TITLE: Packing and isoperimetrics
QUESTION [10 upvotes]: Suppose a manufacturer bottles small units of liquid and ships them via very large trucks.
If the transportation cost nothing, spherical bottles would minimize the packaging cost (isoperimetric inequality); if packaging cost nothing, cubic bottles (say) would minimize the transportation cost, because they would pack on the trucks with no wasted space.
How would the ideal container vary with the relative cost of packaging (measured as the surface area of one bottle) and transportation (measured by the packing density on an infinitely capacious truck)?
[My "applied" formulation notwithstanding, I mean this as a pure mathematics question, so please idealize and ignore any distracting side issues.]
A reasonable conjecture might seem that one gets the correct family of shapes by starting with a tight lattice packing of spheres and then over-inflating and rescaling the spheres until asymptotically they assume the form of Voronoi cells of a lattice packing.  So one would get constant mean curvature surfaces up to the circular disks at the interfaces between adjacent spheres.  Optimal or not, this would give lower bounds.  But how to compute them?

REPLY [8 votes]: This is the problem of wet foams, soap bubble clusters separated by water. As the liquid fraction goes to zero, you get the dry foam or partitioning problem, optimally the hexagonal honeycomb in 2D (Hales) and conjecturally the Weaire-Phelan structure in 3D or the Kelvin structure if all cells must be congruent. As the liquid fraction increases you pass through the optimal sphere packing (proved by Thue in 2D and Hales in 3D) to spherical bubbles floating in a sea of liquid. The transition in 2D from the hexagonal honeycomb to the circle packing is easy to conjecture but would be very hard to prove. In 3D a similar smooth transition beginning with the Kelvin partition would yield the inefficient BCC lattice sphere packing instead of the optimal FCC lattice sphere packing. A smooth transition beginning with the less efficient rhombic dodecahedra would yield the optimal FCC lattice sphere packing. So the optimal wet foam probably jumps discontinuously from one branch to the other at some point.