13 - Regular Polytopes Related to Linear Groups

Summary

Since the automorphism group of each finite regular polytope can be thought of as a permutation group, it can also be regarded as a linear group. However, there are families of regular polytopes, of ranks 3 and 4, whose groups are intimately related to certain linear groups, in that they are, for example, central quotients of special linear groups or the like. In this chapter, we shall discuss these families. While it might be natural to include some chiral polytopes in the discussion, since they arise by the same or similar constructions, in fact we shall not do so.

The chapter is arranged as follows. In Section 13A, we shall describe families of regular polyhedra related to projective linear groups. In Section 13B, we consider various connexions among these polyhedra; in particular, in certain cases two of them mix to form another. We discuss realizations of some of the polyhedra in Section 13C; in principle, we could describe the realization spaces of all the polyhedra, but in practice the general case is rather complicated, and so we confine our attention to primes. In Section 13D, we investigate analogous families of 4-polytopes, whose vertex-figures fall among those of the previous polyhedra. Finally, in Section 13E, we describe various connexions among these 4-polytopes.