Comparing the Weaire-phelan Equal-volume Foam to Kelvin’s Foam

The problem of partitioning space into equal-volume cells, using the least interface area, was considered in 1887 by Sir William Thomson, Lord Kelvin [Kel]. His proposed solution yields a foam with cells of a single shape, tiling space by the translations of the body-centered cubic lattice. In 1993, Denis Weaire and Robert Phelan [WP] proposed a new equal-volume foam with two different cell shapes, which uses less area according to their computer experiments with Ken Brakke’s evolver [Bra]. Mathematically, one difficulty is that neither foam can be described explicitly; even their existence is troublesome. In this note, we will examine these two foams, showing that the Weaire-Phelan foam is in fact more efficient than the Kelvin foam, and we will describe some other interesting candidate foams. More mathematical details, including a proof of existence for the Kelvin foam, are forthcoming in [AKS]. A partition of space is a division of R3 into disjoint cells. We are mainly interested in the surfaces forming the interface between the cells. The partitions we consider will be periodic with respect to some lattice, with some number n of cells in each periodic domain; thus they could be viewed as partitions of a quotient threetorus into n cells. Fixing the volume of each cell to be V, we want to minimize the cost of the equal-volume partition, defined scale-invariantly to be μ := A3/V 2, where A is the average interface area per cell. (Note that A is really half the boundary area of a typical cell, since each interface is shared by two cells.) If we scale to make V = 1, the case of unit-volume cells, then our cost is simply the cube of the total surface area in a periodic domain, divided by n3. If we start with some partition, perhaps polyhedral, and let it relax until the cost is a minimum, at least among nearby partitions, then the resulting stable partition should exhibit the geometry of a cluster of soap bubbles. Thus a stable partition should follow the rules recorded by Plateau [Pla] in 1873 for such clusters, and proved by Jean Taylor [Tay] in 1976 for a certain mathematical model (due to Fred Almgren) of compound bubbles. These Plateau rules state that the interfaces are smooth surfaces of constant mean curvature, except where they meet in threes (at equal 120◦ angles) along smooth arcs. These arcs are, furthermore, allowed to come together (four at a time) at isolated points, where the configuration is tetrahedral; but no other singularities are allowed. The allowed singularities are observed for instance in a cluster of three