Enumeration of Polyominoes, Polyiamonds and Polyhexes for Isohedral Tilings with Rotational Symmetry

We describe computer algorithms that can enumerate and display, for a given n > 0 (in theory, of any size), all n-ominoes, niamonds, and n-hexes that can tile the plane using only rotations; these sets necessarily contain all such tiles that are fundamental domains for p4, p3, and p6 isohedral tilings. We display the outputs for small values of n. This expands on earlier work [3]. 1 Polyominoes, polyiamonds, polyhexes and isohedral tilings Polyominoes, polyiamonds and polyhexes are among the simplest shapes for tiles and are easily produced by computer or by hand. A polyomino (or n-omino) is a planar tile made up of n congruent squares joined at their edges. A polyiamond (or n-iamond) is a planar tile made up of n congruent equilateral triangles joined at their edges. A polyhex (or n-hex) is a planar tile made up of n congruent regular hexagons joined at their edges. There is a rich store of problems concerning these tiles. Most of the problems about these tiles are about their tiling properties. In this work, we focus on isohedral tilings of the plane by these tiles in which the tilings have 3-, 4-, or 6-fold rotational symmetry. An isohedral tiling of the plane is one in which congruent copies of a single tile fill the plane without gaps or overlaps, and the symmetry group of the tiling acts transitively on the tiles. In our discussion, we assume knowledge of the lattice of rotation centers of the symmetry groups p3, p4, and p6.[1, 2]