由hat和turtle瓷砖对应的非周期平铺导出的两类凹多边形瓷砖集
Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles
AI总结：
本文利用Type 5族凸五边形单调瓷砖，通过两种透视法导出四种凹多边形瓷砖，证明其对应非周期平铺的簇，并探讨配对瓷砖集与ASPmr的关联。
AI中文摘要：
利用属于Type 5族的凸五边形单调瓷砖，我们研究了hat瓷砖、turtle瓷砖和Tile$(1, 1)$之间的关系。通过应用Sugimoto透视法和Amfirifma透视法，我们获得了四种凹多边形：AH-tile、BH-tile、AT-tile和BT-tile，在考虑的条件下每种都具有Heesch数1。我们证明这些多边形对应于用于生成非周期平铺$\mathscr{T}_h$和$\mathscr{T}_s$的簇。我们进一步讨论了从这些多边形中选择的配对组成的瓷砖集可能对应于$\textit{ASPmr}\{\text{A-tile}, \text{B-tile}\}$的可能性。
英文摘要：
Using a convex pentagonal monotile belonging to the Type 5 family, we investigate the relationships among the hat tile, turtle tile, and Tile$(1, 1)$. By applying Sugimoto's Perspective and Amfirifma's Perspective, we obtain four types of concave polygons, AH-tile, BH-tile, AT-tile, and BT-tile, each having Heesch number 1 under the conditions considered. We show that these polygons correspond to clusters used to generate the non-periodic tilings $\mathscr{T}_h$ and $\mathscr{T}_s$. We further discuss the possibility that tile sets consisting of pairs selected from these polygons may correspond to $\textit{ASPmr}\{\text{A-tile}, \text{B-tile}\}$.
