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Texan conservatives, 'Monumental' Math Proof Solves Triple Bubble Problem and More

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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.

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.”

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, 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.”

Shadow Bubbles
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.

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.

Merrill Sherman/Quanta Magazine
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.

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.

A Dark Art
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.

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 the University of California, 
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.”

https://www.quantamagazine.org/monumental-math-proof-solves-triple-bubble-
problem-and-more-20221006/

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Texan conservatives, 'Monumental' Math Proof Solves Triple Bubble Problem and More zinn <zinn@reno.us> - 2022-10-15 08:00 +0000

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