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?
Best Answer
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%.