An integer representation for periodic tilings of the plane by regular polygons

We describe a representation for periodic tilings of the plane by regular polygons. Our approach is to represent explicitly a small subset of seed vertices from which we systematically generate all elements of the tiling by translations. We represent a tiling concretely by a (2 + n) x4 integer matrix containing lattice coordinates for two translation vectors and n seed vertices. We discuss several properties of this representation and describe how to exploit the representation elegantly and efficiently for reconstruction, rendering, and automatic crystallographic classification by symmetry detection. (C) 2021 Elsevier Ltd. All rights reserved.

Automated summary

The paper introduces a representation method for periodic tilings of the plane using regular polygons, where all elements of the tiling are systematically generated from a small subset of seed vertices by translations. The concrete representation of a tiling is done using a (2 + n) x4 integer matrix containing lattice coordinates for two translation vectors and n seed vertices. Various properties of this representation are discussed, along with how to efficiently exploit it for reconstruction, rendering, and automatic crystallographic classification by symmetry detection.