Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles

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 ${\mathcal{T}}_h$ and ${\mathcal{T}}_s$. We further discuss the possibility that tile sets consisting of pairs selected from these polygons may correspond to $\mathit{\text{ASPmr}}\{\text{A-tile},\text{B-tile}\}$.