 TOPICS  # 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).

Anisohedral Tiling

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## References

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." http://www.srcf.ucam.org/~jsm28/tiling/.Sloane, N. J. A. Sequences A075201 and A075205 in "The On-Line Encyclopedia of Integer Sequences."

Isohedral Tiling

## Cite this as:

Weisstein, Eric W. "Isohedral Tiling." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IsohedralTiling.html