Isohedral Tiling
Let 
be the group of symmetries which
map a monohedral tiling 
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
. If 
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 with 
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. 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.
Keating, K. and Vince, A. "Isohedral Polyomino Tiling of the Plane." Disc. Comput. Geom. 21, 615-630, 1999.
Myers, J. "Polyomino Tiling." https://www.srcf.ucam.org/~jsm28/tiling/.
Sloane, N. J. A. Sequences A075201 and A075205 in "The On-Line Encyclopedia of Integer Sequences."
Weisstein, Eric W. "Isohedral Tiling." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IsohedralTiling.html
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