The Classification of Regular Solids

In [4], Farran and Robertson extended the classical concept of regularity from convex polytopes to convex bodies in general. A convex body that is regular in this new sense is called a regular solid. Thus the set ^ of all regular polytopes is a subset of the set 5^ of all regular solids. Farran and Robertson constructed a projection P&R~> &R a d observed a close resemblance between this construction and a procedure of Kostant [7]. In the present terminology, this associates either one or two regular solids (that are not polytopes) to some (but not all) regular polytopes. As a consequence, certain conjectures arise on the determination of the set £fR, as shown in [4]. In this paper we prove these conjectures, obtaining a complete classification of regular solids. The key to this classification is Dadok's work [3] on polar representations. For any regular n-solid B in E, the symmetry group G of B may be regarded as a subgroup of the orthogonal group O(n). The inclusion n:G->O(ri) is a representation that turns out to be irreducible and polar in the sense of Dadok. We may therefore use Dadok's classification of irreducible polar representations to associate with each regular solid B a certain symmetric space whose Weyl group enables us to assign a regular polytope P to B. This yields the above projection P^R -*• &R m a f ° r m t n a t explains the analogy with Kostant's work and provides a complete description of SfR. In Section 1 we give a brief review of notation and terminology. Section 2 is devoted to the classificatipn theorem, and in Section 3 we re-examine the more general problem of classifying 'perfect' solids. The definition of regularity given in Section 1 is equivalent to that of [4]. Its more elegant form results from a reformulation of the concept of maximality for flags. We thank A. J. Breda d'Azevedo for this improvement.