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.
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.
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.
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.
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 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.
Melinda
Green's geometry page. Green makes models of regular sponges
(infinite non-convex generalizations of Platonic solids) out of plastic
"Polydron" pieces.
Hebesphenomegacoronaonna stick in space! Space Station Science picture of
the day. In case you don't remember what a hebesphenomegacorona is, it's
one of the Johnson solids: convex polyhedra with regular-polygon faces.
Hecatohedra.
John Conway discusses the possible symmetry groups of hundred-sided polyhedra.
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).
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.
The
Platonic solids. With Java viewers for interactive manipulation. Peter Alfeld, Utah.
Platonic
solids and Euler's formula. Vishal Lama shows how the formula can be
used to show that the familiar five Platonic solids are the only ones
possible.
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.
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).
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.
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).
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 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.)
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.
Solution
of Conway-Radin-Sadun problem.
Dissections of combinations of regular dodecahedra, regular icosahedra,
and related polyhedra into rhombs that tile space. By Dehn's solution to
Hilbert's third problem this is impossible for individual dodecahedra
and icosahedra, but Conway,
Radin, and Sadun showed that certain combinations could work.
Now Izidor Hafner shows how.
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 and Stella4d,
Windows software for visualizing regular and semi-regular polyhedra and
their stellations in three and four dimensions, morphing them into each other, drawing unfolded nets for
making paper models, and exporting polyhedra to various 3d design packages.
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...
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.
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.
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.
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.
