In recreational mathematics

Recreational mathematics

Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...

, a polyhex is a polyform

Polyform

In recreational mathematics, a polyform is a plane figure constructed by joining together identical basic polygons. The basic polygon is often a convex plane-filling polygon, such as a square or a triangle. More specific names have been given to polyforms resulting from specific basic polygons, as...

 with a regular hexagon (or 'hex' for short) as the base form.

As with polyomino

Polyomino

A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling with a connected interior....

es, 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, … ; 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 .

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

See also

• Tessellation
  Tessellation
  A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
  
  
• Percolation theory
  Percolation theory
  In mathematics, percolation theory describes the behavior of connected clusters in a random graph. The applications of percolation theory to materials science and other domains are discussed in the article percolation.-Introduction:...
  
  
• Polyiamond
  Polyiamond
  A polyiamond is a polyform whose base form is an equilateral triangle. The word polyiamond is a back-formation from diamond, because this word is often used to describe the shape of a pair of equilateral triangles placed base to base, and the initial "di-" looked like a Greek prefix meaning...
  
   - tilings with equilateral triangles
• Polyomino
  Polyomino
  A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling with a connected interior....
  
   - tilings with squares
• Polycyclic aromatic hydrocarbon
  Polycyclic aromatic hydrocarbon
  Polycyclic aromatic hydrocarbons , also known as poly-aromatic hydrocarbons or polynuclear aromatic hydrocarbons, are potent atmospheric pollutants that consist of fused aromatic rings and do not contain heteroatoms or carry substituents. Naphthalene is the simplest example of a PAH...
  
  - hydrocarbons whose structure is based on polyhexes