Beating Kelvin's Partition of Space

In 1887, Lord Kelvin asked how to partition space into cells of volume 1 such that the total area of the interfaces between the cells is a minimum. The best partition Kelvin could come up with was made of slightly curved 14-sided polyhedra. Two of Kelvin's tetrakaidecahedra are pictured here:

Kelvin's partition

For over a century, nobody could improve on Kelvin's partition. Then in 1993, Denis Weaire and Robert Phelan came up with a partition of space into two kinds of cells (of equal volume, of course) that beat Kelvin's partition by 0.3% in area. Ironically, an image with the same topological structure was in Linus Pauling's 1960 chemistry book sitting on my father's bookshelf while I tried to beat Kelvin sitting 10 feet away.

Several views of the Weaire-Phelan partition are shown below. A fundamental region of 8 different colored cells is shown. Two cells (green and blue) are dodecahedra, and the other six are 14-sided with two opposite hexagonal faces and 12 pentagonal faces. The 14-sided cells stack into three sets of orthogonal columns, and the dodecahedra fit into the interstices between the columns.

Weaire-Phelan A Weaire-Phelan B Weaire-Phelan C Weaire-Phelan D Weaire-Phelan E Weaire-Phelan bunch

There is no proof that the Weiare-Phelan partition is optimal, or that Kelvin's partition is optimal for a single shape of cell. R. Kusner and J. Sullivan have proved analytically that the polyhedral Weaire-Phelan foam beats any foam with the Kelvin topology, removing any possible doubts about numerical inaccuracies.

The area calculations and these images were made with the Surface Evolver program, with the datafiles twointor.fe for Kelvin's partition, and phelanc.fe for the Weaire-Phelan partition.

Paper models for building Weaire-Phelan clusters are available from Stardust, or by free download from Thomas Girsewald.

References:

W. Thomson, Lord Kelvin, "On the division of space with minimum partitional area", Phil. Mag. vol. 24 (1887), 503.

D. Weaire and R. Phelan, "A counterexample to Kelvin's conjecture on minimal surfaces", Phil. Mag. Lett. vol. 69 (1994), 107-110.

R. Kusner and J. Sullivan, "Comparing the Weaire-Phelan equal-volume foam to Kelvin's foam," Forma 11:3, 1996, pp 233-242.

All of the above and others are reprinted in:

D. Weaire et al., The Kelvin Problem, Taylor & Francis, 1996.


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