# 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.

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 $18.57752$.