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 $(2+n)\times 4$ 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.

中文翻译：

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用规则多边形对平面进行定期平铺的整数表示

我们用规则多边形描述平面的定期平铺的表示。我们的方法是明确表示种子顶点的一小部分，从中我们通过翻译系统地生成平铺的所有元素。我们具体地用$（2+ñ）\times 4$ 包含两个平移向量和的晶格坐标的整数矩阵 $ñ$ 种子顶点。我们讨论了此表示的几个属性，并描述了如何通过对称检测优雅，有效地利用该表示进行重构，渲染和自动晶体学分类。