When the Hitomi space telescope successfully reached orbit in early 2016, astronomers thought their string of bad luck was over. A predecessor, launched in 2000, had crashed into the ocean. A follow-up mission leaked helium; full operations lasted only a few weeks. Hitomi, an X-ray satellite with the unique ability to sort some of the universe’s most energetic photons by energy, represented astrophysicists’ latest chance to demystify the behavior of colossal galaxies. The moment it became operational, the spacecraft trained its eye on a cluster of galaxies in the constellation Perseus.

Scientists had long wondered why these energetically “active” galaxies — filled with gas measuring millions of degrees — don’t churn out stars at a more ferocious rate. As their hot clouds blast X-rays into space, the gas should cool, like a hot poker losing its glow. In time, the gas should settle and clump together, forming stars. “There should be huge galaxy building going on in the middle of these systems,” said Mark Voit, an astronomer at Michigan State University. But “we see nothing of the sort.” Neither do the galaxies heat up. Something appears to be keeping them Goldilocks warm.

Astrophysicists had long assumed that the supermassive black hole at the heart of each of the galaxies was micromanaging the thermostat. But how could such a tiny speck, comparable in size to our own solar system, drive the behavior of a galaxy hundreds of thousands of light-years across?

Researchers suspected that vast jets of energy, expelled from either side of the black hole, could stir up turbulence in the clouds to keep everything warm. Hitomi was launched in part to spot that turbulence.

Yet when Hitomi turned to Perseus, the gas clouds seemed too quiet, putting stress on the idea that turbulence was enough to do the job. Data collected in just the first few days of observations challenged a leading idea in astrophysics.

Then disaster struck.

Before researchers could turn Hitomi toward other galaxies for confirmation, the spacecraft tumbled out of control and spun itself into pieces, a victim of bad luck and ruinous engineering. A brief observation of Perseus was the main result to come out of the $300 million mission. “It’s a sad story,” said Voit. “We’ve been eager to have this capability to do good X-ray spectroscopy for decades.”

Now a new study has challenged Hitomi’s conclusions. By sidestepping the hard-to-get X-ray data, it finds that turbulence might still be responsible for the Goldilocks nature of these active galaxies.

If you want to study X-ray-spewing gas, a dedicated X-ray satellite is a nice thing to have. But researchers have come up with a way to probe for turbulence without the use of X-rays.

Yuan Li, a theoretical astrophysicist at the University of California, Berkeley, had originally planned to study special bubbles that lurk near the central supermassive black holes of massive galaxies. She wanted to check if the bubbles’ shapes matched her theories. Since the bubbles are cooler than the surrounding gas, they shine in visible light, which means you don’t need a fancy X-ray satellite to see these denser “filaments.” She searched the existing data and found observations of filaments at the center of three nearby galaxy clusters — Perseus, Abell 2597 and Virgo.

When she got the data, she realized there was also information about which parts of the filaments were moving in which directions. She believed that if she tracked how the bubbles move, she could discern what was happening in the invisible gas — a strategy akin to using fluttering leaves to map the wind. “It’s a whole new way of looking at this issue,” Voit said.

The major question was turbulence, a specific pattern of movement where big bursts of energy break up into smaller whorls, which then split into smaller eddies, distributing energy all the way down to the scale of individual particles. Turbulence could explain how energy was getting from the black hole’s jets — enormous fountains of particles, thousands of light-years long — into particle-level jitters. Turbulence could be how the black hole kept the galaxy warm.

Li and her colleagues searched for the fingerprints of turbulence by coming up with a long list of pairs of locations in the filaments. Some of the pairs were close together, others farther apart. For each pair, they compared the relative speed of the gas at each point to the distance between the points.

If a filament was all moving in a uniform direction — falling toward the black hole, for instance — then points at all distances apart would appear to be moving at the same speed. But if points distant from one another moved relatively quickly, while nearby points moved slowly, then the filaments would be moving in complicated ways, with eddies of all sizes.

That’s exactly what the team found, as they describe in research to be published in *The Astrophysical Journal Letters*. The more distant two spots are, the faster they move.

Li marveled at how it was precisely the kind of intricate swirling we might expect from turbulent gas clouds pushing the filaments around. “It was almost a little disturbing how small the uncertainties were,” she said. “The data itself is truly beautiful.”

The existence of movements ranging from small to large represents, according to Li, a smoking gun indicating that turbulence is responsible for taking galactic-level energy and grinding it down to the particle level. “This [research] is telling us that black holes can drive turbulence,” she said.

Yet others would like to see more detailed modeling of the cascade process before they agree with the conclusions. “It’s interesting,” said Brian McNamara, an astrophysicist at the University of Waterloo in Canada, “but I think they need to go deeper.”

Li strengthens her case by tracing this energy flow back to the central black hole. The largest swirls of filament (and presumably the largest gas swirls) measure a few thousand light-years in size, matching the lengthy jets. If some other, larger cosmic spoon had stirred up the turbulence instead, the team would have found larger eddies.

As for the precise role of turbulence, debate continues. In textbook turbulence, energy cascades from large to small in a particular way — eddies of a given size generate whorls of a certain type. In Li’s filaments, however, the smaller eddies don’t turn quite quickly enough to satisfy the classic definition. Energy fails to reach the small parts of the filaments as efficiently as it should if turbulence alone is doing the distributing.

One turbulence-saving explanation lies in the poorly understood nature of the gas clouds. The clouds are made of a plasma of charged particles so thin that any one particle might drift for dozens of light-years without encountering another. In such wispy veils, energy may well disappear into heat earlier than theory predicts.

Or the appearance of turbulence could be an illusion. Just because the galaxies have large-scale and small-scale motion, said Christopher Reynolds, an astrophysicist at the University of Cambridge, that doesn’t guarantee that turbulence is turning the first into the second. The slower, finer motion could represent thumping sound waves from the black hole — shock waves produced when the jets slam into the surrounding gas.

Li, for her part, believes that true turbulence generates at least some heat. “The real picture is more complicated than what we had previously thought,” she said. While classical turbulence comes from incessant stirring, she pointed out, black hole jets switch on and off, which could also explain the unconventional energy cascade.

Regardless of how much heat comes down to turbulence, researchers hope to further study filaments for their role in tying the whole galactic cycle together. Jets feed the hot gas, which condenses into the cooler clumps, which may eventually fall into the black hole to power future jets. How each step of this cycle really works remains anyone’s guess, but the filaments are likely one of the keys to how, in galaxy after galaxy, the central black hole puts out just enough energy to avoid cooling, but not so much as to trigger runaway heating.

Nailing down the details of that cycle will take both computer simulations and wider observations of filaments. In addition, X-ray images of the invisible gas itself should directly reveal what’s going on. For that, astrophysicists will have to wait for Hitomi’s replacement, currently scheduled for launch in 2021. Fingers crossed.

*Editor’s note: Li has received funding from the Simons Foundation, which also funds this **editorially independent magazine**.*

Experienced mathematicians warn up-and-comers to stay away from the Collatz conjecture. It’s a siren song, they say: Fall under its trance and you may never do meaningful work again.

The Collatz conjecture is quite possibly the simplest unsolved problem in mathematics — which is exactly what makes it so treacherously alluring.

“This is a really dangerous problem. People become obsessed with it and it really is impossible,” said Jeffrey Lagarias, a mathematician at the University of Michigan and an expert on the Collatz conjecture.

Earlier this year one of the top mathematicians in the world dared to confront the problem — and came away with one of the most significant results on the Collatz conjecture in decades.

On September 8, Terence Tao posted a proof showing that — at the very least — the Collatz conjecture is “almost” true for “almost” all numbers. While Tao’s result is not a full proof of the conjecture, it is a major advance on a problem that doesn’t give up its secrets easily.

“I wasn’t expecting to solve this problem completely,” said Tao, a mathematician at the University of California, Los Angeles. “But what I did was more than I expected.”

Lothar Collatz likely posed the eponymous conjecture in the 1930s. The problem sounds like a party trick. Pick a number, any number. If it’s odd, multiply it by 3 and add 1. If it’s even, divide it by 2. Now you have a new number. Apply the same rules to the new number. The conjecture is about what happens as you keep repeating the process.

Intuition might suggest that the number you start with affects the number you end up with. Maybe some numbers eventually spiral all the way down to 1. Maybe others go marching off to infinity.

But Collatz predicted that’s not the case. He conjectured that if you start with a positive whole number and run this process long enough, all starting values will lead to 1. And once you hit 1, the rules of the Collatz conjecture confine you to a loop: 1, 4, 2, 1, 4, 2, 1, on and on forever.

Over the years, many problem solvers have been drawn to the beguiling simplicity of the Collatz conjecture, or the “3*x *+ 1 problem,” as it’s also known. Mathematicians have tested quintillions of examples (that’s 18 zeros) without finding a single exception to Collatz’s prediction. You can even try a few examples yourself with any of the many “Collatz calculators” online. The internet is awash in unfounded amateur proofs that claim to have resolved the problem one way or the other.

“You just need to know multiplying by 3 and dividing by 2 and you can start playing around with it right away. It’s very tempting to try,” said Marc Chamberland, a mathematician at Grinnell College who produced a popular YouTube video on the problem called “The Simplest Impossible Problem.”

But legitimate proofs are rare.

In the 1970s, mathematicians showed that almost all Collatz sequences — the list of numbers you get as you repeat the process — eventually reach a number that’s smaller than where you started — weak evidence, but evidence nonetheless, that almost all Collatz sequences incline toward 1. From 1994 until Tao’s result this year, Ivan Korec held the record for showing just how much smaller these numbers get. Other results have similarly picked at the problem without coming close to addressing the core concern.

“We really don’t understand the Collatz question well at all, so there hasn’t been much significant work on it,” said Kannan Soundararajan, a mathematician at Stanford University who has worked on the conjecture.

The futility of these efforts has led many mathematicians to conclude that the conjecture is simply beyond the reach of current understanding — and that they’re better off spending their research time elsewhere.

“Collatz is a notoriously difficult problem — so much so that mathematicians tend to preface every discussion of it with a warning not to waste time working on it,” said Joshua Cooper of the University of South Carolina in an email.

Lagarias first became intrigued by the conjecture as a student at least 40 years ago. For decades he has served as the unofficial curator of all things Collatz. He’s amassed a library of papers related to the problem, and in 2010 he published some of them as a book titled *The Ultimate Challenge: The 3*x* + 1 Problem*.

“Now I know lots more about the problem, and I’d say it’s still impossible,” Lagarias said.

For Tao, this goal had the same flavor as investigating whether you always eventually get the same number (1) from the Collatz process no matter what number you feed in. As a result, he recognized that techniques for studying PDEs could apply to the Collatz conjecture.

One particularly useful technique involves a statistical way of studying the long-term behavior of a small number of starting values (like a small number of initial configurations of the water in a pond) and extrapolating from there to the long-term behavior of all possible starting configurations of the pond.

In the context of the Collatz conjecture, imagine starting with a large sample of numbers. Your goal is to study how these numbers behave when you apply the Collatz process. If close to 100% of the numbers in the sample end up either exactly at 1 or very close to 1, you might conclude that almost all numbers behave the same way.

But for the conclusion to be valid, you’d have to construct your sample very carefully. The challenge is akin to generating a sample of voters in a presidential poll. To extrapolate accurately from the poll to the population as a whole, you’d need to weight the sample with the correct proportion of Republicans and Democrats, women and men, and so on.

