Toil and trouble over double bubbles.

What soap bubbles do naturally can create a lot of bother for mathematicians.

Consider the case when two bubbles join to form a double bubble--a sight familiar to any soap-bubble aficionado. The bubbles share a disk-shaped wall, and this wall meets the individual bubble walls at 120o. Mathematicians call this configuration the standard double bubble (see illustration).

But this structure isn't the only candidate for the most economical way of packaging a pair of identical volumes. For instance, one bubble may ring the other--like an inner tube fitting snugly around a peanut's waist--to form a two-chambered torus bubble (see illustration).

Now, Joel Hass of the University of California, Davis, and Roger Schlafly of Real Software in Soquel, Calif., have proved mathematically that the standard double bubble triumphs over the torus bubble as the two-chambered geometric structure of least surface area. Nature's design also turns out to be the mathematician's answer to the most economical way of enclosing and separating two given volumes of space.

Hass described the long-sought proof this week at the Burlington (Vt.) Mathfest, a joint meeting of the American Mathematical Society and the Mathematical Association of America.

Mathematicians have long known that the circle is the shortest way to enclose a region of a given area and that the sphere is the most efficient way to enclose a given volume. They suspected that the standard double bubble was the answer for two given volumes, but they couldn't say so with mathematical certainty.

The campaign to settle the question began when Frank Morgan of Williams College in Williamstown, Mass., suggested to a group of undergraduate students that they tackle the two-dimensional case. "It was one of many unsolved problems in the mathematics of soap bubbles," Morgan says.

In 1991, Joel S. Foisy, now a graduate student at Duke University in Durham, N.C., and his coworkers proved that the standard double bubble in two dimensions--that is, two circles squeezed up against each other--has the least possible perimeter. There are no bizarrely curved configurations that do any better.

Joseph D. Masters, a graduate student at the University of Texas at Austin, then established that the same structure works as the smallest-perimeter arrangement on the surface of a sphere.

Going to three dimensions, Michael L. Hutchings, a graduate student at Harvard University, proved a conjecture that narrowed the candidates to the standard double bubble and the torus bubble.

But a torus bubble has innumerable possible shapes, any one of which could possibly beat the standard double bubble. How could all these possibilities be ruled out?

Hass and Schlafly decided to try to solve the problem by using a computer to check all the torus bubble configurations. They established criteria for comparing the surface areas of the various enclosures and wrote a program to conduct the search.

Normally, mathematicians don't use computers to obtain mathematical proofs, partly because computers generally make slight rounding-off errors when they do calculations. But Hass and Schlafly found a way of circumventing this deficiency.

In the end, they showed that no torus bubble does better than the standard double bubble. "Computer calculations [were] essential to our proof that equal-volume double bubbles minimize area," Hass says.

This result still leaves open the question of what happens when the two volumes are unequal. Preliminary findings suggest that the standard double bubble also wins in this case, but not all possibilities have been checked yet.

The situation for triple bubbles--in both two and three dimensions--is much further from being settled. And mathematicians haven't yet proved that the hexagonal honeycomb is the most efficient way of partitioning a region into equal-area units (SN: 3/5/94, p.149).

Nature probably has the answers already, but mathematicians need to establish them with certainty, and such efforts often entail a lot of toil and trouble.