Regular and perfect solids

Abstract

An n-solid is a compact convex subset B of E^m whose affine hull is n-dimensional, for some m ≥ n ≥ -1. The boundary of B is composed of faces which are solids of a lower dimension. A flag of B is a sequence (A₀,...,A_r) of distinct proper faces of B such that A_j-1 is contained in A_j, j = 1,...,r. A flag is said to be maximal if it is not contained in any other flat of B. If the symmetry group of B is transitive on the set of maximal flags of B, then we say that B is regular.

Two solids B and C are symmetry equivalent if the actions of their symmetry groups GB and GC on their face-lattices FB and FC, respectively, are equivalent. A solid B is said to be perfect if B is similar to C whenever B is symmetry equivalent to C.

The aim of this thesis is two-fold. First, the regular solids are classified. This classification is based on the projection of the adjoint action of a compact semisimple Lie group G on its Lie algebra g to the Weyl group action of g. Secondly, a contribution to the solution of the more general problem of classifying perfect n-solids is given. The cases n ≤ 3 are already completely understood. The case n = 4 is solved, thus proving Rostami's conjecture that all prime perfect 4-polytopes are Wythoffian up to polarity. (DX183769)

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