Anisohedral Tiling

A plane tiling is said to be isohedral if the symmetry group of the tiling acts transitively on the tiles, and -isohedral if the tiles fall into n orbits under the action of the symmetry group of the tiling. A -anisohedral tiling is a tiling which permits no -isohedral tiling with .

The numbers of anisohedral polyominoes with , 9, 10, ... are 1, 9, 44, 108, 222, ... (OEIS A075206), the first few of which are illustrated above (Myers).

REFERENCES

Berglund, J. "Is There a -Anisohedral Tile for ?" Amer. Math. Monthly 100, 585-588, 1993.

Berglund, J. "Anisohedral Tilings Page." http://www.angelfire.com/mn3/anisohedral/.Grünbaum, B. and Shephard, G. C. §9.4 in Tilings and Patterns. New York: W. H. Freeman, 1986.

Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory. Washington, DC: Math. Assoc. Amer., 1991.

Myers, J. "Polyomino Tiling." http://www.srcf.ucam.org/~jsm28/tiling/.Sloane, N. J. A. Sequence A075206 in "The On-Line Encyclopedia of Integer Sequences."