‘Monumental’ Math Proof Solves Triple Bubble Problem and More

Calculating the shape of bubble clusters has been done for thousands of years. In nature, soap bubble clusters tend to snap into the lowest energy state, the one that reduces the total surface area of their walls. One of the hardest problems in geometry is checking whether soap bubbles are getting the job done. It took mathematicians until the late 19th century to prove that the sphere is the best single bubble, even though Zenodorus had claimed it more than 2,000 years ago.

The easiest way to solve the bubble problem is to list the numbers for the volumes and then ask how to separate them. Calculating the shape of the bubble walls requires a wide range of different shapes. The best way to minimize surface area is to split one of the volumes across multiple bubbles.

In the simpler setting of the two-dimensional plane, no one knows the best way to keep a collection of areas separate. Emanuel Milman of the Technion in Haifa, Israel said that as the number of bubbles grows, you can't really even get a plausible idea of what's going on.

John Sullivan, now of the Technical University of Berlin, realized 25 years ago that there is a guiding conjecture to be had. Sullivan found that a particular way to enclose the volumes that is more beautiful than any other is possible if the number of volumes is more than the dimensions. He thought that the shadow cluster should minimize the surface area.

In the decade that followed, mathematicians wrote a series of papers proving Sullivan's hypothesis when you're trying to fit only two volumes. If you blow a double bubble in the park on a sunny day, you'll need two spherical pieces with a flat or spherical wall between them.

The mathematician Frank Morgan of Williams College thought it would take another hundred years to prove Sullivan's theory.

The mathematicians have gotten much more than just a solution to the triple bubble problem. In a paper posted online in May, the University of Texas, Austin, has proved Sullivan's hypothesis for triple bubbles in dimensions three and up and quadruple bubbles in dimensions four and up.

When it comes to six or more bubbles, it has been shown that the best cluster must have many of the key attributes of Sullivan's candidate. The essential structure behind the Sullivan conjecture has been grasped by them.

Morgan wrote in an email that the central Theorem ismonumental. It is a great achievement with lots of new ideas.

Shadow Bubbles

When it comes to small clusters, our experiences with real soap bubbles give us a good idea of what to look for. The triple or quadruple bubbles seem to have spherical walls and tend to form tight clumps instead of a long chain of bubbles.

It is difficult to prove that the features of optimal bubble clusters are true. For example, mathematicians don't know if the walls in a bubble cluster are spherical or flat, because they only know that the average curve stays the same from one point to another. Many other surfaces, such as cylinders and unduloids, have the same property. The surfaces are a complete zoo.

Sullivan noticed in the 1990s that when the number of volumes is greater than the dimensions, there is a candidate cluster that seems to beat the rest.

We can use Sullivan's approach to create a three-bubble cluster in the flat plane, so that we can see how such a candidate is built. The points are all the same distance from each other. Imagine that each point is the center of a tiny bubble, living on the surface of the sphere. When the four bubbles on the sphere start bumping into each other, inflate them until they fill the entire surface. We end up with a cluster of four bubbles that make the sphere look like it's puffed out.

The sphere is placed on top of the infinite flat plane as if it is a ball on the floor. The ball is transparent and there is a light at the north pole. Shadows will be projected on the floor by the walls of the bubbles. Three of the four bubbles on the sphere will project down to shadow bubbles on the floor, while the fourth will project down to theinfinite expanse of floor outside the cluster of three shadow bubbles.

The three-bubble cluster is dependent on how we positioned the sphere when we put it on the floor. The three bubbles on the floor will have different areas if we spin the sphere so a different point moves to the north pole. There is a single way to position the sphere so that the three shadow bubbles are in the same area.

We are free to do this process in any way we please. There is a limit to the number of bubbles we can have. We couldn't make a four bubble cluster in the plane. It is not possible to place many equidistant points on a sphere, even with higher-dimensional spheres. Sullivan only creates 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. Sullivan-style bubble clusters don't exist outside the parameters.

