Why are honeycomb cells hexagonal?

A hexagon tiles a flat surface with no gaps, and of the three shapes that do - triangle, square, hexagon - it encloses the most area for the least wall. Wax is expensive; bees consume something like six kilograms of honey to make one of wax. Building in hexagons is the cheapest possible way to store the most honey.

That this is genuinely optimal was suspected for two thousand years and proved only in 1999, when Thomas Hales demonstrated the honeycomb conjecture: no arrangement of any shapes, regular or not, does better. The bees were right, and it took mathematics until the end of the twentieth century to confirm it.

How they do it is a separate and slightly deflating question. Bees appear to build cells that start out circular, and the hexagonal geometry emerges as the soft wax is warmed by the bees and pulled into shape by surface tension where the walls meet - three walls meeting at 120 degrees is what liquid films do naturally, as anyone who has looked at soap foam has seen. The bees supply the effort and the physics supplies the geometry.