Polyhex (mathematics)


In recreational mathematics, a polyhex is a polyform with a regular hexagon as the base form.
As with polyominoes, polyhexes may be enumerated as free polyhexes, fixed polyhexes and one-sided polyhexes. 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, … ; 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.
nFreeFree with holesFree without holesOne - sidedFixed
110111
210113
3303311
47071044
52202233186
682181147814
733323316203652
81448131435282116689
965726765051294277359
10304904043008660639362671

Tessellation properties

All of the polyhexes with fewer than five hexagons can form at least one regular plane tiling. In addition, the plane tilings of the dihex and straight polyhexes are invariant under 180 degrees rotation or reflection parallel or perpendicular to the long axis of the dihex, and the hexagon tiling and some other polyhexes are invariant under 60, 120 or 180 degree rotation.
In addition, the hexagon is a hexiamond, so all polyhexes are also distinct polyiamonds. Also, as an equilateral triangle is a hexagon and three smaller equilateral triangles it is possible to superimpose a large polyiamond on any polyhex, giving two polyiamonds corresponding to each polyhex. This is used as the basis of an infinite division of a hexagon into smaller and smaller hexagons or into hexagons and triangles.
Of the polyhexes shown in the table, 2 have 6-fold rotation and reflection symmetry, the monohex and the hexahex with a hole, 3 others have 3-fold rotation and 3-fold reflection symmetry, 9 others have 2-fold rotation and reflection, 8 have just two fold rotation, 16 just have 2-fold reflection and the other 78 are asymmetrical. The tilings of most of the reflection-symmetrical polyhexes are also invariant under glide reflections of the same order by the length of the polyhex. No polyhex has an order of symmetry greater than six for reflection, rotation or glide.