Chiral polytopes of full rank exist only in ranks 4 and 5

Abstract

Skeletal chiral polytopes of rank n in a Euclidean space must have as ambient space $${\mathbb {E}}^d$$ for some $$d \ge n$$ if they are finite, or some $$d \ge n-1$$ if they are infinite. If the dimension attains the lower bound just mentioned, we say that the polytope is of full rank. In this article it is proven that a chiral polytope of full rank can only have rank 4 or 5.