Geometric vs. Combinatorial Symmetry of Polyhedra
Egon Schulte

Abstract: We outline the complete enumeration of the regular and chiral polyhedra in Euclidean 3-space. There are 48 regular polyhedra (also known as the Grunbaum-Dress polyhedra), as well as several infinite families of chiral polyhedra. All polyhedra, but 18 regular ones, are infinite, expanding over all of space. A typical such polyhedron has convex or non-convex, planar or skew, and finite or infinite (helical or zig-zag) polygonal faces and vertex-figures.