Bees build hexagons because six-sided cells tile a surface using the least wall
Regular hexagons fit together with no gaps, three meeting at every corner, and a hexagonal grid keeps its lines as short as possible when filling a large area. That is why honeycomb cells use less wax than other shapes would and hold up so well under compression.
The name comes from the Greek words for six and corner. The interior angles of any simple hexagon add up to 720 degrees; in a regular one, each is 120 degrees, a third of a full circle. Its side equals the radius of the circle drawn around it, and the long diagonals through the centre are exactly twice a side, so lines from the centre divide it neatly into six equilateral triangles.
The proportions follow from that. A regular hexagon whose long diagonal measures 1 is about 0.866 across its flats, the cosine of 30 degrees, giving a height-to-width ratio of 1 to roughly 1.1547. Its area is about 2.598 times the square of its side. It has six rotational and six mirror symmetries, together forming the dihedral group D6 with its 16 subgroups, and John Conway sorted the resulting family of hexagons into nine symmetry types.
Tiling reaches beyond the regular case. Any hexagon whose opposite sides are parallel and equal can cover the plane just by sliding copies across, and any irregular hexagon meeting the Conway criterion will tile it as well. Hexagonal patterns show up everywhere from beehives to the Giant's Causeway, and the Voronoi diagram of a triangular lattice is itself a honeycomb.
Hexagons also anchor some surprising theorems. Pascal's theorem, nicknamed the Hexagrammum Mysticum, says that if you draw any hexagon inside a conic section and extend each pair of opposite sides until they meet, the three crossing points always fall on a single straight line. And in a regular hexagon with corners A to F, any point P on the surrounding circle between B and C has distances to E and F that add up to the same total as its distances to A, B, C and D.
Source: Hexagon
