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Mathematicians Roll the Dice and Get Rock-Paper-Scissors

Mathematicians Roll the Dice and Get Rock-Paper-Scissors

In their paper, posted online in late November 2022, a key part of the proof involves showing that, for the most part, it doesn’t make sense to talk about whether a single die is strong or weak. Buffett’s dice, none of which is the strongest of the pack, are not that unusual: If you pick a die at random, the Polymath project showed, it’s likely to beat about half of the other dice and lose to the other half. “Almost every die is pretty average,” Gowers said.

The project diverged from the AIM team’s original model in one respect: To simplify some technicalities, the project declared that the order of the numbers on a die matters—so, for example, 122556 and 152562 would be considered two different dice. But the Polymath result, combined with the AIM team’s experimental evidence, creates a strong presumption that the conjecture is also true in the original model, Gowers said.

“I was absolutely delighted that they came up with this proof,” Conrey said.

When it came to collections of four or more dice, the AIM team had predicted similar behavior to that of three dice: For example, if A beats B, B beats C, and C beats D, then there should be a roughly 50-50 probability that D beats A, approaching exactly 50-50 as the number of sides on the dice approaches infinity.

To test the conjecture, the researchers simulated head-to-head tournaments for sets of four dice with 50, 100, 150, and 200 sides. The simulations didn’t obey their predictions quite as closely as in the case of three dice but were still close enough to bolster their belief in the conjecture. But though the researchers didn’t realize it, these small discrepancies carried a different message: For sets of four or more dice, their conjecture is false.

“We really wanted [the conjecture] to be true, because that would be cool,” Conrey said.

In the case of four dice, Elisabetta Cornacchia of the Swiss Federal Institute of Technology Lausanne and Jan Hązła of the African Institute for Mathematical Sciences in Kigali, Rwanda, showed in a paper posted online in late 2020 that if A beats B, B beats C, and C beats D, then D has a slightly better than 50 percent chance of beating A—probably somewhere around 52 percent, Hązła said. (As with the Polymath paper, Cornacchia and Hązła used a slightly different model than in the AIM paper.)

Cornacchia and Hązła’s finding emerges from the fact that although, as a rule, a single die will be neither strong nor weak, a pair of dice can sometimes have common areas of strength. If you pick two dice at random, Cornacchia and Hązła showed, there’s a decent probability that the dice will be correlated: They’ll tend to beat or lose to the same dice. “If I ask you to create two dice which are close to each other, it turns out that this is possible,” Hązła said. These small pockets of correlation nudge tournament outcomes away from symmetry as soon as there are at least four dice in the picture.

The recent papers are not the end of the story. Cornacchia and Hązła’s paper only begins to uncover precisely how correlations between dice unbalance the symmetry of tournaments. In the meantime, though, we know now that there are plenty of sets of intransitive dice out there—maybe even one that’s subtle enough to trick Bill Gates into choosing first.

Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.

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.

A Wheel Made of ‘Odd Matter’ Spontaneously Rolls Uphill

A Wheel Made of ‘Odd Matter’ Spontaneously Rolls Uphill

In a physics lab in Amsterdam, there’s a wheel that can spontaneously roll uphill by wiggling.

This “odd wheel” looks simple: just six small motors linked together by plastic arms and rubber bands to form a ring about 6 inches in diameter. When the motors are powered on, it starts writhing, executing complicated squashing and stretching motions and occasionally flinging itself into the air, all the while slowly making its way up a bumpy foam ramp.

“I find it very playful,” said Ricard Alert, a biophysicist at the Max Planck Institute for the Physics of Complex Systems in Dresden, Germany, who was not involved in making the wheel. “I liked it a lot.”

The odd wheel’s unorthodox mode of travel exemplifies a recent trend: Physicists are finding ways to get useful collective behavior to spontaneously emerge in robots assembled from simple parts that obey simple rules. “I’ve been calling it robophysics,” said Daniel Goldman, a physicist at the Georgia Institute of Technology.

