polyiamonds

Polyiamonds are plane figures formed by joining congruent equilateral triangles edge to edge on the triangular lattice. A polyiamond with n triangles uses n unit triangles as its cells and has a boundary along the lattice edges. The resulting region may be simply connected or may contain holes, depending on how the triangles are arranged.

Three common notions are used to classify polyiamonds by equivalence. Free polyiamonds identify two shapes if one can be moved, rotated, or reflected to coincide with the other. One-sided polyiamonds identify shapes up to translations and rotations, but treat mirror images as distinct. Fixed polyiamonds distinguish shapes even if they differ only by position or orientation, so there is no identification under translations, rotations, or reflections.

Polyiamonds are the triangular-lattice analog of polyominoes and are a subject of combinatorial enumeration. Counting polyiamonds of a given size can be done under the different equivalence notions (free, one-sided, fixed), and researchers use computer algorithms and Burnside’s lemma to derive counts for small and larger n. The study includes questions about which polyiamonds tile the plane, tile hexagonal regions, or tile other shapes without gaps or overlaps.

As a natural counterpart to polyominoes, polyiamonds appear in recreational mathematics and tiling theory, and they serve as a framework for exploring symmetry, packing, and enumeration on the triangular lattice.