A 'Monumental' Math Proof Solves the Triple Bubble Problem

Erica Klarreich

When it comes to understanding the shape of bubble clusters, mathematicians have been playing catch-up to our physical intuitions for millennia. Soap bubble clusters in nature often seem to immediately snap into the lowest-energy state, the one that minimizes the total surface area of their walls (including the walls between bubbles). But checking whether soap bubbles are getting this task right--or just predicting what large bubble clusters should look like--is one of the hardest problems in geometry. It took mathematicians until the late 19th century to prove that the sphere is the best single bubble, even though the Greek mathematician Zenodorus had asserted this more than 2,000 years earlier.

Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research develop-ments and trends in mathe-matics and the physical and life sciences.

The bubble problem is simple enough to state: You start with a list of numbers for the volumes, and then ask how to separately enclose those volumes of air using the least surface area. But to solve this problem, mathematicians must consider a wide range of different possible shapes for the bubble walls. And if the assignment is to enclose, say, five volumes, we don't even have the luxury of limiting our attention to clusters of five bubbles--perhaps the best way to minimize surface area involves splitting one of the volumes across multiple bubbles.

Even in the simpler setting of the two-dimensional plane (where you're trying to enclose a collection of areas while minimizing the perimeter), no one knows the best way to enclose, say, nine or 10 areas. As the number of bubbles grows, "quickly, you can't really even get any plausible conjecture," said Emanuel Milman of the Technion in Haifa, Israel.

But more than a quarter century ago, John Sullivan, now of the Technical University of Berlin, realized that in certain cases, there is a guiding conjecture to be had. Bubble problems make sense in any dimension, and Sullivan found that as long as the number of volumes you're trying to enclose is at most one greater than the dimension, there's a particular way to enclose the volumes that is, in a certain sense, more beautiful than any other--a sort of shadow of a perfectly symmetric bubble cluster on a sphere. This shadow cluster, he conjectured, should be the one that minimizes surface area.

Over the decade that followed, mathematicians wrote a series of groundbreaking papers proving Sullivan's conjecture when you're trying to enclose only two volumes. Here, the solution is the familiar double bubble you may have blown in the park on a sunny day, made of two spherical pieces with a flat or spherical wall between them (depending on whether the two bubbles have the same or different volumes).

But proving Sullivan's conjecture for three volumes, the mathematician Frank Morgan of Williams College speculated in 2007, "could well take another hundred years."

Jehan Azad

Angela Watercutter

Lily Hay Newman

Sabrina Weiss

Now, mathematicians have been spared that long wait--and have gotten far more than just a solution to the triple bubble problem. In a paper posted online in May 2022, Milman and Joe Neeman, of the University of Texas, Austin, have proved Sullivan's conjecture for triple bubbles in dimensions three and up and quadruple bubbles in dimensions four and up, with a follow-up paper on quintuple bubbles in dimensions five and up in the works.

And when it comes to six or more bubbles, Milman and Neeman have shown that the best cluster must have many of the key attributes of Sullivan's candidate, potentially starting mathematicians on the road to proving the conjecture for these cases too. "My impression is that they have grasped the essential structure behind the Sullivan conjecture," said Francesco Maggi of the University of Texas, Austin.

Milman and Neeman's central theorem is "monumental," Morgan wrote in an email. "It's a brilliant accomplishment with lots of new ideas."

Our experiences with real soap bubbles offer tempting intuitions about what optimal bubble clusters should look like, at least when it comes to small clusters. The triple or quadruple bubbles we blow through soapy wands seem to have spherical walls (and occasionally flat ones) and tend to form tight clumps rather than, say, a long chain of bubbles.

But it's not so easy to prove that these really are the features of optimal bubble clusters. For example, mathematicians don't know whether the walls in a minimizing bubble cluster are always spherical or flat--they only know that the walls have "constant mean curvature," which means the average curvature stays the same from one point to another. Spheres and flat surfaces have this property, but so do many other surfaces, such as cylinders and wavy shapes called unduloids. Surfaces with constant mean curvature are "a complete zoo," Milman said.

But in the 1990s, Sullivan recognized that when the number of volumes you want to enclose is at most one greater than the dimension, there's a candidate cluster that seems to outshine the rest--one (and only one) cluster that has the features we tend to see in small clusters of real soap bubbles.

