Principles of Nature: towards a new visual language
What about the hexagon?
Although it is the third regular polygon which can tile the plane without gaps and can be dissected into six equilateral triangles, it cannot be divided into an integral number of component self-similar shapes (as can the square and equilateral triangle). However, if the area of a hexagon of side-length one is taken as a unit of area, it is apparent that a hexagon of side-length two has four times the area, and so on. In other words the areas of hexagons also change with the ‘square of the side’.
In Euclidean geometry, the square and the equilateral triangle are regular polygons which cannot be dissected into an integral number of smaller regular polygons of fewer sides: the square cannot be dissected into component equilateral triangles, nor can the equilateral triangle be dissected into regular polygons of fewer sides than three (since none exists in the geometry of the flat plane). Hexagons (6-sided regular polygons) on the other hand, can be dissected into equilateral triangles (3-sided regular polygons). This establishes a certain connection between the square and equilateral triangle in Euclidean geometry, but by the same token, a simultaneous connection and distinction of the hexagon.
There is a strongly suggested connection here to the chemistry of carbon molecules known as organic chemistry. Not only is carbon an ubiquitous element on Earth but it is also able to form different sorts of bonds with a wide range of other elements and radicals leading to the extremely rich chemistry of carbon molecules. Carbon has an atomic number of six (the sixth element) and can form itself into hexagonal rings of six carbon atoms. This benzene ring occurs in many organic compounds and is fundamental to many biochemical processes.
The hexagon remains crucial in that it is one of only three regular polygons which can tile the Euclidean plane. As such it is of fundamental importance to number theory (including the so-called 'square numbers'). ' Hexagons also enclose more space than the square or the equilateral triangle for the same expenditure of perimetric material, as any honey bee knows.