Taylor-Socolar Hexagonal Tilings as Model Sets
Taylor-Socolar Hexagonal Tilings as Model Sets
复制标题
Taylor-Socolar 六角形平铺作为模型集
DOI：
10.3390/sym5010001
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
R. Moody
中科院分区：
文献类型：
--
作者：
Jeong;R. Moody
Taylor-Socole瓷砖是平面的规则六边形瓷砖，但其区别在于以非周期的方式由两种颜色的六边形组成。我们将Taylor-Socole平铺放在一个代数环境中，这使得人们可以直接将它们视为模型集，并理解相应的平铺外壳及其通用和奇异部分。虽然平铺最初是通过匹配规则和替换获得的，但我们的方法将平铺设置在切割和项目方案的框架中，并研究平铺如何与相应的内部空间相关。一块瓷砖的整组瓷砖的中心在平面上形成一个格子Q。如果XQ表示所有以Q为中心的Taylor-Socole平铺的集合，则XQ在壳的标准局部拓扑下形成一个自然壳，并且是一个关于Q作用的动力系统.Q的q-进完成Q是XQ的一个自然因子，并且自然映射XQ→Q是双射的，除非在/Q中的稠密测度点上.我们证明了XQ由三个平移的Li类组成.其中有两个Li类是非常小的，即XQ中的可数q轨道。另一个是极小动力系统，它满射到/q，在奇点上有不同的2：1、6：1和12：1。我们进一步发展了由瓷砖中心坐标决定瓷砖的奇偶性的公式。最后，我们证明了奇偶平铺的壳可以用壳XQ来识别；更准确地说，这两个壳是相互局部可导的。
The Taylor–Socolar tilings are regular hexagonal tilings of the plane but are distinguished in being comprised of hexagons of two colors in an aperiodic way. We place the Taylor–Socolar tilings into an algebraic setting, which allows one to see them directly as model sets and to understand the corresponding tiling hull along with its generic and singular parts. Although the tilings were originally obtained by matching rules and by substitution, our approach sets the tilings into the framework of a cut and project scheme and studies how the tilings relate to the corresponding internal space. The centers of the entire set of tiles of one tiling form a lattice Q in the plane. If XQ denotes the set of all Taylor–Socolar tilings with centers on Q, then XQ forms a natural hull under the standard local topology of hulls and is a dynamical system for the action of Q.The Q-adic completion Q of Q is a natural factor of XQ and the natural mapping XQ → Q is bijective except at a dense set of points of measure 0 in /Q. We show that XQ consists of three LI classes under translation. Two of these LI classes are very small, namely countable Q-orbits in XQ. The other is a minimal dynamical system, which maps surjectively to /Q and which is variously 2 : 1, 6 : 1, and 12 : 1 at the singular points. We further develop the formula of what determines the parity of the tiles of a tiling in terms of the coordinates of its tile centers. Finally we show that the hull of the parity tilings can be identified with the hull XQ; more precisely the two hulls are mutually locally derivable.