Anisohedral tiling

In geometry, a shape is said to be anisohedral if it admits a tiling, but no such tiling is isohedral (tile-transitive); that is, in any tiling by that shape there are two tiles that are not equivalent under any symmetry of the tiling. A tiling by an anisohedral tile is referred to as an anisohedral tiling.

The second part of Hilbert's eighteenth problem asked whether there exists an anisohedral polyhedron in Euclidean 3-space; Grünbaum and Shephard suggest that Hilbert was assuming that no such tile existed in the plane. Reinhardt answered Hilbert's problem in 1928 by finding examples of such polyhedra, and asserted that his proof that no such tiles exist in the plane would appear soon. However, Heesch then gave an example of an anisohedral tile in the plane in 1935.

Reinhardt had previously considered the question of anisohedral convex polygons, showing that there were no anisohedral convex hexagons but being unable to show there were no such convex pentagons, while finding the five types of convex pentagon tiling the plane isohedrally. Kershner gave three types of anisohedral convex pentagon in 1968; one of these tiles using only direct isometries without reflections or glide reflections, so answering a question of Heesch.

The problem of anisohedral tiling has been generalised by saying that the isohedral number of a tile is the least number of orbits (equivalence classes) of tiles in any tiling of that tile under the action of the symmetry group of that tiling, and that a tile with isohedral number k is k-anisohedral. Berglund asked whether there exist k-anisohedral tiles for all k, giving examples for k ≤ 4 (examples of 2-anisohedral and 3-anisohedral tiles being previously known, while the 4-anisohedral tile given was the first such published tile). Goodman-Strauss considered this in the context of general questions about how complex the behaviour of a given tile or set of tiles can be, noting a 10-anisohedral example of Myers. Grünbaum and Shephard had previously raised a slight variation on the same question.

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