The problem of locomotion—one of the most elementary behaviors of living things—has long preoccupied biologists and engineers alike. When animals encounter obstacles and rugged terrain, we instinctively take these challenges in stride, but how we do this is not so simple. Engineers have struggled to build robots that won’t collapse or lurch forward when navigating real-world environments, and they can’t possibly program a robot to anticipate all the challenges it might encounter.

The odd wheel, developed by the physicists Corentin Coulais of the University of Amsterdam and Vincenzo Vitelli of the University of Chicago and collaborators and described in a recent preprint, embodies a very different approach to locomotion. The wheel’s uphill movement emerges from simple oscillatory motion in each of its component parts. Although these parts know nothing about the environment, the wheel as a whole automatically adjusts its wiggling motion to compensate for uneven terrain.

Energy generated during each cyclical oscillation of the odd wheel allows it to push off against the ground and roll upward and over obstacles. (Another version of the wheel with only six motors was studied in a recent paper.)Video: Corentin Coulais

The physicists also created an “odd ball” that always bounces to one side and an “odd wall” that controls where it absorbs energy from an impact. The objects all stem from the same equation describing an asymmetric relationship between stretching and squashing motions that the researchers identified two years ago.

“These are indeed behaviors you would not expect,” said Auke Ijspeert, a bioroboticist at the Swiss Federal Institute of Technology Lausanne. Coulais and Vitelli declined to comment while their latest paper is under peer review.

In addition to guiding the design of more robust robots, the new research may prompt insights into the physics of living systems and inspire the development of novel materials.

Odd Matter

The odd wheel grew out of Coulais and Vitelli’s past work on the physics of “active matter”—an umbrella term for systems whose constituent parts consume energy from the environment, such as swarms of bacteria, flocks of birds and certain artificial materials. The energy supply engenders rich behavior, but it also leads to instabilities that make active matter difficult to control.

Vincenzo Vitelli

Vincenzo Vitelli of the University of Chicago.Courtesy of Kristen Norman

Physicists have historically focused on systems that conserve energy, which must obey principles of reciprocity: If there’s a way for such a system to gain energy by moving from A to B, any process that takes the system from B back to A must cost an equal amount of energy. But with a constant influx of energy from within, this constraint no longer applies.

In a 2020 paper in Nature Physics, Vitelli and several collaborators began to investigate active solids with nonreciprocal mechanical properties. They developed a theoretical framework in which nonreciprocity manifested in the relationships between different kinds of stretching and squashing motions. “That to me was just a beautiful mathematical framework,” said Nikta Fakhri, a biophysicist at the Massachusetts Institute of Technology.

How Realistic Is the Celestial Navigation in Moon Knight?

How Realistic Is the Celestial Navigation in Moon Knight?

Our planet also changes positions. In six months, the Earth will go from one side of the sun to the other. This is a change in distance of almost 300 million kilometers, and it’s enough to cause a noticeable apparent position change for some of the nearest stars. In fact, parallax is an important tool for measuring the distance to these stars. (Here are the other ways to measure stellar distances.)

So, yes, constellations change—but not that much.

Finding Your Longitude

Here’s how to find your longitude with a clock and a star chart. Let’s start with the star chart. Suppose there is a star on that chart that will always be directly above a point in Greenwich, England, at 4 am local time, which we would call Greenwich Mean Time. (I didn’t pick Greenwich at random. The prime meridian, or the 0 degree longitude line, runs right through the Royal Observatory Greenwich, so it’s good for measurements.)

Now let’s imagine that you are in another location and trying to figure out where you are by using that same star. You will need to know what time it is when that star appears directly overhead at your location. Hence the clock.

Checking the time reveals that, where you are, that star appears directly overhead at 1 am, instead of 4 am—three hours earlier than Greenwich. That means you are three out of 24 hours to the west of Geenwich. If you want to convert that to degrees, it would be (3/24) × 360 = 45 degrees. That would put you on a longitude line that runs through Greenland and Brazil. (Things can get a bit more complicated than this, since you likely wouldn’t have a star directly overhead, but you get the idea.)

