Prove that honeycomb structures are the most geometrically efficient structure

Question

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?

Answer 1

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%.

Answer 2

It is called geometrically efficient because it is densely packed.

Also read:

1. Sphere Packing Wikipedia

Answer 3

This question can be explained and understood with the aid of the physics of soap bubbles. :=)

One problem when thinking on such questions is that one often thinks the walls as "rubber sheets". For the surface (and in arbtrary cross sections ways) minimisation the surface tension has to be thought constant.

So always when three lamellae join in a common "corner" the orthogonal cross section is 120 degrees. (thee identical forces in one point) Excuse this "corner" and orthogonal, I am not aware of geometry in English.