Madden, Timo Michael
(1994)
Regular and perfect solids.
*University of Southampton, Doctoral Thesis*.

## 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 (*A*_{0},...,*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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