# Polyhex (mathematics)

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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 -hexes for = 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 A038144; the number of free polyhexes without holes is given by A018190; the number of fixed polyhexes is given by A001207; the number of one-sided polyhexes is given by A006535.^{[1]}^{[2]}

The Monohex: | |

The Dihex: | |

The 3 Trihexes: | |

The 7 Tetrahexes: | |

The 22 Pentahexes: | |

The 82 Hexahexes: |

## See also

- Tessellation
- Percolation theory
- Polyiamond - tilings with equilateral triangles
- Polyomino - tilings with squares
- Polycyclic aromatic hydrocarbon - hydrocarbons whose structure is based on polyhexes

## References

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