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A New Proof Moves the Needle on a Sticky Geometry Problem

A New Proof Moves the Needle on a Sticky Geometry Problem

The original version of this story appeared in Quanta Magazine.

In 1917, the Japanese mathematician Sōichi Kakeya posed what at first seemed like nothing more than a fun exercise in geometry. Lay an infinitely thin, inch-long needle on a flat surface, then rotate it so that it points in every direction in turn. What’s the smallest area the needle can sweep out?

If you simply spin it around its center, you’ll get a circle. But it’s possible to move the needle in inventive ways, so that you carve out a much smaller amount of space. Mathematicians have since posed a related version of this question, called the Kakeya conjecture. In their attempts to solve it, they have uncovered surprising connections to harmonic analysis, number theory, and even physics.

“Somehow, this geometry of lines pointing in many different directions is ubiquitous in a large portion of mathematics,” said Jonathan Hickman of the University of Edinburgh.

But it’s also something that mathematicians still don’t fully understand. In the past few years, they’ve proved variations of the Kakeya conjecture in easier settings, but the question remains unsolved in normal, three-dimensional space. For some time, it seemed as if all progress had stalled on that version of the conjecture, even though it has numerous mathematical consequences.

Now, two mathematicians have moved the needle, so to speak. Their new proof strikes down a major obstacle that has stood for decades—rekindling hope that a solution might finally be in sight.

What’s the Small Deal?

Kakeya was interested in sets in the plane that contain a line segment of length 1 in every direction. There are many examples of such sets, the simplest being a disk with a diameter of 1. Kakeya wanted to know what the smallest such set would look like.

He proposed a triangle with slightly caved-in sides, called a deltoid, which has half the area of the disk. It turned out, however, that it’s possible to do much, much better.

The deltoid to the right is half the size of the circle, though both needles rotate through every direction.Video: Merrill Sherman/Quanta Magazine

In 1919, just a couple of years after Kakeya posed his problem, the Russian mathematician Abram Besicovitch showed that if you arrange your needles in a very particular way, you can construct a thorny-looking set that has an arbitrarily small area. (Due to World War I and the Russian Revolution, his result wouldn’t reach the rest of the mathematical world for a number of years.)

To see how this might work, take a triangle and split it along its base into thinner triangular pieces. Then slide those pieces around so that they overlap as much as possible but protrude in slightly different directions. By repeating the process over and over again—subdividing your triangle into thinner and thinner fragments and carefully rearranging them in space—you can make your set as small as you want. In the infinite limit, you can obtain a set that mathematically has no area but can still, paradoxically, accommodate a needle pointing in any direction.

“That’s kind of surprising and counterintuitive,” said Ruixiang Zhang of the University of California, Berkeley. “It’s a set that’s very pathological.”

When You Drop a Rock Overboard, What Happens to the Water Level?

When You Drop a Rock Overboard, What Happens to the Water Level?

Physics questions are the most fun when people don’t immediately agree on the answer. What feels intuitive or obvious—sometimes isn’t. We can argue over the solution for hours of entertainment, and we might even learn something in the end.

Here’s one of these seemingly obvious questions that’s been around a long time: Suppose a large rock is on a boat that is floating in a very small pond. If the rock is dumped overboard, will the water level of the pond rise, fall, or remain unchanged?

Go ahead and debate it with your friends and family. While you convince them that your answer is correct, here is a picture of my boat with a rock in it:

measuring beaker in bigger measuring beaker

Photograph: Rhett Allain

OK, it’s not actually a boat, it’s part of a plastic bottle. Also, the “rock” is a lead weight and the “pond” is a beaker. But this way we can see what happens to the water level when we drop an object into it.

When a boat is floating on water, two forces are acting on it. First, there is the downward-pulling gravitational force, which is equal to the mass of the boat and everything on it (m) times the gravitational field (g = 9.8 newtons per kilogram). We often call this product the “weight.”

The other force is the upward-pushing buoyancy interaction with the water. Two things are true about this buoyancy force. First, if the boat is floating, then the upward buoyancy must be equal to the weight of the boat. Second, the buoyancy force is equal to the weight of the water displaced by the boat.

We can calculate this buoyancy force by taking the volume of the water displaced (Vd) and using the density of water (ρw) along with the gravitational field (g).

Fb equal pw time Vd times g

Illustration: Rhett Allain

The New Math of Wrinkling Patterns

The New Math of Wrinkling Patterns

A few minutes into a 2018 talk at the University of Michigan, Ian Tobasco picked up a large piece of paper and crumpled it into a seemingly disordered ball of chaos. He held it up for the audience to see, squeezed it for good measure, then spread it out again.

“I get a wild mass of folds that emerge, and that’s the puzzle,” he said. “What selects this pattern from another, more orderly pattern?”

