Regular Polytopes of Nearly Full Rank

An abstract regular polytope Pmathcal{P} of rank n can only be realized faithfully in Euclidean space mathbbE^dmathbb{E}^{d} of dimension d if d≥n when Pmathcal{P} is finite, or d≥n−1 when Pmathcal{P} is infinite (that is, Pmathcal{P} is an apeirotope). In case of equality, the realization P of Pmathcal{P} is said to be of full rank. If there is a faithful realization P of Pmathcal{P} of dimension d=n+1 or d=n (as Pmathcal {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.