Determine whether AT-tile and BT-tile admit tilings other than the hat-generated non-periodic tiling

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

Background

The AT-tile and BT-tile each have Heesch number 1, and together they are known to generate Th\mathscr{T}_h with mutually reversed tile densities. The complete set of tilings admitted by this pair remains unresolved.

References

A tile set consisting of two types of concave polygons, AT-tile and BT-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

However, because the number of combinations to be examined is enormous, it remains unclear how far our current approach can be extended or whether it will ultimately yield the desired results.

— 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 7, Conclusion