Determine whether AH-tile and BH-tile admit tilings other than the hat-generated non-periodic tiling

Determine whether the tile set consisting of the two concave polygons AH-tile and BH-tile admits tilings other than the non-periodic tiling \(\mathscr{T}_h\).

Background

The paper establishes that AH-tile and BH-tile each have Heesch number 1 and that their pair can generate the non-periodic tiling Th\mathscr{T}_h. It does not determine whether additional tilings, potentially including periodic tilings, can be generated by the same two-tile set.

References

A tile set consisting of two types of concave polygons, AH-tile and BH-tile, is known to generate $\mathscr{T}_h$, but whether it admits other tilings remains unknown.

— Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles  (2609.09603 - Sugimoto, 9 Sep 2026) in Section 5.3, Properties of tile sets containing AH-tile, BH-tile, AT-tile, and BT-tile

Based on the summary above, the following question arises: ``Is there a tile set consisting of two types of concave polygons, obtained by selecting two tile types from AH-tile, BH-tile, AT-tile, and BT-tile, all of which have Heesch number 1, that is an aperiodic tile set (i.e., an aperiodic set of tiles)?''

— Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles  (2609.09603 - Sugimoto, 9 Sep 2026) in Section 5.3, Properties of tile sets containing AH-tile, BH-tile, AT-tile, and BT-tile