Polyiamond

Polyiamonds are plane geometric figures formed by joining equilateral triangles edge to edge. An n-iamond consists of n triangles arranged so that each triangle shares full edges with its neighbors, and the set is connected via edge adjacency. They are the triangular-lattice analogue of polyominoes, which are built from unit squares.

Objects of study include free polyiamonds (considering rotations and reflections as the same), one-sided polyiamonds (reflections distinct but rotations identified), and fixed polyiamonds (distinct by all isometries of the plane).

Polyiamonds can be convex or concave and may enclose holes in larger configurations. The enumeration of polyiamonds by order n (the number of constituent triangles) is a classic problem in combinatorial geometry. Counts are known for small n and grow rapidly; methods use Burnside's lemma, generating functions, and computer enumeration.

Historically, polyiamonds have appeared in recreational mathematics and tiling theory as a natural triangular-grid counterpart to polyominoes. They are used in puzzles, tiling designs, and in studies of lattice animals and polyform enumeration.

Examples: n=1 yields a single triangle; n=2 yields a rhombus of two triangles; higher orders produce a variety of shapes, with dozens of free polyiamonds existing at modest n.