Kelvin's problem : What space-filling arrangement of similar cells of equal volume has minimal surface area ?

(The presentation of the problem and the 3 first illustrations have been taken from Mathworld and Wikipedia)

This questions arises naturally in the theory of foams when the liquid content is small. The problem has been solved in 2D in 1999 by Hales, T. C (see arXive) : honeycombs have minimal perimeter among tilling with cells of equal areas. In 3D, Kelvin (Thomson 1887) proposed that the solution was a 14-sided truncated octahedron having a very slight curvature of the hexagonal faces. The isoperimetric quotient of the uncurved truncated octahedron (see below) is approximativelly equal to 0.753367

Kelvin's tilling

while Kelvin's slightly curved variant has a slightly less optimal quotient of 0.757. Despite one hundred years of failed attempts and Weyl's (1952) opinion that the curved truncated octahedron could not be improved upon, Weaire and Phelan (1994) discovered a space-filling unit cell consisting of six 14-sided polyhedra and two 12-sided polyhedra with irregular faces and only hexagonal faces remaining planar. This structure has an isoperimetric quotient of 0.764, or approximately 0.3% less that Kelvin's cell (see below).

As it has been reported by Weaire, this strange structure has been identified with the help of the very efficient software Evolver (developped by Kennet A. Brakke). Starting from an initial configuration (a good one !) Weaire and Phelan used the surface minimization algorithm (based on a descent approach) provided by Evolver in order to find the closer local minimizer.

Weaire and Phelan structure
Initial densities

We present in this work a completely different approach in order to identify optimal partitions. Surprisingly, this method is based on a volumic formulation of Kelvin's problem. This approach leads us to a relaxation which is very efficient in identifying global minimizers. We present below the densities that we found. Second figures are simply obtained by a periodic reconstruction of those densities. Finally, we use Evolver in a last step in order to find a precise description of those minimizing structures.

We perform the following computations with ~ 1e6 parameters and we minimized a nonconvex functional under linear equality constraints.

Click on a picture to see an animated view

Optimal periodic tilings obtained by gamma Γ-convergence

Let us precise that our method does not require any initial guess (see one initial configuration in the previous picture) : in that sense our method is (at least experimentally) globally convergent. We observe that looking for 8 densities we are able to recover exactly Weaire and Phelan structure. With 16 cells we recover Kelvin's tilling.

Optimization of 16 densities in a cube with periodic conditions

Optimization of 8 densities in a cube with periodic conditions

We also carried out those computations with a cell number from 8 to 20. We give in the table below the optimal values that we obtained :

Number of cells Assymptotic perimeters (for cells of volume 1)
8 2.644175
10 2.692954
11 2.695891
12 2.671376
13 2.683315
14 2.694757
15 2.68954
16 2.653171
17 2.675445
18 2.681586
19 2.680236
20 2.655404
Summary of our results