Next, if you are in the northern hemisphere, you can use the North Star to calculate your latitude and determine your exact location on the planet, which is where those latitude and longitude lines cross. Hopefully, it’s not in the middle of the Atlantic Ocean.

What’s Wrong with Moon Knight?

Now it’s time to talk about Moon Knight. (Some spoilers ahead.) In episode 3, Moon Knight, the earthly avatar of Khonshu, has teamed up with Marc’s wife, Layla. They are trying to find the tomb of the Egyptian god Ammit. If Ammit is freed, she will do some bad stuff to the human race, so they really want to get there first. They put together parts of a burial shroud to form an ancient star chart, and want to use this to find the location of the tomb, which is just like celestial navigation.

But there is a problem: This map was made 2,000 years ago, so the arrangement of the constellations is wrong. The stars have since moved to new positions. Since Moon Knight is the avatar of Khonshu, he uses his powers to move the stars in the sky back into the pattern shown when the map was created. Problem solved. Moon Knight and Layla are able to get to Ammit’s tomb.

Could Anyone Do Luke’s Plank Flip From ‘Return of the Jedi’?

Could Anyone Do Luke’s Plank Flip From ‘Return of the Jedi’?

It’s May 4, so happy Star Wars Day—may the fourth be with you!

One of the iconic scenes from Star Wars: Return of the Jedi is the battle on Tatooine at the Sarlacc Pit, the home of a massive creature that just waits to eat the things that fall into its sand hole. (No spoiler alert: It’s been almost 30 years since Return of the Jedi hit the theaters. If you haven’t seen it by now, you probably aren’t going to.)

Luke Skywalker is being held captive by Jabba the Hutt’s guards. They’re on a skiff above the Sarlacc Pit, and Luke is standing on a plank, about to be pushed into the creature’s maw. R2-D2 is some distance away on Jabba’s sail barge—and he has been keeping Luke’s lightsaber. Now for the best part: At just the right moment, R2 launches Luke’s lightsaber so that it flies across the pit for Luke to catch. As that happens, Luke jumps off the plank and spins around. He catches the edge of the plank and uses it to springboard himself into a flip back onto the skiff. Now the battle begins.

I’m going to look at these two motions—the lightsaber toss and the plank flip—and see if it’s possible for an ordinary human to do this, or if you have to be a Jedi like Luke. But I am going to make one big assumption about this scene, and you might not like it. I’m going to assume that the planet Tatooine has the same surface gravity as Earth, so that g = 9.8 newtons per kilogram. This would mean that a jumping human and a thrown lightsaber would follow similar trajectories on both planets.

Oh, I get it: Tatooine is not the same as Earth. However, in the movie it looks a lot like Earth (you know why), and this allows me to make some actual calculations. Let’s do it.

Motion of a Lightsaber

I’m going to start with the lightsaber that R2-D2 launches towards Luke. What can we figure from this part of the action sequence? Well, let’s start with some data.

First I’m going to get the total flight time as the lightsaber moves from R2 to Luke. The simplest way to do this is to use a video analysis program; my favorite is Tracker. With this, I can mark the video frame that shows the weapon leaving R2-D2’s head (which is kind of weird when you think about it) and then mark the frame where it gets to Luke. This gives a flight time of 3.84 seconds.

I’m going to assume that’s not the actual flight time. Why? First, it’s a pretty long time for the lightsaber to be in the air. Also, there’s quite a bit happening during that shot. In the sequence seen in the movie, R2-D2 shoots the saber and we see it rising. Cut to Luke doing a front flip onto the skiff. Cut to Luke landing, then a shot of the lightsaber falling towards him. The final shot shows Luke’s hand catching the weapon. That’s a lot of cuts, and so it might not be a real-time sequence. Don’t worry, that’s fine. That’s what movie directors do.