Numbers have their own “demographic” characteristics. There are odd and even numbers, of course, and numbers that are multiples of 3, and numbers that differ from each other in even subtler ways. When you construct a sample of numbers, you can weight it toward containing certain kinds of numbers and not others — and the better you choose your weights, the more accurately you’ll be able to draw conclusions about numbers as a whole.

Tao’s challenge was much harder than just figuring out how to create an initial sample of numbers with the proper weights. At each step in the Collatz process, the numbers you’re working with change. One obvious change is that almost all numbers in the sample get smaller.

Another, maybe less obvious change is that the numbers might start to clump together. For example, you could begin with a nice, uniform distribution like the numbers from 1 to 1 million. But five Collatz iterations later, the numbers are likely to be concentrated in a few small intervals on the number line. In other words, you may start out with a good sample, but five steps later it’s hopelessly skewed.

“Ordinarily one would expect the distribution after the iteration to be completely different from the one you started with,” said Tao in an email.

Tao’s key insight was figuring out how to choose a sample of numbers that largely retains its original weights throughout the Collatz process.

For example, Tao’s starting sample is weighted to contain no multiples of 3, since the Collatz process quickly weeds out multiples of 3 anyway. Some of the other weights Tao came up with are more complicated. He weights his starting sample toward numbers that have a remainder of 1 after being divided by 3, and away from numbers that have a remainder of 2 after being divided by 3.

The result is that the sample Tao starts with maintains its character even as the Collatz process proceeds.

“He found some way to continue this process further, so that after some number of steps you still know what’s going on,” Soundararajan said. “When I first saw the paper, I was very excited and thought that it was very striking.”

Tao used this weighting technique to prove that almost all Collatz starting values — 99% or more — eventually reach a value that is quite close to 1. This allowed him to draw conclusions along the lines of 99% of starting values greater than 1 quadrillion eventually reach a value below 200.

It is arguably the strongest result in the long history of the conjecture.

“It’s a great advance in our knowledge of what’s happening on this problem,” said Lagarias. “It’s certainly the best result in a very long time.”

Tao’s method is almost certainly incapable of getting all the way to a full proof of the Collatz conjecture. The reason is that his starting sample still skews a little after each step in the process. The skewing is minimal as long as the sample still contains many different values that are far from 1. But as the Collatz process continues and the numbers in the sample draw closer to 1, the small skewing effect becomes more and more pronounced — the same way that a slight miscalculation in a poll doesn’t matter much when the sample size is large but has an outsize effect when the sample size is small.

Any proof of the full conjecture would likely depend on a different approach. As a result, Tao’s work is both a triumph and a warning to the Collatz curious: Just when you think you might have cornered the problem, it slips away.

“You can get as close as you want to the Collatz conjecture, but it’s still out of reach,” Tao said.

The quest to understand what’s happening inside the minds and brains of animals has taken neuroscientists down many surprising paths: from peering directly into living brains, to controlling neurons with bursts of light, to building intricate contraptions and virtual reality environments.

In 2013, it took the neurobiologist Bob Datta and his colleagues at Harvard Medical School to a Best Buy down the street from their lab.

At the electronics store, they found what they needed: an Xbox Kinect, a gaming device that senses a player’s motions. The scientists wanted to monitor in exhaustive detail the body movements of the mice they were studying, but none of the usual laboratory techniques seemed up to the task. So Datta’s group turned to the toy, using it to collect three-dimensional motor information from the animals as they explored their environment. The device essentially rendered them as clouds of points in space, and the team then analyzed the rhythmic movement of those points.

Armed with pen, paper and stopwatch, scientists have been quantifying animal behavior in the wild (and in their labs) for decades, watching their subjects sleep and play and forage and mate. They’ve tallied observations and delineated patterns and come up with organizational frameworks to systematize and explain those trends. (The biologists Nikolaas Tinbergen, Konrad Lorenz and Karl von Frisch won a Nobel Prize in 1973 for independently performing these kinds of experiments with fish, birds and insects.)

The inventories of behaviors arising from this work could get extremely detailed: A description of a mouse’s grooming in a 1973 *Nature* article involved a “flurry of forelimbs below face” and “large synchronous but asymmetric strokes of forelimbs over top of head,” with estimates of how likely such gestures might be under different circumstances. Researchers needed to capture all that detail because they couldn’t know which aspects of the observed behaviors might turn out to be important.

Some scientists have taken the opposite tack, reducing animals’ behavioral variability to its bare bones by putting them in controlled laboratory settings and allowing them to make only simple binary decisions, like whether to turn left or right in a maze. Such simplifications have sometimes been useful and informative, but artificial restrictions also compromise researchers’ understanding of natural behaviors and can cause them to overlook important signals. “Having a good grasp on the behavior is really the limiting factor for this research,” said Ann Kennedy, a postdoctoral researcher in theoretical neuroscience at the California Institute of Technology.

That’s why scientists have set out to modernize the field by “thinking about behavior more quantitatively,” according to Talmo Pereira, a graduate student in the labs of Murthy and Joshua Shaevitz at Princeton. And a change that has been instrumental in that makeover has been the automation of both data collection and data analysis.

Image-capture technology has always been crucial for tracking the poses of animals in motion. In the 1800s, Eadweard Muybridge used stop-motion photography to tease apart the mechanics of horses running and people dancing. The photos made it easier and more accurate to mark, say, where an animal’s legs were frame by frame, or how its head was oriented. When video technology arrived, researchers were able to take more precise measurements — but these still tended to be based on coarse quantities, such an animal’s speed or its average position. Tracking every movement through three dimensions was impossible. And all the video annotations still had to be laboriously logged into a computer by hand, a process that wasn’t much of an improvement on the older method of drawing in notebooks.

In the 1980s, researchers started adapting computer vision algorithms, which were already being used to find edges and contours in images, for animal behavior problems like tracing the outlines of flies on a surface. Over the next few decades, systems were developed to label the location of an animal in each frame of a video, to differentiate among multiple organisms, and even to start identifying certain body parts and orientations.

Still, these programs weren’t nearly as efficient as scientists needed them to be. “There were a few glimmers of what the future was to hold,” said Iain Couzin, director of the Max Planck Institute of Animal Behavior in Germany. “But nothing really sophisticated could happen until very, very recently, until the advent of deep learning.”

With deep learning, researchers have started to train neural networks to track the joints and major body parts of almost any animal — insects, mice, bats, fish — in every frame of a video. All that’s needed is a handful of labeled frames (for some algorithms, as few as 10 will do). The output appears as colored points transposed over the animal’s body, identifying its nose, tail, ears, legs, feet, wings, spine and so on.

The number of programs that do this has exploded in the past couple of years, fueled not only by progress in machine learning, but by parallel work on mapping human motion by moviemakers, animators and the gaming industry.

Of course, for the kinds of motion capture relevant to Hollywood and Silicon Valley, it’s easy for people to wear bodysuits studded with markers that the systems can easily spot and follow. That data can then be used to build detailed models of poses and movements. But bodysuit solutions weren’t really an option in the world of animal studies.

Five years ago, Jonathan Whitlock, a neuroscientist at the Norwegian University of Science and Technology, started hunting for another way to mark the mice he studied. He tried anything he could think of: He and his colleagues shaved the animals’ fur and tagged them with infrared reflective ink. They dabbed a suspension of glass beads, usually used in reflective road paint, onto the animals’ backs. They daubed glowing ink and polish on the animals’ joints. The list goes on, but to no avail: Sometimes the markers simply weren’t bright enough to be tracked, and sometimes they made the mice anxious, disrupting their behavior.

Eventually, Whitlock’s team settled on using tiny pieces of reflective tape stuck to three points along the animal’s back to reconstruct the movements of the spine, and a tiny helmet with four additional pieces of tape to track head movements. “That alone already was sufficient to open up a whole new world for us,” Whitlock said.

But many researchers wanted to move past using markers at all, and they wanted to track more than seven points on their animals. So by combining insights gained from previous work, both on animals and humans, multiple labs have created easy-to-use systems that are now seeing widespread application.

The first of these systems came online last year. DeepLabCut was developed by the Harvard neuroscientists Mackenzie Mathis and Alexander Mathis, who repurposed a neural network that was already trained to classify thousands of objects. Other methods followed in rapid succession: LEAP (Leap Estimates Animal Pose), developed by Pereira and others in the labs of Murthy and Shaevitz; SLEAP, the same team’s forthcoming software for tracking the body-part locations of multiple interacting animals at once; and the Couzin group’s DeepPoseKit, published a few months ago.

Attempts to answer these questions have long relied on the observer’s intuition — “immaculate perception,” as ethologists (animal behaviorists) jokingly call it. But intuition is hobbled by inherent biases, a lack of reproducibility, and difficulty in generalizing.

The zoologist Ilan Golani at Tel Aviv University has spent much of the past six decades in search of a less arbitrary way to describe and analyze behavior — one involving a fundamental unit of behavior akin to the atom in chemistry. He didn’t want behaviors to be tagged simply as courting or feeding. He wanted those characterizations to arise “naturally,” from a common set of rules grounded in an animal’s anatomy. Golani has his own model of what those units and rules should look like, but he thinks the field is still far from arriving at a consensus about it.

Other researchers take the opposite position, that machine learning and deep learning could bring the field to a consensus sooner. But while DeepLabCut, LEAP and the other cutting-edge pose-tracking algorithms rely on supervised learning — they’re trained to infer the locations of body parts from hand-labeled data — scientists hope to find and analyze the building blocks of behavior with unsupervised learning techniques. An unsupervised approach holds the promise of revealing the hidden structure of behaviors on its own, without humans dictating every step and introducing biases.

An intriguing example of this appeared in 2008, when researchers identified four building blocks of worm movement that could be added together to capture almost all the motions in the animal’s repertoire. Dubbed the “eigenworm,” this compact representation offered a quantitative way to think about behavioral dynamics.

Datta took this approach to a whole new level with his Xbox Kinect hack in 2013, and he was quickly rewarded for it. When he and his colleagues looked at the data describing the movements of the mice, they were surprised to immediately see an overarching structure within it. The dynamics of the animals’ three-dimensional behavior seemed to segment naturally into small chunks that lasted for 300 milliseconds on average. “This is just in the data. I’m showing you raw data,” Datta said. “It’s just a fundamental feature of the mouse’s behavior.”

Those chunks, he thought, looked an awful lot like what you might expect a unit of behavior to look like — like syllables, strung together through a set of rules, or grammar. He and his team built a deep neural network that identified those syllables by dividing up the animal’s activity in a way that led to the best predictions of future behavior. The algorithm, called Motion Sequencing (MoSeq), spat out syllables that the researchers would later name “run forward” or “down and dart” or “get out!” In a typical experiment, a mouse would use 40 to 50 of them, only some of which corresponded to behaviors for which humans have names.

“Their algorithms can pull out behaviors that we don’t have words for,” Whitlock said.

Now researchers are trying to determine the biological or ecological significance of these previously overlooked behaviors. They’re studying how the behaviors vary between individuals or sexes or species, how behavior breaks down with age or disease, and how it develops during learning or in the course of evolution. They’re using these automatic classifications to discern the behavioral effects of different gene mutations and medical treatments, and to characterize social interactions.

And they’re starting to make the first connections to the brain and its internal states.

Datta and his colleagues discovered that in the striatum, a brain region responsible for motor planning and other functions, different sets of neurons fire to represent the different syllables identified by MoSeq. So “we know that this grammar is directly regulated by the brain,” Datta said. “It’s not just an epiphenomenon, it’s an actual thing the brain controls.”

