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This paper addresses the question of whether a single tile with nearestneighbor matching rules can force a tiling in which the tiles fall into a largenumber of isohedral classes. A single tile is exhibited that can fill theEuclidean plane only with a tiling that contains k distinct isohedral sets oftiles, where k can be made arbitrarily large. It is shown that the constructioncannot work for a simply connected 2D tile with matching rules for adjacenttiles enforced by shape alone. It is also shown that any of the followingmodifications allows the construction to work: (1) coloring the edges of thetiling and imposing rules on which colors can touch; (2) allowing the tile tobe multiply connected; (3) requiring maximum density rather than space-filling;(4) allowing the tile to have a thickness in the third dimension.

Hexagonal parquet tilings: k-isohedral monotiles with arbitrarily large k