The geometry of chiral or regular polyhedra in ordinary spaceEgon SchulteNortheastern University
May 7,

ABSTRACT


Symmetric polyhedra have been investigated since antiquity. With the passage oftime, the concept of a polyhedron has undergone a number of changes which have brought to light new classes of highlysymmetric polyhedra. Coxeter's "Regular Polytopes" book and his various other writings treat the Platonic solids, the KeplerPoinsot polyhedra and the PetrieCoxeter in great detail, and cover what might be called the classical theory. A lot has happened in this area in the past 30 years. In particular, around 1980, the class of regular polyhedra in Euclidean 3space was considerably extended by Grunbaum and Dress, and an alternative approach to the full classification was later described by McMullen and the speaker. This talk presents the complete enumeration of chiral polyhedra in Euclidean 3space. Chiral, or irreflexibly regular, polyhedra are nearly regular polyhedra; their geometric symmetry groups have two orbits on the flags (regularpolyhedra have just one orbit), such that adjacent flags are in distinct orbits.There are several infinite families of chiral polyhedra, each with finite skew, or infinite helical, polygonal faces, and with finite skew vertexfigures. Their geometry and combinatorics are rather complicated. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

