Counting perfect colourings of plane regular tilings

TL;DR

This paper investigates the enumeration of perfect colourings in plane regular tilings, focusing on symmetries and employing group theory to address combinatorial questions in Euclidean and hyperbolic geometries.

Contribution

It introduces a group theoretical approach to count perfect colourings of regular tilings, distinguishing between all symmetries and orientation-preserving symmetries.

Findings

Provides formulas for counting perfect colourings

Distinguishes colourings with different symmetry considerations

Applies methods to Euclidean and hyperbolic tilings

Abstract

A first step in investigating colour symmetries of periodic and nonperiodic patterns is determining the number of colours which allow perfect colourings of the pattern under consideration. A perfect colouring is one where each symmetry of the uncoloured pattern induces a global permutation of the colours. Two cases are distinguished: Either perfect colourings with respect to all symmetries, or with respect to orientation preserving symmetries only (no reflections). For the important class of colourings of regular tilings (and some Laves tilings) of the Euclidean or hyperbolic plane, this mainly combinatorial question is addressed here using group theoretical methods.