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Soap Bubbles and Mathematics

Frank Morgan - Photo by Jeff Bauer of Citco

Let’s test your intuition about bubbles.

The standard “double bubble” consists of a little sphere and a big sphere, with a surface between them. Do you think that (a) the surface is flat, (b) the big bubble pushes up into the little bubble, or (c) the little bubble pushes down into the big bubble?

Frank Morgan (Williams College) asked his audience at the MAA Carriage House on April 28 eight such questions. Part of the MAA’s NSA-funded Distinguished Lecture Series, Morgan’s “Soap Bubbles and Mathematics” beguiled attendees with its gameshow flavor, its displays of bubble-blowing prowess, and its insights into the mathematical process. 

The answer to that double bubble question?

“The little bubble pushes into the big bubble,” Morgan said. “And the reason is there’s more pressure in the little bubble.”

Bubbles may seem like kid stuff, but they’re everywhere: They make bread fluffy, mattresses supportive, fire extinguishers effective. They’re also, Morgan emphasized, a topic of serious mathematical study. Morgan considers the proof of the double bubble conjecture his greatest mathematical achievement, and he took pains in his talk to connect the work of both 2015 Abel Prize winners—John Nash, Jr. and Louis Nirenberg—to soap bubble geometry.

Morgan traced the history of this mathematics all the way back to Zenodorus, who proved in the second century B.C. that a circle is the most efficient way to enclose a given area. It wasn’t until 1884, however, that Hermann Schwarz established the analogous result in three dimensions, that a sphere has the least surface area for a given volume. Area minimization is the principle dictating soap bubble behavior, Morgan explained.

“When you have a cluster of soap bubbles coming together,” he said, “they look for the least area way to enclose and separate three...or more given volumes of air.” 

Analysis of such clusters turns out to require some sophisticated mathematics, including what Morgan called the “notorious geometric measure theory.” As he indicated milestones in the development of strategies to understand bubbles, Morgan played video clips of the mathematicians involved.  

One of these mathematicians was Jean Taylor, whose 1976 paper “The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces” definitively answered the hardest question Morgan posed during his lecture: How many different ways can soap films come together?

“In math...the answer is always one, zero, or infinity,” Morgan quipped, “but in this case the answer is two—what? two!—a very unanticipated kind of answer.”

Indeed, what Taylor proved is that (1) soap bubbles meet in threes along a curve at an angle of 120 degrees and (2) these curves meet in fours at a vertex at the tetrahedral angle of approximately 109.47 degrees.

Though Morgan joked about spending the remainder of his talk “going over some of the technical details” of Taylor’s paper, after only a brief overview of her argument he moved on to other topics, among them the double bubble.  

He floated some alternatives to the standard double bubble, which, until Morgan and his co-authors proved it in 2002, was merely suspected to be the most efficient way to enclose two volumes of air.

Two separate bubbles are wasteful, he noted; if they come together they can share the common wall. A bubble inside a bubble also offers no advantage, since containing the smaller bubble makes the larger bubble unnecessarily—and inefficiently—large.

“So that’s why you never see a bubble inside a bubble,” Morgan said, even as he dipped a pair of wands into his bucket of soap solution and succeeded in blowing one.

Within an instant, however, the smaller bubble popped out and glommed on to the outside of the larger one in that familiar double bubble configuration. “It prefers that shape,” Morgan said.

Proof of the double bubble conjecture established the standard double bubble as the most efficient—more efficient than, say, John Sullivan’s bubble-with-a-bubble-around-its-waist—but it leaves open the question of whether other arrangements are stable and therefore might occur in nature.

“I love this question because mathematicians have no idea how to solve it but a kindergarten student could answer it tomorrow by just blowing a different double bubble,” Morgan said.

Morgan ended his talk with a championship round of sorts. Attendees who had correctly answered more than two of Morgan’s earlier questions tried to outdo one another in the identification of the best—that is, most perimeter-efficient—planar five-bubble, six-bubble, seven-bubble, and eight-bubble. 

Even those finalists eliminated in the first round went home with prizes. “Good attempt,” Morgan said as he handed each of them a miniature jar of bubble solution. “We have little research kits so you can get better at this.”

Blowing bubbles may seem a frivolous pastime, but Morgan gave listeners choice words for anyone chiding them for production of soapy spheres. 

“If you want to understand the universe,” he said mid-talk, “you should start out by understanding the soap bubble.” —Katharine Merow