Intriguingly, the neural representation of a given syllable wasn’t always the same. It instead changed to reflect the sequence in which the syllable was embedded. By looking at the activity of the neurons, for instance, Datta could tell whether a certain syllable was part of a very fixed or very variable sequence. “At the highest level,” he said, “what that tells you is that the striatum isn’t just encoding what behavior gets expressed. It’s also telling you something about the context in which it’s expressed.”

He supported this hypothesis further by testing what happened when the striatum no longer worked properly. The syllables themselves remained intact, but the grammar became scrambled, the sequences of actions seemingly more random and less adaptive.

Other researchers are looking at what’s going on in the brain on longer timescales. Gordon Berman, a theoretical biophysicist at Emory University, uses an unsupervised analysis technique called Motion Mapper to model behavior. The model, which places behaviors within a hierarchy, can predict hierarchical neural activity in the brain, as demonstrated in a paper published by a team of researchers at the University of Vienna two weeks ago. (Berman says that “an aspirational goal” would be to someday use Motion Mapper to predict social interactions among animals as well.)

And then there’s Murthy and her team, and their search for hidden internal states. They had previously created a model that used measurements of the flies’ movements to predict when, how and what the male fly would sing. They discovered, for example, that as the distance between the male and female flies decreased, the male was likelier to produce a particular type of song.

In the work recently published in *Nature Neuroscience*, the scientists extended this model to include potential hidden internal states in the male flies that might improve predictions about which songs the flies would produce. The team uncovered three states, which they dubbed “Close,” “Chasing” and “Whatever.” By activating various neurons and examining the results with their model, they discovered that a set of neurons that had been thought to control song production instead controlled the fly’s state. “It’s a different interpretation of what the neuron is doing in the service of the fly’s behavior,” Murthy said.

They’re now building on these findings with SLEAP. “It’ll be really exciting to see what kind of hidden states this type of model is able to tease out when we incorporate higher-resolution pose tracking,” Pereira said.

The scientists are careful to note that these techniques should enhance and complement traditional behavioral studies, not replace them. They also agree that much work needs to be done before core universal principles of behavior will start to emerge. Additional machine learning models will be needed, for example, to correlate the behavioral data with other complex types of information.

“This is very much a first step in terms of thinking about this problem,” Datta said. He has no doubt that “some kid is going to come up with a much better way of doing this.” Still, “what’s nice about this is that we’re getting away from the place where ethologists were, where people were arguing with each other and yelling at each other over whether my description is better than yours. Now we have a yardstick.”

“We are getting to a point where the methods are keeping up with our questions,” Murthy said. “That roadblock has just been lifted. So I think that the sky’s the limit. People can do what they want.”

*Editor’s note: The work by Bob Datta, Jonathan Pillow and Adam Calhoun is funded in part by the Simons Foundation, which also funds this editorially independent magazine.*

*Animated pose-model of a walking fly courtesy of Pierre Karashchuk, Tuthill/Brunton labs, University of Washington; anipose.org*

Compared to the unsolved mysteries of the universe, far less gets said about one of the most profound facts to have crystallized in physics over the past half-century: To an astonishing degree, nature is the way it is because it couldn’t be any different. “There’s just no freedom in the laws of physics that we have,” said Daniel Baumann, a theoretical physicist at the University of Amsterdam.

Since the 1960s, and increasingly in the past decade, physicists like Baumann have used a technique known as the “bootstrap” to infer what the laws of nature must be. This approach assumes that the laws essentially dictate one another through their mutual consistency — that nature “pulls itself up by its own bootstraps.” The idea turns out to explain a huge amount about the universe.

When bootstrapping, physicists determine how elementary particles with different amounts of “spin,” or intrinsic angular momentum, can consistently behave. In doing this, they rediscover the four fundamental forces that shape the universe. Most striking is the case of a particle with two units of spin: As the Nobel Prize winner Steven Weinberg showed in 1964, the existence of a spin-2 particle leads inevitably to general relativity — Albert Einstein’s theory of gravity. Einstein arrived at general relativity through abstract thoughts about falling elevators and warped space and time, but the theory also follows directly from the mathematically consistent behavior of a fundamental particle.

“I find this inevitability of gravity [and other forces] to be one of the deepest and most inspiring facts about nature,” said Laurentiu Rodina, a theoretical physicist at the Institute of Theoretical Physics at CEA Saclay who helped to modernize and generalize Weinberg’s proof in 2014. “Namely, that nature is above all self-consistent.”

A particle’s spin reflects its underlying symmetries, or the ways it can be transformed that leave it unchanged. A spin-1 particle, for instance, returns to the same state after being rotated by one full turn. A spin- particle must complete two full rotations to come back to the same state, while a spin-2 particle looks identical after just half a turn. Elementary particles can only carry 0, , 1, or 2 units of spin.

Or take gluons, particles that convey the strong force that binds atomic nuclei together. Gluons are also massless spin-1 particles, but they represent the case where there are multiple types of the same massless spin-1 particle. Unlike the photon, gluons can satisfy the four-particle interaction equation, meaning that they self-interact. Constraints on these gluon self-interactions match the description given by quantum chromodynamics, the theory of the strong force.

A third scenario involves spin-1 particles that have mass. Mass came about when a symmetry broke during the universe’s birth: A constant — the value of the omnipresent Higgs field — spontaneously shifted from zero to a positive number, imbuing many particles with mass. The breaking of the Higgs symmetry created massive spin-1 particles called W and Z bosons, the carriers of the weak force that’s responsible for radioactive decay.

Then “for spin-2, a miracle happens,” said Adam Falkowski, a theoretical physicist at the Laboratory of Theoretical Physics in Orsay, France. In this case, the solution to the four-particle interaction equation at first appears to be beset with infinities. But physicists find that this interaction can proceed in three different ways, and that mathematical terms related to the three different options perfectly conspire to cancel out the infinities, which permits a solution.

That solution is the graviton: a spin-2 particle that couples to itself and all other particles with equal strength. This evenhandedness leads straight to the central tenet of general relativity: the equivalence principle, Einstein’s postulate that gravity is indistinguishable from acceleration through curved space-time, and that gravitational mass and intrinsic mass are one and the same. Falkowski said of the bootstrap approach, “I find this reasoning much more compelling than the abstract one of Einstein.”

Thus, by thinking through the constraints placed on fundamental particle interactions by basic symmetries, physicists can understand the existence of the strong and weak forces that shape atoms, and the forces of electromagnetism and gravity that sculpt the universe at large.

In addition, bootstrappers find that many different spin-0 particles are possible. The only known example is the Higgs boson, the particle associated with the symmetry-breaking Higgs field that imbues other particles with mass. A hypothetical spin-0 particle called the inflaton may have driven the initial expansion of the universe. These particles’ lack of angular momentum means that fewer symmetries restrict their interactions. Because of this, bootstrappers can infer less about nature’s governing laws, and nature itself has more creative license.

Spin- matter particles also have more freedom. These make up the family of massive particles we call matter, and they are individually differentiated by their masses and couplings to the various forces. Our universe contains, for example, spin- quarks that interact with both gluons and photons, and spin- neutrinos that interact with neither.

The spin spectrum stops at 2 because the infinities in the four-particle interaction equation kill off all massless particles that have higher spin values. Higher-spin states can exist if they’re extremely massive, and such particles do play a role in quantum theories of gravity such as string theory. But higher-spin particles can’t be detected, and they can’t affect the macroscopic world.

Spin- particles could complete the 0, , 1, , 2 pattern, but only if “supersymmetry” is true in the universe — that is, if every force particle with integer spin has a corresponding matter particle with half-integer spin. In recent years, experiments have ruled out many of the simplest versions of supersymmetry. But the gap in the spin spectrum strikes some physicists as a reason to hold out hope that supersymmetry is true and spin- particles exist.

In his work, Baumann applies the bootstrap to the beginning of the universe. A recent *Quanta* article described how he and other physicists used symmetries and other principles to constrain the possibilities for those first moments.

It’s “just aesthetically pleasing,” Baumann said, “that the laws are inevitable — that there is some inevitability of the laws of physics that can be summarized by a short handful of principles that then lead to building blocks that then build up the macroscopic world.”

Ever since I was very young, I have been enamored of elegant mathematics. Like many people of a similar bent, I agreed completely with Eugene Wigner’s famous article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” in which the Nobel Prize-winning physicist discusses how elegant mathematics has been “unreasonably” successful in explaining physical law. Wigner further states: “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” Similar thoughts have been expressed by several famous scientists, including Albert Einstein.

Wigner’s statement is deeply poetic and born of wonder, but is it really true? Does this apparent connection between beautiful mathematics and natural science always hold? And if it does, is it really something that cannot be understood? One possible cause for skepticism is Wigner’s use of the word “miracle.” Used literally, “miracle” implies something magical or mystical, but Wigner was an eminent scientist — is that what he meant? The word “miracle” can also be used in a hyperbolic sense, as in the sentence, “What my smartphone can do is nothing short of a miracle.” What you really mean is that you know, deep down, that the things your smartphone can do have complex technical explanations, but the end result is simple and wondrous. As scientists, we must eschew the magical and try to unpack a more plausible explanation. Let’s try to do this in our new *Insights* puzzle.

The uneven strips are not uniformly uneven, but consist of 10 square patches that we can think of as “speed bumps.” The speed bumps are stacked one on top of another, and each one is 1 unit by 1 unit in size. The speed bumps vary in how uneven they are. A speed bump can cause the vehicle’s cruising velocity to be reduced by any amount from 50% to 95% in 5% increments. All 10 types of speed bumps are present in every uneven strip, in a random arrangement (one possible arrangement of speed bumps is shown in the first purple strip in the figure). The vehicle can sense the roughness of the speed bump directly ahead of it and can move sideways at its cruising velocity of 1 in order to traverse a different speed bump that will slow it down less, if it so chooses. Of course, this will cost it time, and the time penalty will be greater if it moves sideways multiple blocks. After each strip, the cruising velocity returns to 1 on the intervening smooth strip. What strategy should the vehicle adopt to cross the terrain as fast as possible? How long will this be expected to take?

Note that both these cases are approximations with simplifications. The first case approximates a uniformly rough terrain, and the second case approximates a randomly rough terrain. Though we can use math to reason about them, we lose simplicity, elegance and predictive ability as the situations grow more complex. Such situations occur naturally in messier and more complex sciences like biology, psychology, sociology and human behavior. In such cases, we have to rely on large amounts of accurate data to detect trends and make probabilistic predictions — in the words of Peter Norvig, a research director at Google, and his colleagues, what we have here is the unreasonable effectiveness, not of math, but of data. Physics is generally simpler — and so the math remains tractable, and you can often make very accurate predictions. But even in physics, trying to describe the motion of multiple particles in a quantum or even classical system is hopelessly complex. Thus, the more complicated case B is somewhat reminiscent of, though nowhere near as complex as, the case of multiple particles interacting with each other in a quantum vacuum in which virtual particles are created and destroyed at random. In that situation, the math is likely to be full of nonlinearities and thus completely intractable.

