All mathematicians are familiar with the Platonic solids: the tetrahedron, thecube, the octahedron. the dodecahedron, and the icosahedron. These are thefive convex solids all of whose faces are identical regular polygons.
Considerably less well known are the solids obtained when the above conditionsare relaxed.
If the faces are required to be regular polygons, but not all identical(while all vertices are identical, in the sense of having the same incidentpolygons), we obtain the thirteen Archimedean polyhedra. For example, here isthe Truncated Icosahedron 5, 6, 6 (with a pentagon and two hexagonsmeeting at each vertex):
On the other hand, if the faces are required to be identical regular polygons,but the solid is not required to be convex, we obtain the four Kepler-Poinsotpolyhedra. For example, here is the Great Dodecahedron:
These images are taken from TomGettys' Polyhedra Hyperpages. If you can view Virtual Reality files(VRML), you may wish to check out George W. Hart'sVirtual RealityPolyhedra page, and, in any case, you may wish to look at hisPavilion ofPolyhedreality page. Photos of models of all sorts of polyhedra can befound at Fr. MagnusWenninger--Polyhedron Constructions.
-- Steven Weintraub