  This paper addresses the question of whether a single tile with nearest
neighbor matching rules can force a tiling in which the tiles fall into a large
number of isohedral classes. A single tile is exhibited 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.