Why is it that simple idealized laws give fairly accurate results even in systems containing astronomical numbers of particles? Two reasons are summation and symmetry. We can see this by looking at the law of gravitational attraction: *F = Gm _{1}m_{2}/r^{2}*, where

To summarize, mathematically elegant laws of physics may have the form they do because they are idealizations of simple cases, with simple geometry. They do approximately work in the real world, even in large objects, because of symmetry and summation. It’s lucky that most real-world situations can be modeled without discontinuities, though this is not always the case, as we saw in the *Insights* puzzle involving the elliptical pool table and the ellipsoid paradox. Even more problematic are chaotic, messy situations involving hundreds of particles with multiple interactions, where predictive mathematical beauty might not be possible and we have to revert to supercomputer simulations. Seeing mathematical beauty everywhere in the natural sciences, as Wigner did, could simply be the result of cognitive bias, an emotional reaction to aesthetic beauty. In quantum physics, for instance, the mathematics of renormalization, which is crucial for quantum field theories, has been characterized as “ugly” or “not legitimate” by several giants of physics, including Paul Dirac and Richard Feynman. Yet it works well in practice.

What is your take, *Quanta* readers — what do you think of Wigner’s thesis? Why do you think we have an intrinsic affinity for beautiful, elegant mathematics? Is it a miracle, or are we drawn to it for some other more mundane reason?

Happy puzzling!

*Editor’s note: The reader who submits the most interesting, creative or insightful solution (as judged by the columnist) in the comments section will receive a* Quanta Magazine *T-shirt** or one of the two *Quanta* books, *Alice and Bob Meet the Wall of Fire* or *The Prime Number Conspiracy* (winner’s choice). And if you’d like to suggest a favorite puzzle for a future Insights column, submit it as a comment below, clearly marked “NEW PUZZLE SUGGESTION.” (It will not appear online, so solutions to the puzzle above should be submitted separately.)*

*Note that we may hold comments for the first day or two to allow for independent contributions by readers*.

Our brains may seem physically far removed from our guts, but in recent years, research has strongly suggested that the vast communities of microbes concentrated in our digestive tract open lines of communication between the two. The intestinal microbiome has been shown to influence cognition and emotion, affecting moods and the state of psychiatric disorders, and even information processing. But how it could do so has been elusive.

Until recently, studies of the gut-brain relationship have mostly shown only correlations between the state of the microbiome and operations in the brain. But new findings are digging deeper, building on research that demonstrates the microbiome’s involvement in responses to stress. Focusing on fear, and specifically on how fear fades over time, researchers have now tracked how behavior differs in mice with diminished microbiomes. They identified differences in cell wiring, brain activity and gene expression, and they pinpointed a brief window after birth when restoring the microbiome could still prevent the adult behavioral deficits. They even tracked four particular compounds that may help to account for these changes. While it may be too early to predict what therapies could arise once we understand this relationship between the microbiome and the brain, these concrete differences substantiate the theory that the two systems are deeply entwined.

Pinning down these mechanisms of interaction with the brain is a central challenge in microbiome research, said Christopher Lowry, an associate professor of integrative physiology at the University of Colorado, Boulder. “They have some tantalizing leads,” he added.

Coco Chu, the new study’s lead author and a postdoctoral associate at Weill Cornell Medicine, was intrigued by the concept that microbes inhabiting our bodies could affect both our feelings and our actions. Several years ago, she set out to examine these interactions in fine-grained detail with the help of psychiatrists, microbiologists, immunologists and scientists from other fields.

The researchers performed classical behavioral training on mice, some of which had been given antibiotics to dramatically diminish their microbiomes and some of which had been raised in isolation so that they had no microbiome at all. All the mice learned equally well to fear the sound of a tone that was followed by an electric shock. When the scientists discontinued the shocks, the ordinary mice gradually learned not to fear the sound. But in the mice with depleted or nonexistent microbiomes, the fear persisted — they remained more likely to freeze at the sound of the tone than the untreated mice did.

Peering inside the medial prefrontal cortex, an area of the outer brain that processes fear responses, the researchers noticed distinct differences in the mice with impoverished microbiomes: Some genes were expressed less. One type of glial cell never developed properly. Spiny protrusions on the neurons associated with learning grew less plentifully and were eliminated more often. One type of cell showed lower levels of neural activity. It’s as if the mice without healthy microbiomes couldn’t learn to be unafraid, and the researchers could see it on a cellular level.

The researchers also set out to learn how the condition of the microbiome in the gut caused these changes. One possibility was that microbes send signals to the brain through the long vagus nerve, which carries sensations from the digestive tract to the brain stem. But snipping the vagus didn’t alter the behavior of the mice. It also seemed possible that the microbiomes might stir up responses in the immune system that affect the brain, but the numbers and proportions of immune cells in all the mice were similar.

But the researchers did pinpoint four metabolic compounds with neurological effects that were far less common in the blood serum, cerebrospinal fluid and stool of the mice with impaired microbiomes. Some of the compounds were already linked to neurological disorders in humans. The team speculated that the microbiome might produce certain substances in abundance, with some molecules making their way into the brain, according to the microbiologist David Artis, the director of the Jill Roberts Institute for Research in Inflammatory Bowel Disease at Weill Cornell Medicine and the senior author on the study.

In many laboratories, there’s a growing interest in tracking specific bacterial substances that are involved in nervous system signaling, said Melanie Gareau, an associate professor of anatomy, physiology and cell biology at the University of California, Davis. Numerous metabolites and pathways are probably involved in such processes.

Research on other disorders like depression has also pointed to the involvement of particular compounds created by microbes, but there’s still no consensus on which ones contribute to any condition, said Emeran Mayer, a professor of medicine and director of the G. Oppenheimer Center for Neurobiology of Stress and Resilience at the University of California, Los Angeles. And although the intestinal microbiome is clearly altered in many people with brain conditions, it’s often unclear if that change is a cause or an effect, he said. Differences in the microbiome might give rise to neurological problems, but the conditions could also change the microbiome.

There’s disagreement within the field not just about the consequences of diseased microbiomes, but also about healthy ones. “For a long time, we’ve been focused on this idea that we could identify specific types of bacteria that provide either risk or resilience to stress-related disorders, and it may be that it doesn’t have to be a particular microbe,” Lowry said. Even in healthy people, microbiomes vary widely. Particular microbes might not matter if a microbiome has enough diversity — just as there are many kinds of thriving forests, and one individual type of tree may not be necessary.

Still, the study of microbial effects on the nervous system is a young field, and there is even uncertainty around what the effects are. Previous experiments reached inconsistent or contradictory conclusions about whether microbiome changes helped animals to unlearn fear responses. What gives extra weight to the findings from Chu and her colleagues is that they can point to evidence for a specific mechanism causing the behavior they observed. Animal studies like this one are especially helpful in cementing a clear connection between the nervous system and the microbiome, even if they don’t point to treatments for humans, said Kirsten Tillisch, a professor of medicine at the David Geffen School of Medicine at UCLA. “The way that humans process emotion, physical sensation and cognition in the brain is just so different than in animals that it’s just very difficult to translate,” she said.

In theory, the presence of certain microbial substances might help predict who is most vulnerable to disorders like post-traumatic stress disorder. Experiments like these could even identify pathways of communication between the brain and the microbiome that could be targeted by treatments. “That’s always the big hope from these mouse experiments, that we’re getting close to interventions,” Mayer said, and the studies often generate striking results through rigorous methods. But the operations of the human brain aren’t fully reflected in mice. Moreover, the interactions of the brain and the gut microbiome differ in humans and mice, and diet-driven differences between their respective microbiomes add to the disparity.

For humans, interventions targeting the microbiome might be most effective in infancy and childhood, when the microbiome is still developing and early programming takes place in the brain, Mayer said. In this new research, the scientists saw a specific window of time in infancy when mice needed a typical microbiome to extinguish fear normally when they grew up. Mice that were totally isolated from microbes for their first three weeks were then mixed in with mice that had typical microbiomes. The germ-free mice picked up the microbes of the other mice and developed rich microbiomes, but when they grew up and went through the same fear unlearning experiments, they still showed deficits. At only a few weeks old, they were still too old to learn to extinguish their fear normally.

But when microbiomes were restored in newborn mice, who gained rich microbiomes after they were placed with foster parents, the infant mice grew up to behave normally. In the first few weeks after birth, the microbiome appeared to be critical — an insight that fits smoothly into the larger idea that circuits governing fear sensitivity are impressionable during early life, Tillisch said.

The kind of fear unlearning that the researchers tested is a fundamental skill in an evolutionary sense, Artis said. Knowing what merits fear and adapting when it no longer poses a threat can be crucial to survival. An inability to extinguish fear is also present in PTSD and tied to other brain disorders, so deepening scientific knowledge around the mechanisms that influence this circuitry could illuminate core human behaviors and pave the way for potential therapies.

On an evolutionary timescale, human microbiomes have changed as more people have come to live in cities, and brain disorders have become increasingly prominent. The swarms of microbes inhabiting each of us have evolved with our species, and it’s vital that we understand how they impact both physical and mental health, Lowry said. Our environments may affect our nervous systems by way of the microbiome, adding new layers of complexity to the study of health and disease in the brain.

The world officially became a slightly safer place in October, when the World Health Organization declared that polio’s type 3 strain had been eradicated. This strain — joining type 2, which was eradicated in 2015 — no longer exists anywhere in the world, outside of highly secure laboratories. (Type 1 is the only strain still at large.) Thanks to the hard work of thousands of dedicated individuals, these two strains will no longer cause devastating paralysis or death.

While it was once just a dream, permanently ending diseases has been within our power since 1980, when smallpox was eradicated after an intense campaign. This victory has saved roughly 200 million people who would otherwise have succumbed to the disease since then.

But other attempts to rid the world of diseases have not gone as smoothly. Doctors have been working on ending polio for 31 years, initially hoping it would be completely gone by 2000. Now, due to difficulties tracking the disease, the target eradication date for the remaining type 1 strain is 2023. Another pathogen nearing eradication is a parasite known as Guinea worm, but again, problems have complicated that campaign, and others as well.

So what was it about smallpox that made it so much simpler to eradicate? What makes an organism eradicable in the first place?

Fundamentally, if we want to get rid of a pathogen, we must have a way of stopping its transmission. Halt the spread, and you can isolate those infected without anyone else getting sick. Do a thorough enough job, and there won’t be any new cases anywhere in the world — the disease is eradicated. Theoretically, this process can take many forms. The deployment of an effective vaccine robs a disease of future hosts. Eliminating a key vector takes out the means of infection. And for a bacterial pathogen, antibiotic treatments can target the disease itself. But theory doesn’t always translate to practice in the real world.

For a sense of what actually works, smallpox provides the perfect case study: It turns out to be almost ideally suited to eradication. First, it’s a virus that only affects people, not animals. Wipe it out in humans, and that’s it, you’re done. (We’re not actually sure why smallpox is so choosy, and we’re unlikely to find out anytime soon, since little research today involves the deadly pathogen — and even then, it focuses on treatments and vaccine research over fundamental biology.)

Second, the disease makes its presence clearly and unambiguously known. It produces a rash that’s easy to identify and distinct from rashes caused by other diseases. And infections are not asymptomatic: You can’t be infected and contagious but still appear healthy. (Again, it’s not clear why this is.) These traits make it easier to track new cases and quickly stop outbreaks.

Third, smallpox has a highly effective vaccine, made from a virus closely related to smallpox called the vaccinia virus. Because the vaccine contains a live virus, the immune system produces a rapid, strong and lasting response. The vaccine can even stop a smallpox infection in its tracks. “You can vaccinate somebody who is already developing smallpox up to six days after they have been infected,” said Larry Brilliant, an epidemiologist and former WHO physician who took part in the smallpox eradication campaign. The vaccine made it easier to halt new transmissions and protect healthy people, even if responders arrived at a smallpox outbreak that was already underway.