To get a feel for how such a candidate is built, let's use Sullivan's approach to create a three-bubble cluster in the flat plane (so our "bubbles" will be regions in the plane rather than three-dimensional objects). We start by choosing four points on a sphere that are all the same distance from each other. Now imagine that each of these four points is the center of a tiny bubble, living only on the surface of the sphere (so that each bubble is a small disk). Inflate the four bubbles on the sphere until they start bumping into each other, and then keep inflating until they collectively fill out the entire surface. We end up with a symmetric cluster of four bubbles that makes the sphere look like a puffed-out tetrahedron.

Next, we place this sphere on top of an infinite flat plane, as if the sphere is a ball resting on an endless floor. Imagine that the ball is transparent and there's a lantern at the north pole. The walls of the four bubbles will project shadows on the floor, forming the walls of a bubble cluster there. Of the four bubbles on the sphere, three will project down to shadow bubbles on the floor; the fourth bubble (the one containing the north pole) will project down to the infinite expanse of floor outside the cluster of three shadow bubbles.

Jehan Azad

Angela Watercutter

Lily Hay Newman

Sabrina Weiss

The particular three-bubble cluster we get depends on how we happened to position the sphere when we put it on the floor. If we spin the sphere so a different point moves to the lantern at the north pole, we'll typically get a different shadow, and the three bubbles on the floor will have different areas. Mathematicians have proved that for any three numbers you choose for the areas, there is essentially a single way to position the sphere so the three shadow bubbles will have precisely those areas.

We're free to carry out this process in any dimension (though higher-dimensional shadows are harder to visualize). But there's a limit to how many bubbles we can have in our shadow cluster. In the example above, we couldn't have made a four-bubble cluster in the plane. That would have required starting with five points on the sphere that are all the same distance from each other--but it's impossible to place that many equidistant points on a sphere (though you can do it with higher-dimensional spheres). Sullivan's procedure only works to create clusters of up to three bubbles in two-dimensional space, four bubbles in three-dimensional space, five bubbles in four-dimensional space, and so on. Outside those parameter ranges, Sullivan-style bubble clusters just don't exist.

But within those parameters, Sullivan's procedure gives us bubble clusters in settings far beyond what our physical intuition can comprehend. "It's impossible to visualize what is a 15-bubble in [23-dimensional space]," Maggi said. "How do you even dream of describing such an object?"

Yet Sullivan's bubble candidates inherit from their spherical progenitors a unique collection of properties reminiscent of the bubbles we see in nature. Their walls are all spherical or flat, and wherever three walls meet, they form 120-degree angles, as in a symmetric Y shape. Each of the volumes you're trying to enclose lies in a single region, instead of being split across multiple regions. And every bubble touches every other (and the exterior), forming a tight cluster. Mathematicians have shown that Sullivan's bubbles are the only clusters that satisfy all these properties.

Jehan Azad

Angela Watercutter

Lily Hay Newman

Sabrina Weiss

When Sullivan hypothesized that these should be the clusters that minimize surface area, he was essentially saying, "Let's assume beauty," Maggi said.

But bubble researchers have good reason to be wary of assuming that just because a proposed solution is beautiful, it is correct. "There are very famous problems ... where you would expect symmetry for the minimizers, and symmetry spectacularly fails," Maggi said.

For example, there's the closely related problem of filling infinite space with equal-volume bubbles in a way that minimizes surface area. In 1887, the British mathematician and physicist Lord Kelvin suggested that the solution might be an elegant honeycomb-like structure. For more than a century, many mathematicians believed this was the likely answer--until 1993, when a pair of physicists identified a better, though less symmetric, option. "Mathematics is full ... of examples where this kind of weird thing happens," Maggi said.

When Sullivan announced his conjecture in 1995, the double-bubble portion of it had already been floating around for a century. Mathematicians had solved the 2D double-bubble problem two years earlier, and in the decade that followed, they solved it in three-dimensional space and then in higher dimensions. But when it came to the next case of Sullivan's conjecture--triple bubbles--they could prove the conjecture only in the two-dimensional plane, where the interfaces between bubbles are particularly simple.

