Regular Polygonal Complexes in Space, I

TL;DR

This paper classifies regular polygonal complexes in three-dimensional space, analyzing their symmetry groups and geometric properties, and identifies complexes with face mirrors and specific mirror vectors, laying groundwork for complete enumeration.

Contribution

It provides the foundational structure results, symmetry group analysis, and classification of certain regular polygonal complexes in 3-space, advancing the understanding of their geometric and algebraic properties.

Findings

Established basic structure results for symmetry groups.

Characterized complexes with face mirrors as 2-skeletons of regular 4-apeirotopes.

Enumerated simply flag-transitive complexes with mirror vector (1,2).

Abstract

A polygonal complex in euclidean 3-space is a discrete polyhedron-like structure with finite or infinite polygons as faces and finite graphs as vertex-figures, such that a fixed number r of faces surround each edge. It is said to be regular if its symmetry group is transitive on the flags. The present paper and its successor describe a complete classification of regular polygonal complexes in 3-space. In particular, the present paper establishes basic structure results for the symmetry groups, discusses geometric and algebraic aspects of operations on their generators, characterizes the complexes with face mirrors as the 2-skeletons of the regular 4-apeirotopes in 3-space, and fully enumerates the simply flag-transitive complexes with mirror vector (1,2). The second paper will complete the enumeration.