# Prove that Honeycomb Structures are the Most Geometrically Efficient Structure

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 $109^o 28' 16''$ $\left(= 180^o - cos^{-1}\left(\frac{1}{3}\right)\right)$

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

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## migrated from physics.stackexchange.comFeb 23 '11 at 8:26

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not really physics that I can tell. probably belongs in math.stackexchange – Mark Eichenlaub Feb 23 '11 at 3:47
i remember reading somewhere that the bees didnt use the best possible (but very close and optimized wrt some other variable). – yoyo Feb 23 '11 at 16:01

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%.

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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.

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It is called geometrically efficient because it is densely packed.