Select Page
Can You Really Run on Top of a Train, Like in the Movies?

Can You Really Run on Top of a Train, Like in the Movies?

Just because you see something done in a movie, that doesn’t mean you should try it yourself. Take, for example, a human running on top of a moving train. For starters, you can’t be sure it’s real. In early Westerns, they used moving backdrops to make fake trains look like they were in motion. Now there’s CGI. Or they might speed the film up to make a real train look faster than it really is.

So here’s a question for you: Is it possible to run on a train roof and leap from one car to the next? Or will the train zoom ahead of you while you’re in the air, so that you land behind where you took off? Or worse, would you end up falling between the cars because the gap is moving forward, lengthening the distance you have to traverse? This, my friend, is why stunt actors study physics.

Framing the Action

What is physics anyway? Basically it’s a set of models of the real world, which we can use to calculate forces and predict how the position and velocity of things will change. However, we can’t find the position or velocity of anything without a reference frame.

Suppose I’m standing in a room, holding a ball, and I want to describe its location. I can use Cartesian coordinates for a 3D space to give the ball an (x, y, z) value. But these numbers depend on the origin and orientation of my axes. It seems natural to use a corner of the room as the origin, with x and y axes running along the base of two adjacent walls and the z axis running vertically upward. Using this system (with units in meters), I find that the ball is at the point (1, 1, 1).

What if my pal Bob is there, and he measures the ball’s location in a different way? Maybe he puts the origin where the ball starts, in my hand, giving it an initial position of (0, 0, 0). That seems logical too. We could argue about who’s right, but that would be silly. We just have different frames of reference, and they’re both arbitrary. (Don’t worry, we’ll get back to trains.)

Now I toss that ball straight up in the air. After a short time interval of 0.1 second, my coordinate system has the ball at the location (1, 1, 2), meaning it’s 1 meter higher. Bob also has a new location, (0, 0, 1). But notice that in both systems, the ball rose by 1 meter in the z direction. So we would agree that the ball has an upward velocity of 10 meters per second.

A Moving Reference Frame

Now suppose I take that ball on a train traveling at 10 meters per second (22.4 miles per hour). I again toss the ball straight up—what will happen? I’m inside the railcar, so I use a coordinate system that moves along with the train. In this moving reference frame, I am stationary. Bob is standing on the side of the tracks (he can see the ball through the windows), so he uses a stationary coordinate system, in which I am moving.

Graphic of a moving train formula

Courtesy of Rhett Allain

Never-Repeating Patterns of Tiles Can Safeguard Quantum Information

Never-Repeating Patterns of Tiles Can Safeguard Quantum Information

This extreme fragility might make quantum computing sound hopeless. But in 1995, the applied mathematician Peter Shor discovered a clever way to store quantum information. His encoding had two key properties. First, it could tolerate errors that only affected individual qubits. Second, it came with a procedure for correcting errors as they occurred, preventing them from piling up and derailing a computation. Shor’s discovery was the first example of a quantum error-correcting code, and its two key properties are the defining features of all such codes.

The first property stems from a simple principle: Secret information is less vulnerable when it’s divided up. Spy networks employ a similar strategy. Each spy knows very little about the network as a whole, so the organization remains safe even if any individual is captured. But quantum error-correcting codes take this logic to the extreme. In a quantum spy network, no single spy would know anything at all, yet together they’d know a lot.

Each quantum error-correcting code is a specific recipe for distributing quantum information across many qubits in a collective superposition state. This procedure effectively transforms a cluster of physical qubits into a single virtual qubit. Repeat the process many times with a large array of qubits, and you’ll get many virtual qubits that you can use to perform computations.

The physical qubits that make up each virtual qubit are like those oblivious quantum spies. Measure any one of them and you’ll learn nothing about the state of the virtual qubit it’s a part of—a property called local indistinguishability. Since each physical qubit encodes no information, errors in single qubits won’t ruin a computation. The information that matters is somehow everywhere, yet nowhere in particular.

“You can’t pin it down to any individual qubit,” Cubitt said.

All quantum error-correcting codes can absorb at least one error without any effect on the encoded information, but they will all eventually succumb as errors accumulate. That’s where the second property of quantum error-correcting codes kicks in—the actual error correction. This is closely related to local indistinguishability: Because errors in individual qubits don’t destroy any information, it’s always possible to reverse any error using established procedures specific to each code.

Taken for a Ride

Zhi Li, a postdoc at the Perimeter Institute for Theoretical Physics in Waterloo, Canada, was well versed in the theory of quantum error correction. But the subject was far from his mind when he struck up a conversation with his colleague Latham Boyle. It was the fall of 2022, and the two physicists were on an evening shuttle from Waterloo to Toronto. Boyle, an expert in aperiodic tilings who lived in Toronto at the time and is now at the University of Edinburgh, was a familiar face on those shuttle rides, which often got stuck in heavy traffic.

“Normally they could be very miserable,” Boyle said. “This was like the greatest one of all time.”

Before that fateful evening, Li and Boyle knew of each other’s work, but their research areas didn’t directly overlap, and they’d never had a one-on-one conversation. But like countless researchers in unrelated fields, Li was curious about aperiodic tilings. “It’s very hard to be not interested,” he said.

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.