Polyhex

In mathematics and combinatorics, a polyhex is a polyomino on a hexagonal grid. It consists of a finite, connected union of hexagonal cells, joined edge-to-edge. The size n of a polyhex is defined as the number of hexagons it contains. Polyhexes generalize the idea of polyominoes from square to hexagonal tilings and are studied in lattice enumeration and tiling theory.

Several variants exist for counting polyhexes. Fixed polyhexes distinguish shapes by translations on the plane, so two copies that differ only by position are considered different. Free polyhexes identify shapes up to the full symmetry group of the hexagonal lattice (rotations and reflections). One-sided polyhexes identify shapes up to rotations but distinguish reflections.

Enumerating polyhexes is a standard problem in lattice combinatorics. The numbers of distinct polyhexes grow exponentially with size, and counts are known for small n. The precise asymptotic growth rate is not known in a closed form, and researchers use specialized algorithms to generate and classify polyhexes, often employing canonical representations to avoid duplicates under symmetry and translation.

Typical properties of polyhexes include the perimeter, defined as the number of boundary edges exposed to the exterior, and geometric measures such as diameter. The interior forms a connected cluster, while the boundary is determined by exposed edges. Symmetries of the hexagonal lattice are used to reduce duplicates in enumeration.

Polyhexes appear in studies of tilings, statistical physics, and lattice models. They provide discrete models of cluster formation on hexagonal lattices and relate to concepts such as lattice animals, percolation, and self-avoiding configurations on hex grids. For small n, simple examples include a single hex (n=1) and two hexes sharing an edge (n=2).