He then held up a second large piece of paper—this one pre-folded into a famous origami pattern of parallelograms known as the Miura-ori—and pressed it flat. The force he used on each sheet of paper was about the same, he said, but the outcomes couldn’t have been more different. The Miura-ori was divided neatly into geometric regions; the crumpled ball was a mess of jagged lines.

“You get the feeling that this,” he said, pointing to the scattered arrangement of creases on the crumpled sheet, “is just a random disordered version of this.” He indicated the neat, orderly Miura-ori. “But we haven’t put our finger on whether or not that’s true.”

Making that connection would require nothing less than establishing universal mathematical rules of elastic patterns. Tobasco has been working on this for years, studying equations that describe thin elastic materials—stuff that responds to a deformation by trying to spring back to its original shape. Poke a balloon hard enough and a starburst pattern of radial wrinkles will form; remove your finger and they will smooth out again. Squeeze a crumpled ball of paper and it will expand when you release it (though it won’t completely uncrumple). Engineers and physicists have studied how these patterns emerge under certain circumstances, but to a mathematician those practical results suggest a more fundamental question: Is it possible to understand, in general, what selects one pattern rather than another?

In January 2021, Tobasco published a paper that answered that question in the affirmative—at least in the case of a smooth, curved, elastic sheet pressed into flatness (a situation that offers a clear way to explore the question). His equations predict how seemingly random wrinkles contain “orderly” domains, which have a repeating, identifiable pattern. And he cowrote a paper, published in August, that shows a new physical theory, grounded in rigorous mathematics, that could predict patterns in realistic scenarios.

Notably, Tobasco’s work suggests that wrinkling, in its many guises, can be seen as the solution to a geometric problem. “It is a beautiful piece of mathematical analysis,” said Stefan Müller of the University of Bonn’s Hausdorff Center for Mathematics in Germany.

It elegantly lays out, for the first time, the mathematical rules—and a new understanding—behind this common phenomenon. “The role of the math here was not to prove a conjecture that physicists had already made,” said Robert Kohn, a mathematician at New York University’s Courant Institute, and Tobasco’s graduate school adviser, “but rather to provide a theory where there was previously no systematic understanding.”

Stretching Out

The goal of developing a theory of wrinkles and elastic patterns is an old one. In 1894, in a review in Nature, the mathematician George Greenhill pointed out the difference between theorists (“What are we to think?”) and the useful applications they could figure out (“What are we to do?”).

In the 19th and 20th centuries, scientists largely made progress on the latter, studying problems involving wrinkles in specific objects that are being deformed. Early examples include the problem of forging smooth, curved metal plates for seafaring ships, and trying to connect the formation of mountains to the heating of the Earth’s crust.

DeepMind Has Trained an AI to Control Nuclear Fusion

DeepMind Has Trained an AI to Control Nuclear Fusion

The inside of a tokamak—the donut-shaped vessel designed to contain a nuclear fusion reaction—presents a special kind of chaos. Hydrogen atoms are smashed together at unfathomably high temperatures, creating a whirling, roiling plasma that’s hotter than the surface of the sun. Finding smart ways to control and confine that plasma will be key to unlocking the potential of nuclear fusion, which has been mooted as the clean energy source of the future for decades. At this point, the science underlying fusion seems sound, so what remains is an engineering challenge. “We need to be able to heat this matter up and hold it together for long enough for us to take energy out of it,” says Ambrogio Fasoli, director of the Swiss Plasma Center at École Polytechnique Fédérale de Lausanne.

That’s where DeepMind comes in. The artificial intelligence firm, backed by Google parent company Alphabet, has previously turned its hand to video games and protein folding, and has been working on a joint research project with the Swiss Plasma Center to develop an AI for controlling a nuclear fusion reaction.

In stars, which are also powered by fusion, the sheer gravitational mass is enough to pull hydrogen atoms together and overcome their opposing charges. On Earth, scientists instead use powerful magnetic coils to confine the nuclear fusion reaction, nudging it into the desired position and shaping it like a potter manipulating clay on a wheel. The coils have to be carefully controlled to prevent the plasma from touching the sides of the vessel: this can damage the walls and slow down the fusion reaction. (There’s little risk of an explosion as the fusion reaction cannot survive without magnetic confinement).

But every time researchers want to change the configuration of the plasma and try out different shapes that may yield more power or a cleaner plasma, it necessitates a huge amount of engineering and design work. Conventional systems are computer-controlled and based on models and careful simulations, but they are, Fasoli says, “complex and not always necessarily optimized.”

DeepMind has developed an AI that can control the plasma autonomously. A paper published in the journal Nature describes how researchers from the two groups taught a deep reinforcement learning system to control the 19 magnetic coils inside TCV, the variable-configuration tokamak at the Swiss Plasma Center, which is used to carry out research that will inform the design of bigger fusion reactors in future. “AI, and specifically reinforcement learning, is particularly well suited to the complex problems presented by controlling plasma in a tokamak,” says Martin Riedmiller, control team lead at DeepMind.

