For a long time, tessellation (tiling), covering, and packing have fascinated me. Especially tessellations by convex pentagons have a relationship with roughly half of my life. The time I have been doing mathematical exploration so far was long, but artistic curiosity was always in me. In recent years, I spend a lot of time on artistic activities.
TH-pentagons can form hexagonal flowers. Since such hexagonal flowers and their reflections have identical outlines, tessellations of hexagonal flowers can be formed by employing anterior TH-pentagons and posterior TH-pentagons. The representative Type 5 tessellation and the tessellation discovered by Marjorie Rice in the year 1995 can be generated by incorporating the differences in the contiguities of hexagonal flower units. This wooden model depicts the assemblage of one type of cat tile representing Type 5 tessellation with three types of cat tiles representing the tessellation discovered by Marjorie Rice. It should be noted that the cat tile with a rotation center of order six has been designed by Makoto Nakamura.
The convex pentagons shown in the figure have been termed as "TH-pentagons" by the associated authors. Each such unique convex pentagon comprises 7/3 equilateral triangles and is obtained from a trisected heptiamond (TH). Since the TH-pentagon can generate infinite tessellation patterns, it is known as a "versatile." Based on the TH-pentagon, the cat tiles shown in the figure have been designed. Since the representative Type 5 tessellation of TH-pentagons is an isohedral tiling (a tiling is isohedral if and only if each polygon is surrounded in an identical manner), it needs one type of cat tile. However, the tessellation discovered by Marjorie Rice in the year 1995 is a 3-isohedral tiling, thus requiring three types of cat tiles.