Socolar, JES, *The hexagonal parquet tiling k-isohedral monotiles with arbitrarily large k*,
The Mathematical Intelligencer, vol. 29 no. 2
(2007),
pp. 33-38 [pdf], [doi] .
**Abstract:**

*The interplay between local constraints and global structure of mathematical and physical systems is both subtle and important. This paper shows how to construct a single tile that can fill the Euclidean plane only with a tiling that contains k distinct isohedral sets of tiles, where k can be made arbitrarily large. It is shown that the construction cannot work for a simply connected 2D tile with matching rules for adjacent tiles enforced by shape alone. It is also shown that any of the following modifications allows the construction to work: (1) coloring the edges of the tiling and imposing rules on which colors can touch; (2) allowing the tile to be multiply connected; (3) requiring maximum density rather than space-filling; (4) allowing the tile to have a thickness in the third dimension.*