发表机构
Charles University（查理大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种密度方法，通过分析平面截面中瓷砖频率，证明特定三维棱柱族具有有限Heesch数，并给出6个无限族凹口棱柱Heesch数≥2的计算结果。
AI 中文摘要
在本文中，我们提出了一种限制特定族瓷砖的 Heesch 数的新方法。特别地，我们展示了某一族三维棱柱具有有限的 Heesch 数。为此，我们开启了对由两种原瓷砖构成的比率 $q$ 平铺的结构性研究，在这种平铺中，粗略地说，一种瓷砖出现的频率是另一种的 $q$ 倍。该方法如下：我们取棱柱周围冠状区域在两个垂直方向上的平面截面，并比较这些区域中所得原瓷砖的频率。如果由这些原瓷砖构成的每个平铺的极限比率至少为 $q > 1$，那么足够大的棱柱冠状区域将需要在两个方向上具有不相容的瓷砖数量。我们还提供了一些计算结果，并展示了 $6$ 个无限族的带凹口棱柱，其有限 Heesch 数大于或等于 $2$。最后，我们提供了一些关于具有给定 Heesch 数的某些棱柱不存在的负面计算结果。
英文摘要
In this paper, we provide a novel approach to limiting Heesch numbers of a specific family of tiles. In particular, we show how a certain family of three--dimensional prisms has a finite Heesch number. In so doing, we initiate a structural study of ratio $q$ tilings by two prototiles, in which, loosely speaking, one tile appears $q$ times more often than the other. The method works as follows: we take planar intersections with coronas around the prism in two perpendicular directions and compare the frequencies of the resulting prototiles in those patches. If every tiling by these prototiles has limiting ratio at least $q >1$, then sufficiently large coronas of the prism would require incompatible tile counts in the two directions. We also provide some computational results and exhibit $6$ infinite families of notched prisms that have finite Heesch numbers greater or equal to $2$. Finally, we provide some negative computational results about the non--existence of certain prisms with given Heesch numbers.
