be the group of symmetries which map a monohedral
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 .
has transitivity classes, then is said to be -isohedral. Berglund (1993) gives examples of -isohedral tilings for , 2, and 4.
The numbers of isohedral -polyomino tilings (more specifically, the number of polyominoes
cells that tile the plane by rotation but not by translation) for , 8, 9, ... are 3, 11, 60, 199, 748, ... (OEIS A075201),
the first few of which are illustrated above (Myers).
The numbers of -polyomino
tilings that tile the plane isohedrally without additional restriction for , 2, ... are 1, 1, 2, 5, 12, 35, 104, ... (OEIS A075205).
Berglund, J. "Is There a -Anisohedral Tile for ?" Amer. Math. Monthly100, 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.Keating, K. and Vince, A. "Isohedral Polyomino Tiling
of the Plane." Disc. Comput. Geom.21, 615-630, 1999.Myers,
J. "Polyomino Tiling." http://www.srcf.ucam.org/~jsm28/tiling/.Sloane,
N. J. A. Sequences A075201 and A075205 in "The On-Line Encyclopedia of Integer