The neural network—a type of AI setup designed to mimic the architecture of the human brain—was initially trained in a simulation. It started by observing how changing the settings on each of the 19 coils affected the shape of the plasma inside the vessel. Then it was given different shapes to try to recreate in the plasma. These included a D-shaped cross-section close to what will be used inside ITER (formerly the International Thermonuclear Experimental Reactor), the large-scale experimental tokamak under construction in France, and a snowflake configuration that could help dissipate the intense heat of the reaction more evenly around the vessel.

DeepMind’s neural network was able to manipulate the plasma inside a fusion reactor into a number of different shapes that fusion researchers have been exploring.Illustration: DeepMind & SPC/EPFL 

DeepMind’s AI was able to autonomously figure out how to create these shapes by manipulating the magnetic coils in the right way—both in the simulation, and when the scientists ran the same experiments for real inside the TCV tokamak to validate the simulation. It represents a “significant step,” says Fasoli, one that could influence the design of future tokamaks or even speed up the path to viable fusion reactors. “It’s a very positive result,” says Yasmin Andrew, a fusion specialist at Imperial College London, who was not involved in the research. “It will be interesting to see if they can transfer the technology to a larger tokamak.”

Fusion offered a particular challenge to DeepMind’s scientists because the process is both complex and continuous. Unlike a turn-based game like Go, which the company has famously conquered with its AlphaGo AI, the state of a plasma constantly changes. And to make things even harder, it can’t be continuously measured. It is what AI researchers call an “under–observed system.”

“Sometimes algorithms which are good at these discrete problems struggle with such continuous problems,” says Jonas Buchli, a research scientist at DeepMind. “This was a really big step forward for our algorithm because we could show that this is doable. And we think this is definitely a very, very complex problem to be solved. It is a different kind of complexity than what you have in games.”

Physicists Created Bubbles That Can Last for Over a Year

Physicists Created Bubbles That Can Last for Over a Year

Blowing soap bubbles never fails to delight one’s inner child, perhaps because bubbles are intrinsically ephemeral, bursting after just a few minutes. Now, French physicists have succeeded in creating “everlasting bubbles” out of plastic particles, glycerol, and water, according to a new paper published in the journal Physical Review Fluids. The longest bubble they built survived for a whopping 465 days.

Bubbles have long fascinated physicists. For instance, French physicists in 2016 worked out a theoretical model for the exact mechanism for how soap bubbles form when jets of air hit a soapy film. The researchers found that bubbles only formed above a certain speed, which in turn depends on the width of the jet of air.

In 2018, we reported on how mathematicians at New York University’s Applied Math Lab had fine-tuned the method for blowing the perfect bubble based on a series of experiments with thin, soapy films. The mathematicians concluded that it’s best to use a circular wand with a 1.5-inch (3.8 cm) perimeter and gently blow at a consistent 2.7 inches per second (6.9 cm/s). Blow at higher speeds and the bubble will burst. If you use a smaller or larger wand, the same thing will happen.

And in 2020, physicists determined that a key ingredient for creating gigantic bubbles is mixing in polymers of varying strand lengths. That produces a soap film able to stretch sufficiently thin to make a giant bubble without breaking. The polymer strands become entangled, like a hairball, forming longer strands that don’t want to break apart. In the right combination, a polymer allows a soap film to reach a ‘sweet spot’ that’s viscous but also stretchy—just not so stretchy that it rips apart. Varying the length of the polymer strands resulted in a sturdier soap film.

Scientists are also interested in extending the longevity of bubbles. Bubbles naturally take on the form of a sphere: a volume of air encased in a very thin liquid skin that isolates each bubble in a foam from its neighbors. Bubbles owe their geometry to the phenomenon of surface tension, a force that arises from molecular attraction. The greater the surface area, the more energy is required to maintain a given shape, which is why the bubbles seek to assume the shape with the least surface area: a sphere.

However, most bubbles burst within minutes in a standard atmosphere. Over time, the pull of gravity gradually drains the liquid downward, and at the same time, the liquid component slowly evaporates. As the amount of liquid decreases, the “walls” of the bubbles become very thin, and small bubbles in a foam combine into larger ones. The combination of these two effects is called “coarsening.” Adding some kind of surfactant keeps surface tension from collapsing bubbles by strengthening the thin liquid film walls that separate them. But eventually the inevitable always occurs.

In 2017, French physicists found that a spherical shell made of plastic microspheres can store pressurized gas in a tiny volume. The physicists dubbed the objects “gas marbles.” The objects are related to so-called liquid marbles—droplets of liquid coated with microscopic, liquid-repelling beads, which can roll around on a solid surface without breaking apart. While the mechanical properties of gas marbles have been the subject of several studies, no one had conducted experiments to explore the marbles’ longevity.