Color groups of colorings of \(N\)-planar modules

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...

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Bibliographic Details
Published in:arXiv.org 2016-11
Main Authors: Loquias, Manuel Joseph C, Valdez, Lilibeth D, Ma Lailani B Walo
Format: Article
Language:English
Subjects:
Apexes
Color
Coloring
Group theory
Lattices
Modules
Permutations
Tiling
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1609.03122