Isohedral Tiling

Let $S(T)$ be the group of symmetries which map a Monohedral Tiling $T$ onto itself. The Transitivity Class of a given tile T is then the collection of all tiles to which T can be mapped by one of the symmetries of $S(T)$. If $T$ has $k$ Transitivity Classes, then $T$ is said to be $k$-isohedral. Berglund (1993) gives examples of $k$-isohedral tilings for $k=1$, 2, and 4.

See also Anisohedral Tiling

References

Berglund, J. ``Is There a $k$-Anisohedral Tile for $k\geq 5$?'' Amer. Math. Monthly 100, 585-588, 1993.

Grünbaum, B. and Shephard, G. C. ``The 81 Types of Isohedral Tilings of the Plane.'' Math. Proc. Cambridge Philos. Soc. 82, 177-196, 1977.