Regular Polytopes of Nearly Full Rank

Abstract

An abstract regular polytope $\mathcal{P}$ of rank n can only be realized faithfully in Euclidean space $\mathbb{E}^{d}$ of dimension d if d≥n when $\mathcal{P}$ is finite, or d≥n−1 when $\mathcal{P}$ is infinite (that is, $\mathcal{P}$ is an apeirotope). In case of equality, the realization P of $\mathcal{P}$ is said to be of full rank. If there is a faithful realization P of $\mathcal{P}$ of dimension d=n+1 or d=n (as $\mathcal {P}$ is finite or not), then P is said to be of nearly full rank. In previous papers, all the at most four-dimensional regular polytopes and apeirotopes of nearly full rank have been classified. This paper classifies the regular polytopes and apeirotopes of nearly full rank in all higher dimensions.