# What tesselated three-dimensional shape gives the maximum volume with the minimum surface area?

I recently read an article on the future of buildings. I have long been interested in architecture and it seems to me that this article makes some very good points. It got me thinking about the modular aspect of modern construction. As there have been great improvements in this area in recent years, and a growing public interest, it got me thinking about how this avenue will progress.

I thought about how the development of the intermodal container was a huge change in international goods shipping, and how the development of a similar international standard for modular buildings might do the same for construction. Most modular homes to date are very similar to intermodal containers at least partly owing to their need to be transported.

It seems to me, however, that it would be better to maximize volume relative to surface area in order to reduce materials cost and the mass to volume ratio. Doing so would reduce construction costs and allow for larger buildings. Prompted by all of the biomimicry of late, my immediate thought was that beehives provide an excellent model and that hexagonal prisms would make for good building modules. But, surely there are other options.

I recognize that there are other considerations to take into account, but I figured the most appropriate question to ask was:

What tesselated three-dimensional shape gives the maximum volume with the minimum surface area?

edit - After some more research, I'd like to add a little bit of technical rigor. I am looking for the convex uniform honeycomb that maximizes volume to surface area ratio where "surface" includes internal faces for each cell.

It would also be worthwhile to know which maximizes the volume to edge ratio since the edges would likely be the structually supporting element, and thus, the most expensive material.

-

## migrated from physics.stackexchange.comFeb 20 '13 at 18:12

This question came from our site for active researchers, academics and students of physics.

Whether a tessellation made out of truncated octahedra is optimal was Kelvin's conjecture; this conjecture was open for over 100 years, and was disproved by Weaire and Phelan in 1994.

The best known structure is either the truncated octahedron or the Weaire-Phelan tiling, depending on whether you insist that all the cells be identical or not.

Since you require that the cells be convex polyhedra, this isn't exactly the same problem that Weaire and Phelan solved, but the relationship between the two tilings still holds. Kusner and Sullivan calculate that for tilings with convex polyhedra, Kelvin's structure has cost $A^3/V^2 = 18.7653$ and the Weaire-Phelan structure has cost $18.57752$.

-
Reading about the Kelvin Conjecture in the link you provided pretty well sums up my question. Thanks! Now, off to build a Weaire-Phelan building. :) –  AdamRedwine Feb 20 '13 at 16:15
I don't think the Weaire-Phelan tiling has been proved optimal ... there might be a better tiling with more than two different cell shapes (but considering that Weaire-Phelan only improved on truncated octahedra by 0.3%, any further improvement is likely to be miniscule). –  Peter Shor Feb 20 '13 at 16:20
Yeah, perhaps we should go with the bitruncated cubic honeycomb. There is something to be said for standardized shapes. If you can get a manufacturing process down where you can pump out a ton of truncated octahedral rooms, you can do all sorts of cool stuff. –  AdamRedwine Feb 20 '13 at 16:24