Regular Polytopes of Nearly Full Rank
An abstract regular polytope P of rank n can only be realized faithfully in Euclidean space E(d) of dimension d if d >= n when P is finite, or d >= n - 1 when P is infinite (that is, P is an apeirotope). In case of equality, the realization P of P is said to be of full rank. If there is a faithful realization P of P of dimension d = n + 1 or d = n ( as 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.
nearly full rank abstract regular polytope realization faithful fine schlafli symbol