The fourth reason — and an increasingly relevant one — is not a biological consideration, but a psychological one: Smallpox was a feared disease. People knew it was deadly, and even survivors could be scarred for life. This translated to political support from world governments and local support among populations receiving the vaccination.

All these features in combination enabled us to wipe out one of humanity’s oldest scourges in about a dozen years of intense effort. But if a disease is missing just one or two of these attributes, it can prove much harder to eradicate.

Like smallpox, polio is a disease that only affects humans, and we have an effective vaccine for it. In fact, we have two. But neither is as good as the one for smallpox, and one of them — a live virus vaccine no longer used in the U.S. — has the potential to mutate and cause vaccine-derived polio. In fact, for the last several years, we’ve had more cases of vaccine-derived polio than wild polio infections. (To be clear, no such dangers exist with flu or other typical vaccines.)

Unfortunately, polio differs from smallpox in another crucial way. Approximately 95% of those infected either don’t display any symptoms or only display generic ones such as fever and headache. This means the type of disease tracking that officials used to detect smallpox epidemics is impossible for polio. Instead, health officials take environmental samples to test for polio viruses, with positive results meaning additional vaccine campaigns for the area. The process is repeated until no additional samples contain polio virus. This relatively inefficient approach does work, though, and it’s how we’ve eradicated two of polio’s strains and will hopefully soon take down the third.

Guinea worm is another pathogen whose biology differs enough from smallpox to make it harder, but not impossible, to eradicate. The Guinea worm eradication campaign was championed by former President Jimmy Carter (among others) and launched in 1986, and it has led to a sharp reduction in cases of this water-borne parasite, from around 3.5 million yearly cases to under 30. Like smallpox, Guinea worm provides an obvious and unambiguous sign of infection: After growing in the host’s body for about a year, the worm emerges through the skin through a blister on the lower leg. But Guinea worm eradication has grown more complex over the past five years as doctors have recognized that it’s not a human-specific infection, as had been assumed: Recent studies have demonstrated that dogs, frogs and fish can also transmit the pathogen. This will slow the eradication timeline since the animals can re-contaminate clean water supplies.

But despite these setbacks, the polio and Guinea worm campaigns have shown how viable such goals really are. For future eradication campaigns, yaws and measles are both good potential targets. Yaws is a human-specific infection caused by a spirochete; left untreated, it has the potential to cause serious illness and disability. In 2012, researchers realized that a single dose of antibiotics could treat yaws and break the transmission cycle. Through this method, India became yaws-free in 2016, though the emergence of antibiotic resistance in the spirochete may complicate efforts.

Measles also echoes many of smallpox’s characteristics: It only affects humans, it doesn’t appear asymptomatically, and it has a highly effective vaccine. As a result, worldwide measles deaths have already dropped by approximately 20% since 2000, and the disease has been eliminated in the U.S., meaning it’s no longer constantly present here (despite occasional cases brought in by travelers). But while it’s a good candidate for eradication based solely on its biology, measles doesn’t share smallpox’s crucial psychological factor: People don’t fear it enough. To the contrary, unsubstantiated fears of vaccines are on the rise, harming the global eradication campaign and nearly bringing endemic measles back to the U.S.

So while we may have the biological tools to allow us to eradicate certain diseases, that’s not enough.

“It’s not science, it’s public will,” Brilliant said. “Public will is so critical.”

In January 1916, Karl Schwarzschild, a German physicist who was stationed as a soldier on the eastern front, produced the first exact solution to the equations of general relativity, Albert Einstein’s radical, two-month-old theory of gravity. General relativity portrayed gravity not as an attractive force, as it had long been understood, but rather as the effect of curved space and time. Schwarzschild’s solution revealed the curvature of space-time around a stationary ball of matter.

Curiously, Schwarzschild noticed that if this matter were confined within a small enough radius, there would be a point of infinite curvature and density — a “singularity” — at the center.

Infinities cropping up in physics are usually cause for alarm, and neither Einstein, upon learning of the soldier’s result, nor Schwarzschild himself believed that such objects really exist. But starting in the 1970s, evidence mounted that the universe contains droves of these entities — dubbed “black holes” because their gravity is so strong that nothing going into them, not even light, can come out. The nature of the singularities inside black holes has been a mystery ever since.

Recently, a team of researchers affiliated with Harvard University’s Black Hole Initiative (BHI) made significant progress on this puzzle. Paul Chesler, Ramesh Narayan and Erik Curiel probed the interiors of theoretical black holes that resemble those studied by astronomers, seeking to determine what kind of singularity is found inside. A singularity is not a place where quantities really become infinite, but “a place where general relativity breaks down,” Chesler explained. At such a point, general relativity is thought to give way to a more exact, as yet unknown, quantum-scale description of gravity. But there are three different ways in which Einstein’s theory can go haywire, leading to three different kinds of possible singularities. “Knowing when and where general relativity breaks down is useful in knowing what theory [of quantum gravity] lies beyond it,” Chesler said.

The BHI group built on a major advance achieved in 1963, when the mathematician Roy Kerr solved Einstein’s equations for a spinning black hole — a more realistic situation than the one Schwarzschild took on since practically everything in the universe rotates. This problem was harder than Schwarzschild’s, because rotating objects have bulges in the center and therefore lack spherical symmetry. Kerr’s solution unambiguously described the region outside a spinning black hole, but not its interior.

Kerr’s black hole was still somewhat unrealistic, as it occupied a space devoid of matter. This, the BHI researchers realized, had the effect of making the solution unstable; the addition of even a single particle could drastically change the black hole’s interior space-time geometry. In an attempt to make their model more realistic and more stable, they sprinkled matter of a special kind called an “elementary scalar field” in and around their theoretical black hole. And whereas the original Kerr solution concerned an “eternal” black hole that has always been there, the black holes in their analysis formed from gravitational collapse, like the ones that abound in the cosmos.

First, Chesler, Narayan and Curiel tested their methodology on a charged, non-spinning, spherical black hole formed from the gravitational collapse of matter in an elementary scalar field. They detailed their findings in a paper posted on the scientific preprint site arxiv.org in February. Next, Chesler tackled the more complicated equations pertaining to a similarly formed rotating black hole, reporting his solo results three months later.

Their analyses showed that both types of black holes contain two distinct kinds of singularities. A black hole is encased within a sphere called an event horizon: Once matter or light crosses this invisible boundary and enters the black hole, it cannot escape. Inside the event horizon, charged stationary and rotating black holes are known to have a second spherical surface of no return, called the inner horizon. Chesler and his colleagues found that for the black holes they studied, a “null” singularity inevitably forms at the inner horizon, a finding consistent with prior results. Matter and radiation can pass through this kind of singularity for most of the black hole’s lifetime, Chesler explained, but as time goes on the space-time curvature grows exponentially, “becoming infinite at infinitely late times.”

The physicists most wanted to find out whether their quasi-realistic black holes have a central singularity — a fact that had only been established for certain for simple Schwarzschild black holes. And if there is a central singularity, they wanted to determine whether it is “spacelike” or “timelike.” These terms derive from the fact that once a particle approaches a spacelike singularity, it is not possible to evolve the equations of general relativity forward in time; evolution is only allowed along the space direction. Conversely, a particle approaching a timelike singularity will not inexorably be drawn inside; it still has a possible future and can therefore move forward in time, although its position in space is fixed. Outside observers cannot see spacelike singularities because light waves always move into them and never come out. Light waves can come out of timelike singularities, however, making them visible to outsiders.

Of these two types, a spacelike singularity may be preferable to physicists because general relativity only breaks down at the point of singularity itself. For a timelike singularity, the theory falters everywhere around that point. A physicist has no way of predicting, for instance, whether radiation will emerge from a timelike singularity and what its intensity or amplitude might be.

The group found that for both types of black holes they examined, there is indeed a central singularity, and it is always spacelike. That was assumed to be the case by many, if not most, astrophysicists who held an opinion, Chesler noted, “but it was not known for certain.”

The physicist Amos Ori, a black hole expert at the Technion in Haifa, Israel, said of Chesler’s new paper, “To the best of my knowledge, this is the first time that such a direct derivation has been given for the occurrence of a spacelike singularity inside spinning black holes.”

Gaurav Khanna, a physicist at the University of Massachusetts, Dartmouth, who also investigates black hole singularities, called the BHI team’s studies “great progress — a quantum leap beyond previous efforts in this area.”

While Chesler and his collaborators have strengthened the case that astrophysical black holes have spacelike singularities at their cores, they haven’t proved it yet. Their next step is to make more realistic calculations that go beyond elementary scalar fields and incorporate messier forms of matter and radiation.

Chesler stressed that the singularities that appear in black hole calculations should disappear when physicists craft a quantum theory of gravity that can handle the extreme conditions found at those points. According to Chesler, the act of pushing Einstein’s theory to its limits and seeing exactly how it fails “can guide you in constructing the next theory.”

Most of the universe’s heft, oddly enough, could come in the form of particles billions of times lighter than the electron — a featherweight itself, as particles go. Streaming through the cosmos in thick hordes, these wispy “axion” particles could deliver a collective wallop as the missing dark matter that appears to outweigh all visible matter 6-to-1.

For decades, physicists have searched for the axion’s chief rival: a sluggish and far heavier dark matter candidate known as a WIMP (for “weakly interacting massive particle”). But WIMP experiments remain empty-handed as researchers approach the edges of their search field, while the hunt for the axion is only beginning.

“Dark matter could still be WIMPs, but every day it looks a little bit less likely,” said Ben Safdi, a physicist at the University of Michigan who specializes in dark matter. The axion “is kind of the best dark matter candidate that we have at the moment,” he said, given that others have failed to turn up in experiments.

The Axion Dark Matter Experiment (ADMX) at the University of Washington last year became the first experiment sensitive enough to detect the most likely kind of axion, and the experimental team recently announced the results of their latest search. They did not catch an axion. But, as they reported in a paper currently under review for publication in *Physical Review Letters*, they were able to rule out a swath of possible axion masses four times wider than the mass range they explored in their first run. ADMX is continuing to sweep through the places where an axion is most likely to be hiding. “They’re leading the charge,” Safdi said.

The axion attracts believers because it could solve two enigmas at once. Its invisible presence would explain why the universe acts so much heavier than it looks. And the particle would also show why the two fundamental forces that shape atomic nuclei follow different rulebooks — which is why physicists devised the axion in the first place, in the 1970s.

The puzzle is that the strong nuclear force arranges particles inside the neutron, known as quarks, so that their overall charge seemingly never grows lopsided. This property showcases a peculiar equanimity on the neutron’s part called charge-parity (CP) symmetry: Inverting each quark’s charge and reflecting them all in a mirror doesn’t affect the neutron’s behavior. A neutron with lopsided charge would fail CP symmetry, because reflecting it would flip its electric field relative to its intrinsic angular momentum, an effect similar to looking in a mirror and seeing yourself wearing your sweater on your legs and your jeans on your torso. Real neutrons look the same in a mirror, as experiments have found them to be electrically uniform to at least one part in a billion.

This symmetry would be all well and good if physicists had not discovered in 1964 that the weak nuclear force doesn’t share it: Two-quark particles called neutral kaons decay in ways CP symmetry forbids. Since quarks are involved in both cases, experts would have expected the weak-force symmetry-breaking to extend to the strong force as well. Suddenly the neutron’s impeccable charge distribution became a puzzle — the “strong CP problem.”

The axion represents the leading solution, although the theorists who laid the groundwork for it didn’t immediately see the full picture. “I actually wrote down the equations just sort of fudging it to make it work,” said Helen Quinn, who proposed a way to restore balance to the strong force along with Roberto Peccei in 1977.

