Socolar–Taylor tile

[2] The basic version of the tile is a simple hexagon, with printed designs to enforce a local matching rule, regarding how the tiles may be placed.

[3] One of their papers[2] shows a realization of the tile as a connected set.

[2][3] This is, however, confirmed to be possible in three dimensions, and, in their original paper, Socolar and Taylor suggest a three-dimensional analogue to the monotile.

[1] Taylor and Socolar remark that the 3D monotile aperiodically tiles three-dimensional space.

Physical copies of the three-dimensional tile could not be fitted together without allowing reflections, which would require access to four-dimensional space.