TITLE: Prove that honeycomb structures are the most geometrically efficient structure
QUESTION [6 upvotes]: I was reading this paragraph and it got me thinking:

The closed ends of the honeycomb cells
  are also an example of geometric
  efficiency, albeit three-dimensional
  and little-noticed. The ends are
  trihedral (i.e., composed of three
  planes) sections of rhombic
  dodecahedra, with the dihedral angles
  of all adjacent surfaces measuring
  $120^o$, the angle that minimizes surface
  area for a given volume. (The angle
  formed by the edges at the pyramidal
  apex is approximately $109^\circ 28^\prime 16^{\prime\prime}$ $\left(=
180^\circ - \cos^{-1}\left(\frac13\right)\right)$ 

This is hardly intuitive; is there a proof of this somewhere?

REPLY [12 votes]: If you want to divide space up into uniform volume cells with minimum surface area, the honeycomb is not optimal. Look at the Weaire–Phelan structure. While honeycombs are not quite optimal, they are certainly close enough for bees -- they're suboptimal by only 0.3%.