The strong CP problem boils down to the unexpected value of one constant — an angle labeled θ, or theta — in the equations that describe the strong force. Its value seems to be zero, which makes the neutron’s charges stay in line. But for the many other values θ could take, the quarks stray. After some fudging, Quinn and Peccei promoted θ from a constant to a field that permeates space, with a value that could naturally settle down to zero everywhere. Quinn compares her model to a tilted bowler hat: A ball can start at any angle around the rim, but it will always roll to the bottom. Two other theorists, Steven Weinberg and Frank Wilczek, soon observed that the Peccei-Quinn field requires a particle — an excitation in the field — and the axion was born.

Then, in the 1980s, observations of galaxies’ rotational speeds and other evidence increasingly suggested that a huge amount of the universe’s matter is invisible, interacting with everything else mainly through its gravity. The growing evidence for dark matter prompted Pierre Sikivie, a theoretical physicist now at the University of Florida, to calculate just how invisible the axion might be.

Sikivie explained in an interview that the axion would be something of a spiritual cousin to the photon, but with just a hint of mass. Since the photon — the particle of light and electromagnetism — is governed by Maxwell’s equations, Sikivie tweaked the classical theory to incorporate the axion and found that axions just might pack the universe tightly enough that they could add up to the missing dark matter.

He also calculated that axions wouldn’t be completely undetectable; now and then they would transform into two photons. He realized that saturating an area with a strong magnetic field (and thus lots of photons) would stimulate axion decay, just as photons ease the emission of other photons in lasers. Also like photons, axions are very wavelike, falling on the wavy end of the wave-particle duality. Their minuscule mass makes them extremely low-energy waves, with wavelengths somewhere between a building and a football field in length.

Sikivie realized that the key to coaxing these low-energy axions to turn into photons would be a device that could be tuned to resonate at precisely the same wavelength as the axions. He envisioned a machine called a haloscope that would amplify a signal, essentially ringing like a bell when an axion decayed.

Implementing Sikivie’s notions took more than 30 years, but ADMX is now sensitive enough to detect axions with masses that theorists deem most plausible, even if the particles decay at the lowest theoretical rates. With a potent magnet sitting in an icebox chilled nearly to absolute zero, ADMX slowly adjusts its resonance and scans for axions. How often the magnet might turn an axion into two photons is unknown, but with quadrillions of axions potentially passing through the experiment each second, a detection would become clear quickly.

In its first run, reported last year, the experiment scanned from 0.65 to nearly 0.68 gigahertz looking for excess power from axion-spawned photons; this year the collaboration has continued on to 0.8 gigahertz. These frequencies mean that the experiment has ruled out axions weighing between 187 billion times and 151 billion times less than the electron, with wider ranges to come. “We’re starting to take larger and larger chunks,” said Gianpaolo Carosi, a member of the collaboration.

The group expects to reach at least 2 gigahertz over the next few years and hopes to eventually push all the way to 10 gigahertz, which would correspond to an axion 12 billion times lighter than the electron. Estimates vary, but most theorists say an axion that does double duty as both dark matter and a neutron fixer should fall somewhere within that range.

If it hears nothing but static, ADMX won’t flat out disprove the existence of axions. Frequencies of a few gigahertz match the simplest dark matter schemes, but some theorists have cooked up more intricate recipes. And if dark matter is a mix of axions and something else, axions could span a mass range exceeding 10 orders of magnitude.

But as other promising dark matter candidates fail to materialize, more experimental groups are turning to axions. Some are developing terrestrial magnetic devices like ADMX, while others plan to scan the radio waves coming from nature’s mightiest magnets — neutron stars. Together these teams may someday cover most of the possible frequencies.

A discovery would permanently rewrite the laws of particle physics and cosmology, but today axions remain entirely hypothetical. Quinn just feels humbled that her musings have launched such a formidable search party. “Roberto and I spent a few months cooking up this theory,” she said, “and now the experimentalists have spent 40 years looking for it.”

A thousand seemingly insignificant things change as an organism ages. Beyond the obvious signs like graying hair and memory problems are myriad shifts both subtler and more consequential: Metabolic processes run less smoothly; neurons respond less swiftly; the replication of DNA grows faultier.

But while bodies may seem to just gradually wear out, many researchers believe instead that aging is controlled at the cellular and biochemical level. They find evidence for this in the throng of biological mechanisms that are linked to aging but also conserved across species as distantly related as roundworms and humans. Whole subfields of research have grown up around biologists’ attempts to understand the relationships among the core genes involved in aging, which seem to connect highly disparate biological functions, like metabolism and perception. If scientists can pinpoint which of the changes in these processes induce aging, rather than result from it, it may be possible to intervene and extend the human life span.

So far, research has suggested that severely limiting calorie intake can have a beneficial effect, as can manipulating certain genes in laboratory animals. But recently in *Nature*, Bruce Yankner, a professor of genetics and neurology at Harvard Medical School, and his colleagues reported on a previously overlooked controller of life span: the activity level of neurons in the brain. In a series of experiments on roundworms, mice and human brain tissue, they found that a protein called REST, which controls the expression of many genes related to neural firing, also controls life span. They also showed that boosting the levels of the equivalent of REST in worms lengthens their lives by making their neurons fire more quietly and with more control. How exactly overexcitation of neurons might shorten life span remains to be seen, but the effect is real and its discovery suggests new avenues for understanding the aging process.

In the early days of the molecular study of aging, many people were skeptical that it was even worth looking into. Cynthia Kenyon, a pioneering researcher in this area at the University of California, San Francisco, has described attitudes in the late 1980s: “The ageing field at the time was considered a backwater by many molecular biologists, and the students were not interested, or were even repelled by the idea. Many of my faculty colleagues felt the same way. One told me that I would fall off the edge of the Earth if I studied ageing.”

That was because many scientists thought that aging (more specifically, growing old) must be a fairly boring, passive process at the molecular level — nothing more than the natural result of things wearing out. Evolutionary biologists argued that aging could not be regulated by any complex or evolved mechanism because it occurs after the age of reproduction, when natural selection no longer has a chance to act. However, Kenyon and a handful of colleagues thought that if the processes involved in aging were connected to processes that acted earlier in an organism’s lifetime, the real story might be more interesting than people realized. Through careful, often poorly funded work on *Caenorhabditis elegans*, the laboratory roundworm, they laid the groundwork for what is now a bustling field.

A key early finding was that the inactivation of a gene called *daf-2* was fundamental to extending the life span of the worms. “*daf-2* mutants were the most amazing things I had ever seen. They were active and healthy and they lived more than twice as long as normal,” Kenyon wrote in a reflection on these experiments. “It seemed magical but also a little creepy: they should have been dead, but there they were, moving around.”

This gene and a second one called *daf-16* are both involved in producing these effects in worms. And as scientists came to understand the genes’ activities, it became increasingly clear that aging is not separate from the processes that control an organism’s development before the age of sexual maturity; it makes use of the same biochemical machinery. These genes are important in early life, helping the worms to resist stressful conditions during their youth. As the worms age, modulation of *daf-2* and *daf-16* then influences their health and longevity.

These startling results helped draw attention to the field, and over the next two decades many other discoveries illuminated a mysterious network of signal transduction pathways — where one protein binds another protein, which activates another, which switches off another and so on — that, if disturbed, can fundamentally alter life span. By 1997, researchers had discovered that in worms *daf-2 *is part of a family of receptors that send signals triggered by insulin, the hormone that controls blood sugar, and the structurally similar hormone IGF-1, insulin-like growth factor 1; *daf-16 *was farther down that same chain. Tracing the equivalent pathway in mammals, scientists found that it led to a protein called FoxO, which binds to the DNA in the nucleus, turning a shadowy army of genes on and off.

That it all comes down to the regulation of genes is perhaps not surprising, but it suggests that the processes that control aging and life span are vastly complex, acting on many systems at once in ways that may be hard to pick apart. But sometimes, it’s possible to shine a little light on what’s happening, as in the Yankner group’s new paper.

Figuring out which genes are turned on and off in aging brains has long been one of Yankner’s interests. About 15 years ago, in a paper published in *Nature*, he and his colleagues looked at gene expression data from donated human brains to see how it changes over a lifetime. Some years later, they realized that many of the changes they’d seen were caused by a protein called REST. REST, which turns genes off, was mainly known for its role in the development of the fetal brain: It represses neuronal genes until the young brain is ready for them to be expressed.

But that’s not the only time it’s active. “We discovered in 2014 that [the *REST* gene] is actually reactivated in the aging brain,” Yankner said.

To understand how the REST protein does its job, imagine that the network of neurons in the brain is engaged in something like the party game Telephone. Each neuron is covered with proteins and molecular channels that enable it to fire and pass messages. When one neuron fires, it releases a flood of neurotransmitters that excite or inhibit the firing of the next neuron down the line. REST inhibits the production of some of the proteins and channels involved in this process, reining in the excitation.

In their new study, Yankner and his colleagues report that the brains of long-lived humans have unusually low levels of proteins involved in excitation, at least in comparison with the brains of people who died much younger. This finding suggests that the exceptionally old people probably had less neural firing. To investigate this association in more detail, Yankner’s team turned to *C. elegans*. They compared neural activity in the splendidly long-lived *daf-2* mutants with that of normal worms and saw that firing levels in the *daf-2 *animals were indeed very different.

“They were almost silent. They had very low neural activity compared to normal worms,” Yankner said, noting that neural activity usually increases with age in worms. “This was very interesting, and sort of parallels the gene expression pattern we saw in the extremely old humans.”

When the researchers gave normal roundworms drugs that suppressed excitation, it extended their life spans. Genetic manipulation that suppressed inhibition — the process that keeps neurons from firing — did the reverse. Several other experiments using different methods confirmed their results. The firing itself was somehow controlling life span — and in this case, less firing meant more longevity.

Because REST was plentiful in the brains of long-lived people, the researchers wondered if lab animals without REST would have more neural firing and shorter lives. Sure enough, they found that the brains of elderly mice in which the *Rest* gene had been knocked out were a mess of overexcited neurons, with a tendency toward bursts of activity resembling seizures. Worms with boosted levels of their version of REST (proteins named SPR-3 and SPR-4) had more controlled neural activity and lived longer. But *daf-2* mutant worms deprived of REST were stripped of their longevity.

“It suggests that there is a conserved mechanism from worms to [humans],” Yankner said. “You have this master transcription factor that keeps the brain at what we call a homeostatic or equilibrium level — it doesn’t let it get too excitable — and that prolongs life span. When that gets out of whack, it’s deleterious physiologically.”

What’s more, Yankner and his colleagues found that in worms the life extension effect depended on a very familiar bit of DNA: *daf-16*. This meant that REST’s trail had led the researchers back to that highly important aging pathway, as well as the insulin/IGF-1 system. “That really puts the REST transcription factor somehow squarely into this insulin signaling cascade,” said Thomas Flatt, an evolutionary biologist at the University of Fribourg who studies aging and the immune system. REST appears to be yet another way of feeding the basic molecular activities of the body into the metabolic pathway.

Neural activity has been implicated in life span before, notes Joy Alcedo, a molecular geneticist at Wayne State University who studies the connections between sensory neurons, aging and developmental processes. Previous studies have found that manipulating the activity of even single neurons in *C. elegans* can extend or shorten life span. It’s not yet clear why, but one possibility is that the way the worms respond biochemically to their environment may somehow trip a switch in their hormonal signaling that affects how long they live.

