The geometry of chiral or regular polyhedra in ordinary space

Egon Schulte

Northeastern University

May 7,
refreshments at 3:45pm


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 highly-symmetric polyhedra. Coxeter's "Regular Polytopes" book and his various other writings treat the Platonic solids, the Kepler-Poinsot polyhedra and the Petrie-Coxeter 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 3-space 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 3-space. 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 vertex-figures. Their geometry and combinatorics are rather complicated.

Speaker's Contact Info: schulte(at-sign)

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