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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.

The Physics of Wile E. Coyote’s 10 Billion-Volt Electromagnet

The Physics of Wile E. Coyote’s 10 Billion-Volt Electromagnet

I like to analyze the physics of science fiction, and so I’m going to argue that the Merrie Melodies cartoon “Compressed Hare” takes place in the far future when animals rule the world. I mean, Bugs Bunny and Wile E. Coyote walk on two legs, talk, and build stuff. How would that not be science fiction?

Let me set the scene—and I don’t think we have to worry about spoiler alerts since this episode is 60 years old. The basic idea is, of course, that Wile E. Coyote has decided he should eat the rabbit. After a couple of failed attempts to capture Bugs, he comes up with a new plan. First, he’s going to drop a carrot-shaped piece of iron into Bugs’ rabbit hole. After the carrot is consumed (and I have no idea how that would happen), Wile E. Coyote will turn on a giant electromagnet and pull the rabbit right to him. It’s such a simple and awesome plan, it just has to work, right?

But wait! Here’s the part that I really like: While Wile E. Coyote is assembling his contraption, we see that it comes in a huge crate labeled “One 10,000,000,000 Volt Electric Magnet Do It Yourself Kit.”

In the end, you can probably guess what happens: Bugs doesn’t actually eat the iron carrot, so once the coyote turns on the magnet, it just goes zooming toward him and into his cave. And of course a bunch of other stuff gets attracted to it, too—including a lamppost, a bulldozer, a giant cruise ship, and a rocket.

OK, let’s break down the physics of this massive electromagnet and see if this would have worked if Bugs had fallen for it.

What Is an Electromagnet?

There are essentially two ways to make a constant magnetic field. The first is with a permanent magnet, like those things that stick to your refrigerator door. These are made of some type of ferromagnetic material like iron, nickel, alnico, or neodymium. A ferromagnetic material basically contains regions that act like individual magnets, each with a north and south pole. If all these magnetic domains are aligned, the material will act like a magnet. (There’s some very complicated stuff going on at the atomic level, but let’s not worry about that right now.)

However, in this case Wile E. Coyote has an electromagnet, which creates a magnetic field with an electric current. (Note: We measure electric current in amps, which is not to be confused with voltage, which is measured in volts.) All electric currents produce magnetic fields. Normally, to make an electromagnet you would take some wire and wrap it around a ferromagnetic material, like iron, and turn the current on. The strength of its magnetic field depends on the electric current and the number of loops the wire makes around the core. It’s possible to make an electromagnet without the iron core, but it won’t be as strong.

When the electric current makes a magnetic field, this field then interacts with the magnetic domains in the piece of iron. Now that iron also acts like a magnet—the result is the electromagnet and the induced magnet attract each other.

What About 10 Billion Volts?

I don’t know how the script for this episode came about, but in my mind they had a group of writers working together. Perhaps someone came up with the idea of an electromagnet and an iron carrot and everyone agreed to put that in there. Surely someone raised their hand and said, “You know, we can’t just do an electromagnet. It has to be over-the-top big.” Another writer must have replied, “Let’s put a number there. What about 1 million volts?” Someone else interjected: “Sure, 1 million volts is cool—but what about 10 billion volts?”

What does 10 billion volts even mean for an electromagnet? Remember, the most important thing about an electromagnet is the electric current (in amps), not the voltage (in volts). To make a connection between voltage and current, we need to know the resistance. Resistance is a property that tells you how difficult it is to move electric charges through a wire, and it’s measured in ohms. If we know the resistance of the electromagnet wire, then we can use Ohm’s law to find the current. As an equation, it looks like this:

The World Is Messy. Idealizations Make the Physics Simple

The World Is Messy. Idealizations Make the Physics Simple

Sometimes the universe is just too complicated to analyze.

Heck, if you take a tennis ball and toss it across the room, even that is practically too complicated. After it leaves your hand, the ball has a gravitational interaction with the Earth, which makes it accelerate toward the ground. The ball is spinning as it moves, which means that there could be more frictional drag on one side of the ball than on the other. The ball is also colliding with some of the oxygen and nitrogen molecules in the air—and some of these molecules end up interacting with even more air. The air itself isn’t even constant—the density changes as the ball moves higher, and the air could be in motion. (We normally call that wind.) And once the ball hits the ground, even the floor isn’t perfectly flat. Yes, it looks flat, but it’s on the surface of a spherical planet.

But all is not lost. We can still model this tossed tennis ball. All we need are some idealizations. These are simplifying approximations that turn an impossible problem into a solvable problem.

In the case of the tennis ball, we can assume that all the mass is concentrated at a single point (in other words, that the ball has no actual dimensions) and that the only force acting on it is the constant downward-pulling gravitational force. Why is it OK to ignore all those other interactions? It’s because they just don’t make a significant (or even measurable) difference.

Is this even legal in the court of physics? Well, science is all about the process of building models, including the equation for the trajectory of a tennis ball. At the end of the day, if the experimental observations (where the ball lands) agrees with the model (the prediction of where it will land), then we are good to go. For the tennis ball idealization, everything works very well. In fact, the physics of a tossed ball becomes a test question in an introductory physics class. Other idealizations are harder, like trying to determine the curvature of the Earth just by looking at this super-long terminal in the Atlanta airport. But physicists do this kind of thing all the time.

Perhaps the most famous idealization was done by Galileo Galilei during his study of the nature of motion. He was trying to figure out what would happen to a moving object if you don’t exert a force on it. At the time, just about everyone followed the teachings of Aristotle, who said that if you don’t exert a force on a moving object, it will stop and remain at rest. (Even though his work was around 1,800 years old, people thought Aristotle was too cool to be wrong.)

But Galileo didn’t agree. He thought it would keep moving at a constant speed.

If you want to study an object in motion, you need to measure both position and time so that you can calculate its velocity, or its change in position divided by the change in time. But there is a problem. How do you accurately measure the time for objects moving at high speeds over short distances? If you drop something even from a relatively small height, like 10 meters, it takes fewer than 2 seconds for it to reach the ground. And back around the year 1600, when Galileo was alive, that was a pretty difficult time interval to measure. So, instead, Galileo looked at a ball rolling down a track.