Chiral extensions of regular toroids

TL;DR

This paper constructs new chiral polytopes with facets based on regular cubic tessellations of tori, demonstrating the existence of infinitely many such structures in dimensions three and higher.

Contribution

It introduces a method to build chiral polytopes with specified regular cubic tessellation facets on tori, proving infinite existence in dimensions three and above.

Findings

Existence of infinitely many chiral d-polytopes for all d ≥ 3.

Construction of chiral polytopes with prescribed cubic tessellation facets.

Extension of polytope theory to include chiral structures on toroidal surfaces.

Abstract

Abstract polytopes are combinatorial objects that generalise geometric objects such as convex polytopes, maps on surfaces and tilings of the space. Chiral polytopes are those abstract polytopes that admit full combinatorial rotational symmetry but do not admit reflections. In this paper we build chiral polytopes whose facets (maximal faces) are isomorphic to a prescribed regular cubic tessellation of the $n$-dimensional torus ($n \geq 2$). As a consequence, we prove that for every $d \geq 3$ there exist infinitely many chiral $d$-polytopes.