Colloquium: Soap bubble clusters

Soap bubble clusters and froths model biological cells, metallurgical structures, magnetic domains, liquid crystals, fire-extinguishing foams, bread, cushions, and many other materials and structures. Despite the simplicity of the governing principle of energy or area minimization, the underlying mathematical theory is deep and still not understood, even for rather simple, finite clusters. Only with the advent of geometric measure theory could mathematics treat surfaces which might have unprescribed singularities and topological complexities. In 1884, Schwarz gave a rigorous mathematical proof that a single round soap bubble provides the least-area way to enclose a given volume of air. Similarly, the familiar double bubble provides the absolute least-area way to enclose and separate the two given volumes of air, although the proof did not come until 2000 and has an interesting story, as this Colloquium explains in some detail. Whether a triple soap bubble provides the least-area way to enclose and separate three given volumes of air remains an open conjecture today. Even planar bubble clusters remain mysterious. In about 200 $\mathrm{B.C.}$ Zenodorus essentially proved that a circle provides the least-perimeter way to enclose a single given area. The planar double and triple bubbles were proved minimizing recently. The status of the planar four-bubble remains open today. In most spaces other than Euclidean space, even the best single bubble remains unproven. One exception is Gauss space, which is of much interest to probabilists and should be more familiar to physicists. General “isoperimetric” problems of minimizing area for given volume occur throughout mathematics and play an important role in differential geometry and analysis, including Perelman’s proof of the Poincaré conjecture.

Figure 1

(Color) A soap bubble cluster. Graphics by John M. Sullivan.

Figure 2

Planar soap bubble froth photographed by Olivier Lorderau at Rennes.

Figure 3

(Color) The standard double bubble provides the least-perimeter way to enclose and separate two given volumes of air. Graphics by John M. Sullivan.

Figure 4

It is an unproven conjecture that this standard triple bubble provides the least-perimeter way to enclose and separate three given volumes of air. Photo by Jeremy Ackerman.

Figure 5

Least-perimeter clusters of $N$ unit areas, as computed numerically. Bubbles in the interior are expected to have six neighbors while bubbles on the periphery are expected to have four. The black bubbles have more than the expected number of neighbors and the gray bubbles have fewer. From [4].

Figure 6

(Color online) The three conjectured minimizers in a rectangular box and two other contenders. From 21, Fig. 9.

Figure 7

(Color online) The gyroid, a conjectured periodic minimizer, has been verified by Ros to within 3%. From 20, Fig. 1.

Figure 8

The five types of interior singularities for unoriented area-minimizing surfaces in three dimensions with imposed symmetries. They separate space into two regions, here shaded (or cross-hatched) and unshaded. From 16, Fig. 1.

Figure 9

(Color) The hard part of the double-bubble theorem proof showing that each region does not break up into multiple components. Drawing by Yuan Lai.

Figure 10

A useful decomposition of a double bubble. From 15, Fig. 14.8.1.

Figure 11

(Color) Rotating the two halves of this bubble in opposite directions about a carefully chosen axis $A$ stretches the top, shrinks the bottom, and reduces area. First graphic from John M. Sullivan. Second from 14, Fig. 3.

Figure 12

Case by case, one has to find an axis $A$ and division points so that the instability rotations stretch the bubble rather than ripping it apart. From 15, Fig. 14.8.1.