Polyhex (mathematics)

Template:Commonscat In recreational mathematics, a polyhex is a polyform with a regular hexagon (or 'hex' for short) as the base form.

As with polyominoes, polyhexes may be enumerated as free polyhexes (where rotations and reflections count as the same shape), fixed polyhexes (where different orientations count as distinct) and one-sided polyhexes (where mirror images count as distinct but rotations count as identical). They may also be distinguished according to whether they may contain holes. The number of free ${\displaystyle n}$-hexes for ${\displaystyle n}$ = 1, 2, 3, … is 1, 1, 3, 7, 22, 82, 333, 1448, … (sequence A000228 in OEIS); the number of free polyhexes with holes is given by ; the number of free polyhexes without holes is given by ; the number of fixed polyhexes is given by ; the number of one-sided polyhexes is given by .[1][2]

 The Monohex: The Dihex: The 3 Trihexes: The 7 Tetrahexes: The 22 Pentahexes: The 82 Hexahexes:

References

1. Wolfram Mathworld: Polyhex
2. Glenn C. Rhoads, Planar tilings by polyominoes, polyhexes, and polyiamonds, Journal of Computational and Applied Mathematics 174 (2005), No. 2, pp 329–353