几何 Penrose 平铺由其 1-atlas 刻画
Geometrical Penrose Tilings are characterized by their 1-atlas
AI总结：
本文研究几何 Penrose 平铺，通过替换结构和切割投影两种方法计算图案，证明其由 vertex-atlas 刻画，构成有限型平铺空间，并首次提供完整证明。
摘要：
菱形 Penrose 平铺是用两种装饰菱形对平面的平铺，使得装饰在两块砖的交界处匹配（就像拼图一样）。用动力学术语来说，它们构成了一个有限型的平铺空间。如果我们去掉装饰，根据定义，我们得到了一个 sofic 平铺空间，我们在此称之为几何 Penrose 平铺。在这里，我们展示了如何通过两种不同的方法来计算出现在这些平铺中的给定大小的图案：一种基于 Penrose 平铺的替换结构，另一种基于它们的切割与投影方法定义。我们利用这一点证明了几何 Penrose 平铺是由一小组称为 vertex-atlas 的图案刻画的，即它们构成了一个有限型的平铺空间。据我们所知，尽管这被认为是已知常识，但此前尚未发表过该结果的完整证明。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
