Regular Polygonal Complexes in Space, II

TL;DR

This paper completes the classification of regular polygonal complexes in 3D space, identifying 25 unique complexes beyond regular polyhedra, including 21 simply flag-transitive and 4 as 2-skeletons of regular 4-apeirotopes.

Contribution

It provides a complete enumeration of all regular polygonal complexes in Euclidean 3-space, extending previous work to include complexes with various mirror vectors.

Findings

Identified 25 non-polyhedral regular polygonal complexes in 3-space.

Enumerated all simply flag-transitive complexes with different mirror vectors.

Connected complexes to 4-apeirotopes and face mirror structures.

Abstract

Regular polygonal complexes in euclidean 3-space are discrete polyhedra-like structures with finite or infinite polygons as faces and with finite graphs as vertex-figures, such that their symmetry groups are transitive on the flags. The present paper and its predecessor describe a complete classification of regular polygonal complexes in 3-space. In Part I we established basic structural results for the symmetry groups, discussed operations on their generators, characterized the complexes with face mirrors as the 2-skeletons of the regular 4-apeirotopes in 3-space, and fully enumerated the simply flag-transitive complexes with mirror vector (1,2). In this paper, we complete the enumeration of all regular polygonal complexes and in particular describe the simply flag-transitive complexes for the remaining mirror vectors. It is found that, up to similarity, there are precisely 25 regular polygonal complexes which are not regular polyhedra, namely 21 simply flag-transitive complexes and 4 complexes which are 2-skeletons of regular 4-apeirotopes.