The new study, however, suggests something broader: that overactivity in general is unhealthy. Neuronal overactivity may not feel like anything in particular from the viewpoint of the worm, mouse or human, unless it gets bad enough to provoke seizures. But perhaps over time it may damage neurons.

The new work also ties into the idea that aging may fundamentally involve a loss of biological stability, Flatt said. “A lot of things in aging and life span somehow have to do with homeostasis. Things are being maintained in a proper balance, if you will.” There’s a growing consensus in aging research that what we perceive as the body slowing down may in fact be a failure to preserve various equilibria. Flatt has found that aging flies show higher levels of immune-related molecules, and that this rise contributes to their deaths. Keeping the levels in check, closer to what they might have been when the flies were younger, extends their lives.

The results may help explain the observation that some drugs used for epilepsy extend life span in lab animals, said Nektarios Tavernarakis, a molecular biologist at the University of Crete who wrote a commentary that accompanied Yankner’s recent paper. If overexcitation shortens life span, then medicines that systematically reduce excitation could have the opposite effect. “This new study provides a mechanism,” he said.

In 2014, Yankner’s laboratory also reported that patients with neurodegenerative diseases like Alzheimer’s have lower levels of REST. The early stages of Alzheimer’s, Yankner notes, involve an increase in neural firing in the hippocampus, a part of the brain that deals with memory. He and his colleagues wonder whether the lack of REST contributes to the development of these diseases; they are now searching for potential drugs that boost REST levels to test in lab organisms and eventually patients.

In the meantime, however, it’s not clear that people can do anything to put the new findings about REST to work in extending their longevity. According to Yankner, REST levels in the brain haven’t been tied to any particular moods or states of intellectual activity. It would be a “misconception,” he explained by email, “to correlate amount of thinking with life span.” And while he notes that there is evidence that “meditation and yoga can have a variety of beneficial effects for mental and physical health,” no studies show that they have any bearing on REST levels.

Why exactly do overexcited neurons lead to death? That’s still a mystery. The answer probably lies somewhere downstream of the DAF-16 protein and FoxO, in the genes they turn on and off. They may be increasing the organism’s ability to deal with stress, reworking its energy production to be more efficient, shifting its metabolism into another gear, or performing any number of other changes that together add up a sturdier and longer-lived organism. “It is intriguing that something as transient as the activity state of a neural circuit could have such a major physiological influence on something as protean as life span,” Yankner said.

*This article was reprinted on **Wired.com**.*

Some mathematical patterns are so subtle you could search for a lifetime and never find them. Others are so common that they seem impossible to avoid.

A new proof by Sarah Peluse of the University of Oxford establishes that one particularly important type of numerical sequence is, ultimately, unavoidable: It’s guaranteed to show up in every single sufficiently large collection of numbers, regardless of how the numbers are chosen.

“There’s a sort of indestructibility to these patterns,” said Terence Tao of the University of California, Los Angeles.

Peluse’s proof concerns sequences of numbers called “polynomial progressions.” They are easy to generate — you could create one yourself in short order — and they touch on the interplay between addition and multiplication among the numbers.

For several decades, mathematicians have known that when a collection, or set, of numbers is small (meaning it contains relatively few numbers), the set might not contain any polynomial progressions. They also knew that as a set grows it eventually crosses a threshold, after which it has so many numbers that one of these patterns has to be there, somewhere. It’s like a bowl of alphabet soup — the more letters you have, the more likely it is that the bowl will contain words.

But prior to Peluse’s work, mathematicians didn’t know what that critical threshold was. Her proof provides an answer — a precise formula for determining how big a set needs to be in order to guarantee that it contains certain polynomial progressions.

Previously, mathematicians had only a vague understanding that polynomial progressions are embedded among the whole numbers (1, 2, 3 and so on). Now they know exactly how to find them.

To get a sense of these patterns, consider one that is slightly simpler than the polynomial progressions Peluse worked with. Start with 2 and keep adding 3: 2, 5, 8, 11, 14, and so on. This pattern — where you begin with one number and keep adding another — is called an “arithmetic progression.” It’s one of the most studied, and most prevalent, patterns in math.

There are two main facts to understand about the frequency with which arithmetic progressions appear among the whole numbers.

In 1975, Endre Szemerédi proved one of them. First, he said, decide how long you want your arithmetic progression to be. It could be any such pattern with four terms (2, 5, 8, 11), or seven terms (14, 17, 20, 23, 26, 29, 32), or any number of terms you want. Then he proved that once a set reaches some exact size (which he couldn’t identify), it must contain an arithmetic pattern of that length. In doing so, he codified the intuition that among a large enough collection of numbers there has to be a pattern somewhere.

“Szemerédi basically said that complete disorder is impossible. No matter what set you take, there have to be little inroads of structure inside of it,” said Ben Green of Oxford.

But Szemerédi’s theorem didn’t say anything about how big a collection of numbers needs to be before these patterns become inevitable. He simply said that there exists a set of numbers, of some unknown size, that contains an arithmetic pattern of the length you’re looking for.

More than two decades later, a mathematician put a number on that size — in effect, proving the second main fact about these arithmetic patterns.

In 2001, Timothy Gowers of the University of Cambridge proved that if you want to be guaranteed to find, say, a five-term arithmetic progression, you need a set of numbers that’s at least some exact size — and he identified just what that size is. (Actually stating the size is complicated, as it involves enormous exponential numbers.)

To understand what Gowers did, you have to understand what mathematicians mean when they talk about the “size” of a set of numbers and the idea of “big enough.”

First, choose an interval on the number line, say 1 to 1,000, or something more random like 17 to 1,016. The endpoints of the interval don’t matter — only the length does. Next, determine the proportion of the numbers within that interval that you’re going to put into your set. For example, if you create a set with 100 numbers from 1 to 1,000, the size of your set is 10% of the interval.

Gowers’ proof works regardless of how you choose the numbers in that set. You could take the first 100 odd numbers between 1 and 1,000, the first 100 numbers that end in a 6, or even 100 random numbers. Whatever your method, Gowers proved that once a set takes up enough space (not necessarily 10%) in a long enough interval, it can’t help but include a five-term arithmetic progression. And he did the same for arithmetic progressions of any length you want.

“After Gowers, you know that if you give me an arithmetic progression of any length, then any subset” of numbers of some specified size must necessarily contain that progression, Peluse said.

Peluse’s work parallels Gowers’ achievement, except that she focused on polynomial progressions instead.

In an arithmetic progression, remember, you choose one starting number and keep adding another. In the type of polynomial progressions studied by Peluse, you still pick a starting value, but now you add powers of another number. For example: 2, 2 + 3^{1}, 2 + 3^{2}, 2 + 3^{3}, 2 + 3^{4}. Or 2, 5, 11, 29, 83. (Her progressions also had only one term for each power, a requirement that makes them more manageable.)

These polynomial progressions are closely related to an important type of pattern known as a geometric progression, formed by raising a number to increasingly high powers: 3^{1}, 3^{2}, 3^{3}, 3^{4}. These appear naturally in many areas of math and physics, and they’ve fascinated mathematicians for millennia. Geometric progressions are rare among even large sets of numbers, but if you tweak a geometric progression slightly — by adding a constant, like 2, to each term — you get a polynomial progression. And these seem to be everywhere.

“You can construct large sets of [numbers] that don’t contain geometric progressions. But if you allow yourself more freedom and shift your geometric progression,” creating a polynomial progression, then large sets seem to be forced to contain them, said Sean Prendiville of Lancaster University, who has collaborated with Peluse on polynomial progressions.

In 1996, Vitaly Bergelson and Alexander Leibman proved that once a set of numbers gets big enough, it has to contain polynomial progressions — the polynomial equivalent of Szemerédi’s work. But once again, that left mathematicians with no idea what “big enough” meant.

Peluse answered that question in a counterintuitive way — by thinking about exactly what it would take for a set of numbers not to contain the pattern you’re looking for.

Peluse wanted to determine how big a set needs to be — what percentage of the numbers in an interval it needs to include — in order to ensure that it contains a given polynomial progression. To do that, she had to think about all the ways a set of numbers could avoid containing the progression — and then to prove that past a certain size, even the cleverest avoidance strategies no longer work.

You can think about the problem as a challenge. Imagine that someone asks you to create a set containing half the numbers between 1 and 1,000. You win if the set doesn’t contain the first four terms of a polynomial progression. How would you select the numbers?

Your instinct might be to choose the numbers at random. That instinct is wrong.

“Most sets are in the middle of the bell curve. The number of polynomial progressions they contain is the average number,” Prendiville said. And that average is way more than the zero you want.

It’s similar to how, if you chose a person at random from the world’s population, you’d probably get someone close to average height. If your goal was to find a much rarer 7-footer, you’d need to search in a more targeted way.

So to win the number-picking challenge, you’ll need a more organized way of deciding which ones to include in your set of 500. You might notice, for example, that if you choose only even numbers, you eliminate the possibility that your set contains polynomial progressions with any odd numbers. Progress! Of course, you’ve also increased the likelihood that your set contains polynomial progressions made up of even numbers.

But the point is that by coming up with a structured way of choosing those 500 numbers, you can rule out the possibility that your set will contain certain polynomial progressions. In other words, it takes a pattern to avoid a pattern.

Peluse set out to prove that after reaching a certain size, even the most cleverly constructed sets still have to contain a given polynomial progression. Essentially, she wanted to identify the critical point at which every time you avoid one instance of a polynomial progression, you back into another, as with the odd and even numbers.

To do that, she had to find a way to quantify just how much structure a set contains.

Before Peluse’s paper, many mathematicians had tried to understand when polynomial progressions appear among sets of numbers. The researchers included some of the most accomplished minds in the field, but none of them made significant progress toward figuring out how big a set needs to be in order to contain polynomial progressions of different lengths.

The main impediment was that mathematicians had no idea how to capture, in a precise way, the kinds of structures that might allow a set to avoid containing polynomial progressions. There was one candidate technique. But at the time Peluse started her work, it was completely inapplicable to questions about polynomial progressions.

The technique originated in Gowers’ 2001 work on arithmetic progressions. There, Gowers had created a test, called the “Gowers norm,” which detects specific kinds of structures in a set of numbers. The test generates a single number quantifying the amount of structure in the set — or, to put it differently, it quantifies how far the set is from being a set of random numbers.

“The notion of looking random is not a well-defined mathematical notion,” Green said. Gowers found a way to quantify what this should mean.

A set can be structured to a greater or lesser extent. Sets containing random numbers have no structure at all — and therefore are likely to contain numerical patterns. They have a lower Gowers norm. Sets containing only odd numbers, or only multiples of 10, have a rudimentary structure. It’s easy to prove that past a certain size, sets with these kinds of simple designs also contain different kinds of patterns.

The most difficult sets to work on are those with a very intricate structure. These sets could still look random when in fact they’re constructed according to some kind of very subtle rule. They have a high Gowers norm, and they present the best chance of systematically avoiding patterns as the sets themselves grow larger.

While Gowers used these techniques to answer questions about arithmetic progressions, they were not applicable to questions about polynomial progressions. This is because arithmetic progressions are evenly spaced, while the terms in polynomial progressions jump wildly from one to the next. The Gowers norms were useful for studying polynomial progressions the way a weed whacker is good for stripping paint from a house: the right general idea, even if it’s not suited to the job.

