polyhexes

Polyhexes are polyforms formed by joining hexagonal cells on a regular hexagonal tiling. Each cell is a hexagon, and shapes are connected by shared edges. The term n-hex denotes a polyhex consisting of n hexagons. Polyhexes are the hexagonal-lattice analogue of polyominoes (square cells) and polyiamonds (triangular grid).

Counting and classification are done up to symmetries of the hexagonal lattice: fixed polyhexes (translations considered distinct), one-sided polyhexes (rotations considered the same but reflections distinct), and free polyhexes (rotations and reflections considered the same).

Polyhexes may contain holes; the set of hexagons is connected via edge adjacencies, but cavities are possible in larger shapes. They are finite subsets of the hexagonal lattice closed under adjacency.

Enumeration of polyhexes is a well-studied problem in combinatorics. Exact counts are known only for small n and are obtained by computer search; the asymptotic behavior is known to be exponential in n and is analyzed using methods such as transfer-matrix techniques and Burnside’s lemma.

Polyhexes relate to broader topics in tiling theory and lattice models, and they appear in recreational mathematics and puzzle design. They provide a hexagonal-grid counterpart to well-known square- and triangle-based polyforms, with applications in counting problems, algorithmic generation, and the study of geometric properties on hexagonal lattices.