论文标题
几何penrose瓷砖的特征是其1-atlas
Geometrical Penrose Tilings are characterized by their 1-atlas
论文作者
论文摘要
菱形彭罗斯（Rhombus penrose）的瓷砖是两个装饰的菱形的平面砖，使得装饰匹配两个瓷砖之间的连接（例如在拼图拼图中）。用动态术语，它们形成有限类型的平铺空间。如果我们删除装饰品，则根据定义，我们将在这里称为几何penrose瓷砖。在这里，我们展示了如何通过两种不同的方法计算给定尺寸的模式：一种基于penrose瓷砖的替代结构，另一个基于剪切和投影方法的定义。我们用它来证明几何penrose瓷砖的特征是一组称为顶点 - atlas的图案，即它们形成有限类型的平铺空间。据我们所知，虽然被认为是民间的，但没有完全证明这种结果。
Rhombus Penrose tilings are tilings of the plane by two decorated rhombi such that the decoration match at the junction between two tiles (like in a jigsaw puzzle). In dynamical terms, they form a tiling space of finite type. If we remove the decorations, we get, by definition, a sofic tiling space that we here call geometrical Penrose tilings. Here, we show how to compute the patterns of a given size which appear in these tilings by two different method: one based on the substitutive structure of the Penrose tilings and the other on their definition by the cut and projection method. We use this to prove that the geometrical Penrose tilings are characterized by a small set of patterns called vertex-atlas, i.e., they form a tiling space of finite type. Though considered as folk, no complete proof of this result has been published, to our knowledge.
