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.