Anisohedral tiling

[1] The first part of Hilbert's eighteenth problem asked whether there exists an anisohedral polyhedron in Euclidean 3-space; Grünbaum and Shephard suggest[2] 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.

[3] However, Heesch then gave an example of an anisohedral tile in the plane in 1935.

[2] 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.

[7] Grünbaum and Shephard had previously raised a slight variation on the same question.

A partial tiling of the plane by Heesch's anisohedral tile. There are two symmetry classes of tiles, one containing the blue and green tiles and the other containing the red and yellow tiles. As Heesch proved, this tile cannot tile the plane with only one symmetry class.