晕面拼花：k-具有任意大k值的k-等面体拼花
Hexagonal parquet tilings: k-isohedral monotiles with arbitrarily large k
摘要：
本文讨论的问题是，是否一个单一的瓷砖与最近邻匹配规则可以强制平铺中的瓷砖落入大量的等面体类。一个单一的瓷砖展示，可以填补欧几里德平面只有一个瓷砖，其中包含k个不同的isohedral集的瓷砖，其中k可以任意大。结果表明，该结构不能工作的一个简单连接的2D瓷砖与相邻的瓷砖形状单独执行的匹配规则。它还表明，任何以下修改允许的建设工作：（1）着色的瓷砖的边缘和强加的规则，颜色可以接触;（2）允许瓷砖是多连接;（3）需要最大的密度，而不是空间填充;（4）允许瓷砖有一个厚度在第三维。
英文摘要：
This paper addresses the question of whether a single tile with nearest neighbor matching rules can force a tiling in which the tiles fall into a large number of isohedral classes. A single tile is exhibited that can fill the Euclidean plane only with a tiling that contains k distinct isohedral sets of tiles, where k can be made arbitrarily large. It is shown that the construction cannot work for a simply connected 2D tile with matching rules for adjacent tiles enforced by shape alone. It is also shown that any of the following modifications allows the construction to work: (1) coloring the edges of the tiling and imposing rules on which colors can touch; (2) allowing the tile to be multiply connected; (3) requiring maximum density rather than space-filling; (4) allowing the tile to have a thickness in the third dimension.
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。
