Existence of strongly aperiodic monotiles

Determine whether there exists a strongly aperiodic monotile—either in geometric tilings of Euclidean spaces under the isometry group or in group-tiling settings—such that every tiling by the monotile has trivial stabilizer (no nontrivial symmetry).

Background

The paper’s constructions (e.g., the Roach in Γ and the Hat-derived tilings) achieve mild aperiodicity, with possible residual symmetries such as 3-fold rotations. Achieving strong aperiodicity would require eliminating all nontrivial symmetries in every tiling by a single tile.

Despite significant progress on aperiodic monotiles in various settings, the authors emphasize that no strongly aperiodic monotile is currently known, whether in geometric tilings or group-tiling frameworks, making the existence problem fundamental and unresolved.