Why Double Bubbles Form the Way They Do
Need to entertain a child? Try blowing soap bubbles. Need to keep a mathematician busy? Just ask why bubbles take the shapes they do. Individual soap bubbles, of course, are spherical, and for a very simple reason: Among all surfaces that enclose a given volume, the sphere has the least area (and in the grand scheme of things, nature inclines toward such minima). On the other hand, when two soap bubbles come together, they form a "double bubble," a simple complex of three partial spheres: two on the outside, with the third serving as a wall between the two compartments. Scientists have long considered it obvious that double bubbles behave this way for the same minimum-seeking reason--because no other shape encloses two given volumes with less total surface area. But mathematicians have countered with their usual vexing question: Where's the proof?
Now they have it. An international team of four mathematicians has announced a proof of the double bubble conjecture. By honing a new technique for analyzing the stability of competing shapes, Michael Hutchings of Stanford University, Frank Morgan of Williams College in Williamstown, Massachusetts, and Manuel Ritoré and Antonio Ros at the University of Granada have shown that only the standard shape is truly minimal--any other, supposedly area-minimizing shape can be ever so slightly twisted into a shape with even less area, a contradiction which rules out these other candidates.
What other shape could two bubbles possibly take? One candidate--or class of candidates--has one bubble wrapped around the other like an inner tube. But it could be even worse: Mathematically, there's no objection to splitting a volume into two separate pieces, so it's possible that siphoning off a bit of the central volume and reinstalling it as a "belt" around the inner tube would actually reduce the total surface area. And conceivably, then, siphoning a bit of the inner tube and placing it as a band around the belt would lead to smaller area yet, and so forth. There's not even any obvious reason that the true, area-minimizing double bubble can't have "empty chambers"--enclosed regions that don't belong to either volume.
Just about the only thing that's (relatively) easy to prove is that the solution must have an axis of symmetry--in other words, it can't have lopsided bulges. Hutchings took the first big step toward ruling out the more bizarre possibilities in the early 1990s. He ruled out empty chambers and showed that the larger volume must be a single piece. Besides the standard double bubble, his results limited the possible solutions to ones consisting of a large inner tube around a small central region, perhaps with a set of one or more belts circling the outside. Hutchings also found formulas that provide bounds on the number of belts, as a function of the ratio of the two volumes. In particular, if the two volumes are equal, or even nearly equal, there can be no belts, so the only alternative is a single inner tube around a central region.
Based on Hutchings's work, in 1995 Joel Hass of the University of California (UC), Davis, and Roger Schlafly, now at UC Santa Cruz, proved the double bubble conjecture for the equal-volume case. Their proof used computer calculations to show that any inner tube arrangement can be replaced by another with smaller area. "Ours was a comparison method," Hass explains. He and Schlafly found they could extend their results for volume ratios up to around 7:1, but beyond that the possible configurations to be ruled out became too complicated.
Surprisingly, the general proof requires no computers, just pencil and paper. The key idea consists of finding an "axis of instability" for each inner tube arrangement. Twisting the two volumes around this axis--with a motion rather like wringing out a washcloth--leads to a decrease in surface area, contradicting the shape's ostensible minimality. "We always thought that these remaining cases were unstable," Morgan says. The proof confirms their suspicions, although it leaves open the possibility that some nonminimizing configuration could also be stable. The twisting argument is new and a bit subtle, Morgan notes. The hardest part is figuring out where to position the axis of instability so that the twisting procedure wouldn't change the volumes of the two regions as well as the surface area. "For a while, it was hard to frame the right questions, especially in Spanish."
Although the proof is only now being announced, the main results were established last spring, when Morgan visited Granada during a sabbatical. Since then, a group of undergraduates in a summer research program at Williams College has extended the results to analogs of the double bubble conjecture in higher dimensions. (The two-dimensional double bubble conjecture was proved by an earlier group of undergraduates in 1990.) Ben Reichardt of Stanford, Yuan Lai of the Massachusetts Institute of Technology, and Cory Heilmann and Anita Spielman of Williams College have shown that an axis of instability always exists for nonstandard shapes in the four-dimensional case, and also in higher dimensions under the mild assumption that the larger volume consists of a single, connected region.
What about triple bubbles? Once again, nature provides a relatively simple and obvious answer, but, Hass notes, "we don't know how to get started" proving it. The triple bubble problem is even open in two dimensions, with equal-sized area (for example, what's the least amount of fencing required to create three acre-sized pens, to separate, say, sheep from goats from hippopotami?) And it gets less certain from there, Hass says. "Once you get up to 20 or 30 regions, we don't even have a conjecture."