In her new proof, Peluse used the basic idea of the Gowers norm to create a wholly new way of analyzing the kinds of structures relevant to polynomial progressions. She used this technique, called “degree-lowering,” to prove that for the polynomial progressions she was interested in, the only kinds of structures you actually need to worry about are the blunt, simple ones with a low Gowers norm. That’s because polynomial progressions veer so extremely from one term to the next that they inevitably run through subtler numerical obstacles — like a bull crashing through display tables on its way out of a china shop.

Peluse’s formula is complicated to state. It involves taking a double logarithm of the length of the initial interval from which you choose the numbers in your set. The minimum size she came up with is also not necessarily the true minimum size — future work might find that the critical threshold is even lower. But before her proof, mathematicians had no quantitative understanding at all about when polynomial progressions were guaranteed.

“She’s the first person to show how large sets need to be,” Prendiville said.

Peluse’s proof answers one quantitative question about polynomial progressions. Mathematicians now hope they can use it to answer another — about when polynomial progressions appear in sets composed entirely of prime numbers, which are the most important numbers in mathematics and notoriously resistant to patterns. Before Peluse’s proof, mathematicians had no idea how to approach that question.

“There’s a hope you can import some of the arguments from my paper into the primes setting,” Peluse said.

Our last *Insights *puzzle explored how a smooth, random distribution of objects arises in a classic, deterministic machine called a Galton board or bean machine. We examined the inner workings of this by playing with some puzzles. I also used the Galton board’s probabilistic result to suggest that perhaps the probabilistic equations of quantum mechanics spring from underlying deterministic laws that we may not be privy to. Readers responded spiritedly to both the puzzle questions and to the philosophical proposition. Let’s look at the puzzle questions first.

As seen below, the Galton board consists of an upright board with rows of pegs that create multiple paths along which marbles can roll from top to bottom. Marbles are dropped at the top and take either the rightward or leftward path with equal probability when they encounter a peg. At the bottom, the marbles accumulate in a set of bins, and can reproduce the Pascal’s triangle and the Gaussian distributions.

Imagine that you have a Galton board of the kind shown in the picture above. This one has bins in place of the eighth row and possesses traditional equal-probability pegs. You wish to modify it so that each of the bins at the bottom will collect an equal number of marbles. You know that you will have to replace some of the traditional pegs with new ones that direct the marbles unequally to their left or right side. You can choose pegs that direct a marble entirely to their left, entirely to their right, or in any proportion between these two extremes.

What is the smallest number of pegs you will need to replace, and with what left-right ratios, in order to achieve the goal of complete equality for all bins?

As a bonus question, can you derive and justify a formula that generalizes the above result to a Galton board of any size?

Rob Corlett correctly answered this problem, and Thana Somsirivattana submitted a formal proof. It turns out that just 10 pegs need to be replaced out of the 28. The replacement pegs are simple: They merely route any marble that strikes them to their left (L pegs) or to their right (R pegs). Following the convention of Lionel Lincoln and Rob Corlett, we can call the equal-probability pegs standard (S) pegs. Here is the solution showing how the pegs are arranged and how they route 128 marbles.

The arrangement is nice and regular, with the replacement pegs down the left and right sides.

As for the bonus question, this arrangement generalizes to other cases for which the number of bins is a power of 2. The following formula was recursively derived by Rob Corlett. Here, *p*(*n*) is the number of nonstandard pegs for a board with *n* bins:

*p*(1) = 0*
p*(2) = 0

p

Thus, the number of replacements needed for a 4-bin board is 2*p*(2) + 4 – 2 = 2 × 0 + 2 = 2. The number for a 16-bin setup would be 2 × 10 + 16 – 2 = 34.

If the number of bins is not a power of 2, the problem becomes far more difficult. In the words of Rob Corlett, “This is an incredibly complex problem when you dig into the structure!” Using an algebraic approach, Corlett generalized from the power-of-2 solution and derived the following formula representing an upper bound for the minimum replacements for an *n*-bin board where *n* is not a power of 2:

*p*(*n*) = *p*(*m*)* + p*(*n *–* m*)* + n *– 1

where *m* is the largest power of 2 smaller than *n*.

It turns out that you can do better than this, as cornflower showed using linear programming for boards of different sizes. Cornflower gives examples of minimum solutions for the Galton boards with five, six or seven bins. The minimum number of pegs that need to be replaced for 5-bin boards is five, for 6-bin boards is six and for 7-bin boards is eight.

The numbers are smaller than predicted by the formula because it turns out that in the above cases you can leave the original pegs at the ends of one of the rows. In all these cases, several minimum-replacement configurations are possible, as Rob Corlett showed algebraically. Cornflower also gave a symmetrical solution for a 12-bin setup, which Corlett showed to be unique.

I recommend reading the great work from these two contributors. Kudos!

As in Puzzle 1, start with a traditional Galton board, but this time one with nine bins at the bottom. You have to modify it by changing the minimum number of pegs so that the distribution of marbles at the bottom is as follows: 0, *x*, 2*x*, *x*, 0, *x*, 2*x*, *x*, 0, where *x* represents 1/8 of the total number of marbles.

Rob Corlett gives the following elegant solution, with the eight replacement pegs symmetrically arranged around the middle this time.

In this picture, consider a marble to have had a “drift” of 0 if it was in one of the four positions in row 4 and ended up in the corresponding position in row 8, as shown by the arrows. If it ended up in any other bin, the value of the drift is equal to the square of the distance from the expected bin. Thus, if a marble started from the leftmost position in row 4 and ended up in the bin marked 7, one bin to the left of its expected bin, its drift is 1^{2} = 1. If it ended up in the leftmost bin in the final row (marked 1), then its drift would be 2^{2} = 4. The mean drift for a particular Galton board is the average drift of all the marbles as they move from row 4 to row 8.

What is the average drift of:

1. The original Galton board.

2. The modified Galton board from Puzzle 1.

3. The modified Galton board from Puzzle 2.

The answers were once again correctly given by Rob Corlett. They are 1, 1.5 and 2.5. Corlett offers a clever tip that simplifies the problem: “The trick here is to notice that a ball on row 4 falling to row 8 is effectively the same as a ball coming in at row 1 and falling to row 5.”

Now let’s discuss the philosophical questions, which elicited many interesting reader comments. Before I discuss some of these, I’d like to make two points about where I was coming from.

First, my perspective is that of a scientist interested solely in the causal chain for a given quantum event (for example, a photon striking a specific point at the far end of a double slit experiment). I know there are probabilistic formulas, but they don’t answer what John Bell noneuphoniously called the “beables” — simply, what causes what. Some antecedent must have propelled the photon to that particular point and no other, be it something interior to the photon, the environment, a complex interaction of the two or the state of the universe in the specific fraction of a yoctosecond that the experiment happened to be carried out in. From this point of view, C T Johnson’s statement that “in a team B view, the photon is in a randomizing environment” and Hank Smith’s declaration that “the quantum universe is inherently probabilistic” seem to smack of magic, or else they are logically incoherent. Surely the random choice was not made in zero time, without any antecedents. If you could examine the tiny fraction of a yoctosecond preceding the photon’s choice, surely something happened that resulted in that choice.

So, and this is my second point, I don’t see how pure random choices without antecedents are logically possible in any conceivable universe. Quantum randomness can seem perfect and inherent only because it is caused by antecedents that are orders of magnitude too small for us to ever hope to observe. But the antecedents have to exist. The randomness, like the randomness of a pair of dice and all other random phenomena, are caused by our ignorance. I agree with segrimm, who replied to another reader’s comment, “Sorry, a single event is determined too because our universe acts non-local. The determination is the result of all the previous changes ‘inside and outside’ the local phenomenon. The fact that we cannot predict the outcome doesn’t mean the outcome is not determined.”

By the way, several readers questioned why I named the two antagonistic perspectives team E (Albert Einstein) and team B (Niels Bohr). I was alluding to the debates that took place at the Solvay Conference, which brought together the world’s top physicists in 1927, soon after the birth of quantum mechanics. The leaders of the two opposing viewpoints were Einstein and Bohr, who argued furiously, raising points and counterpoints that gave each other sleepless nights. Finally, Bohr’s perspective of inherent randomness prevailed as part of the emerging Copenhagen interpretation, while Einstein’s view was encapsulated in his famous exclamation, “God does not play dice with the universe.” The prominent physicists on Bohr’s team were pragmatic ones: Werner Heisenberg, Paul Dirac, Wolfgang Pauli and most others. On Einstein’s team were de Broglie and later, Bohm and Bell. Erwin Schrödinger, whose famous equation contributed to the official team B view, later defected and argued forcefully for team E, especially by proposing the classic Schrödinger’s cat thought experiment.

Now let’s take a look at some of the philosophical responses:

Responding to my statement that the many worlds interpretation (MWI) “assumes that the entire universe is being cloned countless times whenever a tiny particle makes a trivially different choice,” TJ_3rd suggested that I read Hugh Everett’s original thesis. Sure, that would be great to do, and thank you for the link! However, given my declared perspective above, I don’t see how that would affect this discussion. As TJ_3rd states, “Everett went into great detail about how his ‘many worlds interpretation’ does not and cannot predict individual outcomes.” But what causes the individual outcome is exactly what I’m curious about. As for my belief that MWI suggests universe “cloning” and particles making “choices,” these are not my words, but those employed by some modern followers of Everett such as David Deutsch, who seem to believe that multiple parallel universes either exist or actually come into being. I know there are other followers who don’t subscribe to this — apparently, the “other” worlds of MWI can have two different interpretations: real or unreal. But if the worlds are unreal, then the Everett procedure is merely a mathematical artifice that allows you to imagine that the quantum evolution enshrined in the Schrödinger equation is the only process that exists, and therefore saves physicists from bothering with the pesky waveform collapses. But this would make sense only if the Schrödinger equation were a true equation of motion. It is not. It is a probabilistic representation, just as a Gaussian distribution is a probabilistic representation of the outcome of a Galton board, and hence must refer to an ensemble, if you exclude the magical inherent indeterminacy that I referred to above.

Jon Richfield points to the fact that there is a difference between determinism and causality and that determinism requires infinite precision of measurement, which is impossible. I agree that, as chaos theory demonstrates, Pierre-Simon Laplace, who championed a completely deterministic world, was wrong even in the classical sense, but while he was wrong predictively, he was not wrong retrospectively. Given most uncomplicated classical results, you can infer with finite precision what the initial conditions needed to be and the paths that led to that result. If you cannot, your laws are incomplete. But I do agree that for the present argument, I am not so much interested in strict determinism as I am in causality.

Zdeněk Skoupý commented that “if we realistically solve the ‘causality’ of the movement of the balls into ever greater detail, then in principle we cannot go to infinity — the end is at the quantum level.” But balls are large enough macroscopic objects that we need not go to the quantum level to determine their behavior. I think that the randomizing physical mechanism that determines which way a ball goes in a practical, real-world Galton board can be completely explained classically — its motion, as a macroscopic object, is effectively sealed off from the quantum world.

Devin Wesley Harper’s idea that something like the Galton board operates to give the double slit results is interesting. However, it must be kept in mind that it has been convincingly demonstrated that the double slit pattern is an interference pattern caused by each photon interfering with itself. Also, the pattern occurs even in a vacuum. Devin, your theory will need to explain these two caveats if it is to be successful. Let us know how it goes.

That’s all for now. Thank you to all who commented. If any of you want to continue this discussion, you are welcome to do so in the comments section here. I will certainly attempt to respond, if only to applaud.

This month’s *Insights* prize goes to Rob Corlett. An honorable mention goes to cornflower — thank you again for your contribution. Congratulations!