Polyhex

In recreational mathematics, a polyhex is a polyform with a regular hexagon (or 'hex' for short) as the base form, constructed by joining together 1 or more hexagons. Specific forms are named by their number of hexagons: monohex, dihex, trihex, tetrahex, etc. They were named by David Klarner who investigated them. Each individual polyhex tile and tessellation polyhexes and can be drawn on a regular hexagonal tiling.

In recreational mathematics, a polyhex is a polyform with a regular hexagon (or 'hex' for short) as the base form, constructed by joining together 1 or more hexagons. Specific forms are named by their number of hexagons: monohex, dihex, trihex, tetrahex, etc. They were named by David Klarner who investigated them. Each individual polyhex tile and tessellation polyhexes and can be drawn on a regular hexagonal tiling. ==Construction rules== The rules for joining hexagons together may vary. Generally, however, the following rules apply: Two hexagons may be joined only along a common edge, and must share the entirety of that edge. No two hexagons may overlap. A polyhex must be connected. Configurations of disconnected basic polygons do not qualify as polyhexes. The mirror image of an asymmetric polyhex is not considered a distinct polyhex (polyhex are "double sided"). == Enumeration== 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, … ; the number of free polyhexes with holes is given by ; the number of free polyhexes without holes is given by ; the number of one-sided polyhexes is given by ; the number of fixed polyhexes is given by . === Symmetry=== Of the polyhexes up to hexahexes, 2 have 6-fold rotation and reflection symmetry (thus also 3-fold and 2-fold symmetry), the monohex and the hexahex with a hole, 3 others have 3-fold rotation (the compact trihex, the propeller tetrahex and the hexahex looking like an equilateral triangle) 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 (most of the tetrahexes, pentahexes or hexahexes) 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. === Monohexes=== There is one monohex. It tiles the plane as a regular hexagonal tiling. ===Dihexes=== There is one free dihex: ===Trihexes=== There are 3 free and two-sided trihexes: ===Tetrahexes=== There are 7 free and two-sided tetrahexes. They are given names, in the order shown: bar, worm, pistol, propeller, arch, bee, and wave. ===Pentahexes=== There are 22 free and two-sided pentahexes: ===Hexahexes=== There are 82 free and two-sided 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 Rep-tile – tilings of shapes that are made of smaller copies of themselves ==References==

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