Regular Apeirotopes of Dimension and Rank 4

Abstract

In previous papers, all the four-dimensional (finite) regular polytopes have been classified, as well as the regular apeirotopes of full rank (that is, of rank 5). Of the two problems in $\mathbb{E}^{4}$thus left open (namely, regular apeirotopes of ranks 3 and 4), this paper describes the regular apeirotopes of rank 4. The methods employed here are somewhat different from those in earlier work; while knowledge of the possible dimension vectors (dim R 0,…,dim R 3) of the mirrors R 0,…,R 3 of the generating reflexions of the symmetry groups plays a role, the crystallographic restriction leads to a considerable emphasis being placed on the vertex-figures.