            Fall 2001     Cannonballs and Honeycomb: Kelvin

If we turn to the next page after the Kepler conjecture in Kepler's Six-Cornered Snowflake, we find a discussion of the structure of the bee's honeycomb. The rhombic dodecahedron was discovered by Kepler through close observation of the honeycomb. The honeycomb is a six-sided prism sealed at one end by three rhombi. By sealing the other end with three additional rhombi, the honeycomb cell is transformed into the rhombic dodecahedron.

Figure 8 During the 18th century, mathematicians made extensive studies of the isoperimetric properties of the honeycomb and believed the honeycomb to be the most efficient design possible. For example, in 1743, C. MacLaurin in his investigation of the rhombic bases of the honeycomb concluded, ``The cells, by being hexagonal, are the most capacious, in proportion to their surface, of any regular figures that leave no interstices between them, and at the same time admit of the most perfect bases.'' However, the obvious answer provided by the honeybee turned out incorrect. Upsetting the prevailing opinion, L. Fejes T\'oth discovered that the three-dimensional honeycomb cell is not the most economical (Figure 8). The most economical form has never been determined.

The cannonball packing of balls leads to honeycomb cells. It is also related to more general foam problems. If we tile space with hollow rhombic dodecahedra, and imagine that each has walls made of a flexible soap film, we have an example of a foam.

### Kelvin

The problem of foams, first raised by Lord Kelvin, is easy to state and hard to solve. How can space be divided into cavities of equal volume so as to minimize the surface area of the boundary? The rhombic dodecahedral example is far from optimal. Lord Kelvin proposed the following solution. Truncated octahedra fill space (see Figure 9). In fact, their cross sections are regular octagons, and the octagons tile the plane except for square holes. Each truncated octahedron contains square plugs, so that the next layer of octahedra plugs the square holes of the previous layer.

Lord Kelvin found that by warping the faces of the truncated octahedra ever so slightly, he could obtain a foam with smaller surface area than the cells of the truncated octahedra. This was Lord Kelvin's proposed solution. It satisfies the conditions Plateau discovered more than a century ago for minimal soap bubbles. Everyone seemed satisfied with Kelvin's solution; only a proof of optimality was missing.

Kelvin and his supporters were wrong, as the physicists D. Phelan and R. Weaire showed in 1994. They produced a foam with cavities of equal volume with a considerably smaller surface area than the Kelvin foam. The Phelan-Weaire foam contains two different types of cavities, one with 14 sides, the other with 12. (Figure 10).

A finite version of the problem might be more tractable. What shape minimizes surface area if the foam contains only finitely many bubbles of equal volume? If there is a single bubble, the problem is the classical isoperimetric problem. The sphere uniquely minimizes the area of a surface enclosing a given volume. The problem of two bubbles, known as the double bubble conjecture, was solved only recently by J. Hass, M. Hutchings, and R. Schlafy.(The double bubble conjecture, Electron. Res. Announc. Amer. Math. Soc. 1 (1995), no. 3, 98-102.) The problem for more than two bubbles is still unsolved.

The Kepler conjecture and the Kelvin problem are both special cases of a more general foam problem. Phelan and Weaire ask us to imagine that the soapy film walls have a measurable thickness. We interpolate between the Kepler and Kelvin problems with a parameter w (measuring the wetness of the film) that gives the fraction of space filled by the thick film walls, and 1-w is the fraction filled by the cavities. If the foam is perfectly dry, then w = 0, and the film walls are surfaces. The Kelvin problem asks for the most efficient design. When the foam becomes sufficiently wet, w is close to 1, and the cavities of the foam can be independently molded. The isoperimetric inequality dictates that they minimize surfaces area by forming into perfect spheres. The Kepler problem asks for the smallest value of w for which every cavity is a perfect sphere.

At this time of renewed interest in Hilbert's problems, many mathematicians are proposing new lists of problems. My submission to Hilbert's millennial list is the Kelvin problem. It has a rich history. Its solution will require new ideas from geometric measure theory. Frank Morgan predicts that the Kelvin problem might take a century to be solved. For starters: does an optimal solution exist?

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