Hexagonal Inflation Tilings and Planar Monotiles
Hexagonal Inflation Tilings and Planar Monotiles
复制标题
六边形膨胀瓷砖和平面单瓷砖
DOI：
10.3390/sym4040581
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
U. Grimm
中科院分区：
文献类型：
--
作者：
M. Baake;F. Gähler;U. Grimm
无论是在理论上还是在晶体学中的应用，具有少量原型的非周期性平铺都特别令人感兴趣。在这个方向上，许多人尝试构建非周期平铺，这些平铺是从具有最近邻匹配规则的单个原型构建的，然后称为单平铺。对平面单块瓷砖的探索之一集中在王氏瓷砖的六边形类似物上。这导致了两次具有有趣结构细节的通货膨胀平铺。两者都具有非周期性局部规则，用模型集结构定义船体。我们对它们进行比较，并阐明它们与经典的半六角形平铺的关系。特别是，我们以更具比较性的方式制定各种已知结果，并用一些关于底层平铺空间的几何和拓扑的新结果来增强它们。
Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which is then called a monotile. One strand of the search for a planar monotile has focused on hexagonal analogues of Wang tiles. This led to two inflation tilings with interesting structural details. Both possess aperiodic local rules that define hulls with a model set structure. We review them in comparison, and clarify their relation with the classic half-hex tiling. In particular, we formulate various known results in a more comparative way, and augment them with some new results on the geometry and the topology of the underlying tiling spaces.