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'Monumental' Math Proof Solves Triple Bubble Problem and More

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'Monumental' Math Proof Solves Triple Bubble Problem and More

By Erica Klarreich

October 6, 2022

The decades-old Sullivan's conjecture, about the best way to minimize
the surface area of a bubble cluster, was thought to be out of reach
for three bubbles and up -- until a new breakthrough result.

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[Multi-Bubb]

Lawrence Lawry/Science Source

[Klarreich_]

Erica Klarreich

Contributing Correspondent

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October 6, 2022

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[Article-De][HP-Design-]

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."

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[john-sulli]

John Sullivan, shown here in 2008, conjectured 27 years ago that
optimal bubble clusters in certain settings are equivalent to the
shadows of symmetric bubbles covering a sphere.

Ulrich Dahl

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.

[emanuel-mi]

Emanuel Milman (left) of the Technion in Haifa, Israel, and Joe
Neeman of the University of Texas, Austin.

Courtesy of Emanuel Milman; Holland Photo Imaging

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."

[Klarreich_]

Erica Klarreich

Contributing Correspondent

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October 6, 2022

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