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
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
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
(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
in order to find the closer local minimizer.
We present in this work
Weaire and Phelan structure
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