nov 1999, 11:33:39 PST.