Color groups of colorings of $N$-planar modules

TL;DR

This paper introduces a method to determine the symmetry groups of colorings in $N$-planar modules, with specific solutions for $N=4$ and $N=6$, and demonstrates applications to lattices and tilings.

Contribution

It provides a novel method for computing color groups of $N$-planar modules, extending to general cases with practical examples.

Findings

Method successfully determines color groups for $N=4$ and $N=6$

Applications shown for rectangular lattices and Ammann-Beenker tiling

Extends to broader classes of $N$-planar modules

Abstract

A submodule of a $\mathbb{Z}$-module determines a coloring of the module where each coset of the submodule is associated to a unique color. Given a submodule coloring of a $\mathbb{Z}$-module, the group formed by the symmetries of the module that induces a permutation of colors is referred to as the color group of the coloring. In this contribution, a method to solve for the color groups of colorings of $N$-planar modules where $N=4$ and $N=6$ are given. Examples of colorings of rectangular lattices and of the vertices of the Ammann-Beenker tiling are given to exhibit how these methods may be extended to the general case.