Then in 2018, Milman and Neeman proved an analogous version of Sullivan's conjecture in a setting known as the Gaussian bubble problem. In this setting, you can think of every point in space as having a monetary value: The origin is the most expensive spot, and the farther you get from the origin, the cheaper land becomes, forming a bell curve. The goal is to create enclosures with preselected prices (instead of preselected volumes), in a way that minimizes the cost of the boundaries of the enclosures (instead of the boundaries' surface area). This Gaussian bubble problem has applications in computer science to rounding schemes and questions of noise sensitivity.

Milman and Neeman submitted their proof to the Annals of Mathematics, arguably mathematics' most prestigious journal (where it was later accepted). But the pair had no intention of calling it a day. Their methods seemed promising for the classic bubble problem too.

They tossed ideas back and forth for several years. "We had a 200-page document of notes," Milman said. At first, it felt as though they were making progress. "But then quickly it turned into, 'We tried this direction--no. We tried [that] direction--no.'" To hedge their bets, both mathematicians pursued other projects as well.

Jehan Azad

Angela Watercutter

Lily Hay Newman

Sabrina Weiss

Then last fall, Milman came up for sabbatical and decided to visit Neeman so the pair could make a concentrated push on the bubble problem. "During sabbatical it's a good time to try high-risk, high-gain types of things," Milman said.

For the first few months, they got nowhere. Finally, they decided to give themselves a slightly easier task than Sullivan's full conjecture. If you give your bubbles one extra dimension of breathing room, you get a bonus: The best bubble cluster will have mirror symmetry across a central plane.

Sullivan's conjecture is about triple bubbles in dimensions two and up, quadruple bubbles in dimensions three and up, and so on. To get the bonus symmetry, Milman and Neeman restricted their attention to triple bubbles in dimensions three and up, quadruple bubbles in dimensions four and up, and so on. "It was really only when we gave up on getting it for the full range of parameters that we really made progress," Neeman said.

With this mirror symmetry at their disposal, Milman and Neeman came up with a perturbation argument that involves slightly inflating the half of the bubble cluster that lies above the mirror and deflating the half that lies below it. This perturbation won't change the volume of the bubbles, but it could change their surface area. Milman and Neeman showed that if the optimal bubble cluster has any walls that are not spherical or flat, there will be a way to choose this perturbation so that it reduces the cluster's surface area--a contradiction, since the optimal cluster already has the least surface area possible.

Using perturbations to study bubbles is far from a new idea, but figuring out which perturbations will detect the important features of a bubble cluster is "a bit of a dark art," Neeman said.

With hindsight, "once you see [Milman and Neeman's perturbations], they look quite natural," said Joel Hass of UC Davis.

But recognizing the perturbations as natural is much easier than coming up with them in the first place, Maggi said. "It's by far not something that you can say, 'Eventually people would have found it,'" he said. "It's really genius at a very remarkable level."

Milman and Neeman were able to use their perturbations to show that the optimal bubble cluster must satisfy all the core traits of Sullivan's clusters, except perhaps one: the stipulation that every bubble must touch every other. This last requirement forced Milman and Neeman to grapple with all the ways bubbles might connect up into a cluster. When it comes to just three or four bubbles, there aren't so many possibilities to consider. But as you increase the number of bubbles, the number of different possible connectivity patterns grows, even faster than exponentially.

Milman and Neeman hoped at first to find an overarching principle that would cover all these cases. But after spending a few months "breaking our heads," Milman said, they decided to content themselves for now with a more ad hoc approach that allowed them to handle triple and quadruple bubbles. They've also announced an unpublished proof that Sullivan's quintuple bubble is optimal, though they haven't yet established that it's the only optimal cluster.

Milman and Neeman's work is "a whole new approach rather than an extension of previous methods," Morgan wrote in an email. It's likely, Maggi predicted, that this approach can be pushed even further--perhaps to clusters of more than five bubbles, or to the cases of Sullivan's conjecture that don't have the mirror symmetry.

No one expects further progress to come easily; but that has never deterred Milman and Neeman. "From my experience," Milman said, "all of the major things that I was fortunate enough to be able to do required just not giving up."

Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.