Another unit is the bar, where 1 bar is equal to 14.5 psi. The value of 1 bar is very close to the pressure of air on Earth. The atmospheric pressure of the air that surrounds you right now is probably 14.5 psi. (Yes, I said “probably” because I don’t want to judge you. Maybe you are reading this from the top of Mount Everest, where the pressure is just 4.9 psi, because there is less air above you pushing down. If so, send me a picture.) In terms of force and area, it is equal to 100,000 newtons per square meter.
Water is also made of tiny moving molecules that act like balls, and those molecules collide with underwater objects (like people), producing pressure. Water has many more molecules than the same volume of air, which means there are more collisions to produce a greater pressure. But just like going to the top of Mount Everest decreases the air pressure, going deeper in water increases the pressure, because gravity pulls downward on the molecules of water. For every 10 meters of depth, the pressure increases by 1 bar, or 14.5 psi. That means that on a dive 20 meters (around 60 feet) below sea level, there would be a water pressure of 43.5 psi, three times greater than the air pressure at Earth’s surface.
(The fact that pressure increases with depth prevents all the ocean’s water from collapsing into an infinitely thin layer. Since the pressure is greater the deeper you go, the water underneath pushes up more than the water above it pushes down. This difference compensates for the downward gravitational force, so the water level stays constant.)
It might sound like 43.5 psi is too much for a person to handle, but it’s actually not that bad. Human bodies are very adaptable to changes in pressure. If you have been to the bottom of a swimming pool, you already know the answer to this pressure problem—your ears. If the water pressure on the outside of your eardrum is greater than the pressure from the air inside your inner ear, the membrane will stretch, and it can really hurt. But there is a nice trick to fix this: If you push air into your middle ear cavity by pinching your nose closed while attempting to blow air out of it, air will be forced into this cavity. With more air in the inner ear, the pressure on both sides of the membrane will be equal and you will feel normal. This is called “equalization,” for hopefully obvious reasons.
There’s actually another air space that you need to equalize while diving—the inside of your scuba mask. Don’t forget to add air to it as you go deeper, or that thing will awkwardly squish your face.
There is one other physics mistake a diver could make. It’s possible to create an enclosed air space in your lungs by holding your breath. Suppose you hold your breath at a depth of 20 meters and then move up to a depth of 10 meters. The pressure inside your lungs will stay the same during this ascent, because you have the same lung volume, and they contain the same amount of air. However, the water pressure outside of them will decrease. The reduced external pressure on your lungs makes it as though they are overinflated. This can cause tears in lung tissue, or even force air into the bloodstream, which is officially bad stuff.
There’s another problem to deal with when you are underwater: floating and sinking. If you want to stay underwater, it’s useful to sink instead of float—to a point. I don’t think anyone wants to sink to such depths that they never return. Also, it’s nice to be able to float when you’re at the surface. Luckily, scuba divers can change their “floatiness” for different situations. This is called buoyancy control.
Things sink when the downward-pulling gravitational force is greater than the upward-pushing buoyancy force. If these two forces are equal, then the object will be neutrally buoyant and neither rise nor sink. It’s like hovering, but in water, and it is essentially what you want to do when scuba diving.
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.”
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.
That might seem like a large volume, but it’s not. It’s almost half of a liter, so that’s half a bottle of soda.
Moles and Particles
These moles aren’t the furry creatures that make holes in the ground. The name comes from molecules (which is apparently too long to write).
Here’s an example to help you understand the idea of a mole. Suppose you run an electric current through water. A water molecule is made of one oxygen atom and two hydrogen atoms. (That’s H2O.) This electric current breaks up the water molecule, and you get hydrogen gas (H2) and oxygen gas (O2).
This is actually a pretty simple experiment. Check it out here:
Since water has twice as many hydrogen atoms as oxygen, you get twice the number of hydrogen molecules. We can see this if we collect the gases from that water: We know the ratio of the molecules, but we don’t know the number. That’s why we use moles. It’s basically just a way to count the uncountable.
Don’t worry, there is indeed a way to find the number of particles in a mole—but you need Avogadro’s number for that. If you have a liter of air at room temperature and normal pressure (we call that atmospheric pressure), then there will be about 0.04 moles. (That would be n in the ideal gas law.) Using Avogadro’s number, we get 2.4 x 1022 particles. You can’t count that high. No one can. But that’s N, the number of particles, in the other version of the ideal gas law.
Just a quick note: You almost always need some kind of constant for an equation with variables representing different things. Just look at the right side of the ideal gas law, where we have pressure multiplied by volume. The units for this left side would be newton-meters, which is the same as a joule, the unit for energy.
On the right side, there is the number of moles and the temperature in Kelvin—those two clearly do not multiply to give units of joules. But you must have the same units on both sides of the equation, otherwise it would be like comparing apples and oranges. That’s where the constant R comes to the rescue. It has units of joules/(mol × Kelvin) so that the mol × Kelvin cancels and you just get joules. Boom: Now both sides have the same units.
Now let’s look at some examples of the ideal gas law using an ordinary rubber balloon.
Inflating a Balloon
What happens when you blow up a balloon? You are clearly adding air into the system. As you do this, the balloon gets bigger, so its volume increases.
What about the temperature and the pressure inside? Let’s just assume they are constant.
I’m going to include arrows next to the variables that change. An up arrow means an increase and a down arrow means a decrease.
The inside of a tokamak—the doughnut-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 in Switzerland.
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 the 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 re-create 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 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.”
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.