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.


The numbers of isohedral n-polyomino tilings (more specifically, the number of polyominoes with n cells that tile the plane by 180 degrees rotation but not by translation) for n=7, 8, 9, ... are 3, 11, 60, 199, 748, ... (OEIS A075201), the first few of which are illustrated above (Myers).

The numbers of n-polyomino tilings that tile the plane isohedrally without additional restriction for n=1, 2, ... are 1, 1, 2, 5, 12, 35, 104, ... (OEIS A075205).

See also

Anisohedral Tiling

Explore with Wolfram|Alpha


Berglund, J. "Is There a k-Anisohedral Tile for k>=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.Keating, K. and Vince, A. "Isohedral Polyomino Tiling of the Plane." Disc. Comput. Geom. 21, 615-630, 1999.Myers, J. "Polyomino Tiling.", N. J. A. Sequences A075201 and A075205 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Isohedral Tiling

Cite this as:

Weisstein, Eric W. "Isohedral Tiling." From MathWorld--A Wolfram Web Resource.

Subject classifications