Posted by hexnet -

A polyhex is a plane figure composed of n regular hexagons joined at their edges, in the manner of a regular hexagonal tessellation.

Polyhexes have perhaps achieved their greatest utility in organic chemistry, where they can be used to represent various configurations of aromatic hydrocarbons, but are also often employed in puzzles, logic games, and other recreational mathematical pursuits. A more speculative application of the first several orders of free polyhexes can be found in Patrick Mulcahy's article The Hexagonal Geometry of the Tree of Life, available as a PDF file in our Hexagonal Library.

In the image above we see the first six orders of free polyhexes, with rotations and reflections not counted as separate forms. Polyhexes can also be enumerated as one-sided, where reflections but not rotations are counted as distinct forms, and as fixed, where both rotations and reflections are counted distinctly. A table of all three enumerations can be found below.

Free One-sided Fixed
1111
2113
33311
471044
52233186
682147814
73336203652
81448282116689
965721294277359
103049060639362671
111435522861901716033
1268310113646218182213
133274826654543039267086
141579689731586358189492795
1576581875153143956918837374
163728681017457008454474080844
171822236628364437939721866153748
18893491036217869651166107217298977
194393916426387877879487527266673134
202166510360124333012532312599804551168
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