### The Geometry of Bubbles and Foams

Published in **Foams, Emulsions, and Cellular Materials**,
(J.-F. Sadoc, ed.), NATO Adv. Sci. Inst. Ser. C,
Kluwer, 1998, pp 379-402.
**Plain-English Abstract**
Soap films, bubble clusters, and foams and froths can be modeled
mathematically as collections of surfaces which minimize their surface
area subject to volume constraints. Each surface in such a solution has
constant mean curvature, and is thus called a CMC surface. We first
examine a more general class of variational problems for surfaces, and
then concentrate on CMC surfaces, where one important result is the
balancing of certain forces. Next we look at existence results for
bubble clusters, and at the singularities in such clusters and in foams;
these are described by Plateau's rules.

These rules for singularities imply that a foam is combinatorially
dual
to some triangulation of space. Triangulations of a surface are related
through their Euler number to the topology of the surface. In three
dimensions, not as much can be said along these lines, but we describe
certain results. Among the most interesting triangulations, from the
point of view of equal-volume foams, are those of the tetrahedrally
close-packed (TCP) structures from chemistry. The TCP structures
observed in nature fit into an infinite mathematical class; we describe
constructions for some infinite families. Finally, we discuss the
application of these ideas to Kelvin's problem of partitioning space.
The counterexample to Kelvin's conjecture found by Weaire and Phelan was
one TCP foam, and the others all seem useful for generating efficient
equal-volume foams.