Sullivan's procedure gives us bubble clusters far beyond what our intuition can comprehend. It is not possible to see what is a 15-bubble in23-dimensional space. How do you imagine describing such a thing?

Sullivan's bubble candidates have a unique collection of properties reminiscent of the bubbles we see in nature. Wherever three walls meet, they form 120-degree angles as in a Y shape. Instead of being split across multiple regions, each volume lies in one region. Every bubble forms a tight cluster. Sullivan's bubbles are the only clusters that satisfy all these properties, according to mathematicians.

Sullivan was saying "let's assume beauty" when he said the clusters should be the ones that minimize surface area.

Bubble researchers have a reason to be wary of assuming that just because a solution is beautiful it is correct. There are problems where you would expect symmetry for the minimizers to fail.

There is a similar problem of filling infinite space with equal-volume bubbles in a way that reduces surface area. An elegant honeycomb-like structure was suggested by the British mathematician and physicist Lord Kelvin in the late 19th century. For more than a century, mathematicians believed this was the most likely answer, until 1993, when a pair of physicists identified a better alternative. There are many examples of weird things happening in mathematics.

A Dark Art

The double-bubble portion of Sullivan's theory had already been floating around for a century. In the decade that followed, mathematicians solved the double-bubble problem in three-dimensional space and in higher dimensions. Only in the two-dimensional plane, where the interface between bubbles are easy to understand, could they prove Sullivan's next case of triple bubbles.

The same version of Sullivan's conjecture was proved by the two men. The origin is the most expensive spot, and the farther away you get from it, the cheaper land becomes. In order to minimize the cost of the boundaries of the enclosures, the goal is to create enclosures with pre selected prices. There are applications in computer science of the bubble problem.

The Annals of Mathematics was where the proof was accepted. The pair didn't intend to call it quits. The method they used seemed promising for the classic bubble problem.

They came up with ideas for a long time. The document of notes was 200 pages long. It felt like they were making progress. It turned into, 'We tried this direction, no.'. Both mathematicians wanted to hedge their bets.

In the fall of 2015, Milman came up for a sabbatical and decided to visit Neeman so the pair could make a concerted push on the bubble problem. It's a good time to try high-risk, high-gain things.

They didn't get anything for the first few months. They decided to give themselves a simpler task than Sullivan did. The best bubble cluster will have mirror symmetry across a central plane if you give it one extradimensional breathing room.

There are triple bubbles in dimensions two and up and quadruple bubbles in dimensions three and up. They restricted their attention to triple bubbles in dimensions three and up, quadruple bubbles in dimensions four and up, and so on. When we gave up on getting it for the full range of parameters, we were able to make progress.

The half of the bubble cluster that lies above the mirror can be inflated by deflating the half that lies below it. The volume of the bubbles won't change, but the surface area might. If the optimal bubble cluster has any walls that are not spherical or flat, there will be a way to reduce the cluster's surface area.

It's not a new idea to use perturbations to study bubbles, but figuring out which are the most important features is a bit of a dark art.

Joel Hass of the University of California, Davis said that the perturbations looked natural once he saw them.

It's much simpler to recognize the perturbations as natural than it is to come up with them. He said that eventually people would have found it. At a very remarkable level, it is truly genius.

In order for the optimal bubble cluster to satisfy all of Sullivan's clusters, it must touch every other. There were many ways bubbles could connect up into a cluster. There aren't many options when it comes to just a few bubbles. As you increase the number of bubbles, the number of different possibilities grows.

They wanted to find an overarching principle that would cover all of them. They decided to content themselves for now with a more ad hoc approach that would allow them to handle triple and quadruple bubbles. They haven't established that Sullivan's quintuple bubble is the only optimal cluster.

Morgan wrote in an email that the work of Milman and Neeman is a whole new approach. It is likely that this approach can be pushed even further if there are more than five bubbles.

They have never been deterred by the fact that no one expects further progress to be easy. All of the major things that I was able to do required just not giving up.