Quotient Representations of Uniform Tilings

TL;DR

This paper investigates the structure of uniform tilings of the Euclidean plane by regular polygons, providing a way to represent them as quotients of regular infinite polyhedra through analysis of flag stabilizers.

Contribution

It introduces a method to describe vertex-transitive tessellations as quotients of regular polyhedra by explicitly determining flag stabilizers.

Findings

Flag stabilizers are explicitly determined for each uniform tiling.

Uniform tilings are represented as quotients of regular infinite polyhedra.

Provides presentations of these tilings as quotients, facilitating their algebraic understanding.

Abstract

Given a flag in each of the vertex-transitive tessellations of the Euclidean plane by regular polygons, we determine the flag stabilizer under the action of the automorphism group of a regular cover. In so doing we give a presentation of these tilings as quotients of regular (infinite) polyhedra.