Constructions of Chiral Polytopes of Small Rank

TL;DR

This paper introduces a general method for constructing new finite chiral polytopes from existing ones, significantly expanding the known examples in ranks 3, 4, and 5.

Contribution

It presents a novel technique to derive new finite chiral polytopes from existing ones of the same rank, broadening the scope of known structures.

Findings

Constructed many new examples of chiral polytopes in ranks 3, 4, and 5.

Developed a general method applicable to existing chiral polytopes.

Enhanced understanding of the structure and diversity of chiral polytopes.

Abstract

An abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. The present paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks 3, 4 and 5.