The 2026 Fields Medal, one of the world’s most prestigious mathematics prizes, has just been awarded to four pioneering young researchers — including mathematician Hong Wang, the third woman ever to win the prize in the 90 years since the award was established.

The Fields Medal is an award presented every four years to mathematicians under the age of 40 for their outstanding discoveries. The International Mathematical Union announced Thursday (July 23) that Wang, a professor at New York University, is among the four recipients of the 2026 medal for her work on a decades-old conjecture about how needles move in 3D spaces. .

The 2026 Fields Medal winners also include Yu Deng of the University of Chicago, who was recognized for connecting microscopic and macroscopic descriptions of gases; John Pardon of Stony Brook University, whose work solved major problems in topology and geometry; and Jacob Tsimerman of the University of Toronto, who developed powerful new techniques in algebraic geometry that have advanced progress on longstanding mathematical puzzles.

Wang joins an exceptionally small group of women to receive the Fields Medal. Iranian mathematician Maryam Mirzakhani was the first, in 2014, followed by Ukrainian mathematician Maryna Viazovska in 2022. No woman received the award at all in the nearly 80 years before Mirzakhani's win.

Wang was recognized for her work on the Kakeya conjecture, a problem that asks how little space is needed to rotate a needle so that it points in every possible direction in three dimensions. Mathematicians had chased a solution for roughly 50 years.

Working with collaborator Joshua Zahl, Wang showed that if you track every direction the needle could point as a bundle of thin tubes, there's a precise trade-off between how thin those tubes are and how much total space they must occupy. Solving the puzzle required tools from harmonic analysis, a field that studies how complex shapes and signals can be broken down into simpler pieces.

The proof wasn't a triumphant moment for Wang, at least not right away. She and Zahl spent months checking their 127-page proof before making it public — and even then, Wang worried that their argument might be unclear. The paper nonetheless earned comparisons to a "once-in-a-century" result and set off a string of honors culminating in the Fields Medal, according to Quanta Magazine.

The 2026 Fields Medals were presented during the International Congress of Mathematicians in Philadelphia. The first Fields Medal was awarded in 1936. So far, 65 men have received the award.

Science word of the day: Cryptology

Pronunciation: Krip-TAH'-leh-jee

What it means: Cryptology is the study of secret codes ‪—‬ both making them (cryptography) and breaking them (cryptanalysis). Once associated with spy networks' secret codes and ciphers, cryptology now often refers to the computer algorithms used to encrypt data to protect passwords and personal information from prying eyes.

How to use it in a sentence: Native American soldiers known as "code talkers" harnessed cryptology in World War II by developing codes and ciphers based on their Indigenous languages.

Can you crack our science word of the day puzzle, Chain Word?

Two physicists have used generative artificial intelligence (AI) to solve a stubborn mathematical problem in physics that had vexed researchers for more than a decade.

Their solution, described July 1 in the Journal of Statistical Mechanics: Theory and Experiment, came about when the physicists chose to revisit a problem they thought they had attempted to solve exhaustively within a topic they knew intimately. This concept, known as jamming, refers to the sudden transition from a fluid system to a rigid-but-disordered one.

The simplest way to understand this idea is to imagine a pool table covered with billiard balls. If you keep adding balls, eventually the table becomes so congested that there is no space for any more and each ball on the table is securely held in place by its neighbors. This is a disordered, completely frozen situation known as a jammed state.

The study authors —‬ ‪Giorgio Parisi, winner of the 2021 Nobel Prize in physics, and Francesco Zamponi, ‪both physicists at the Sapienza University of Rome ‪—‬ and collaborators had mathematically described jamming and offered numerical solutions in a 2014 paper. In the process, they noticed that two parameters — $a$ and $b$ — would mysteriously always add up to 1.

"The parameters $a$ and $b$ dictate exactly how the distribution of contact forces and small gaps [between balls] scales as the physical system hits that critical jamming point," Zamponi told Live Science in an email. "We were quite bothered by the fact that we had never been able to mathematically prove the relation $a+b=1$."

Moreover, separate work by Matthieu Wyart, a physicist at the Swiss Federal Technology Institute (EPFL), took a completely different approach but yielded the same relation. For Zamponi and colleagues, this suggested "entirely new physical concepts" were needed to link their and Wyart's work and simultaneously explain why $a+b=1$.

Fast-forward a decade, and no progress had been made in finding these new concepts nor a reason for why $a+b=1$. Stuck in a rut, Parisi had a thought: perhaps generative AI could offer a fresh perspective. For this, he turned to Anthropic's Claude. After Claude successfully reproduced the 2014 numerical result, Parisi prompted the AI to prove why $a+b=1$.

"Giorgio initially sent me Claude's output while I was traveling, so I ended up reviewing it on an airplane," Zamponi recalled. "As I read through the LaTeX file Claude generated, it became immediately clear that the core idea was correct … That moment significantly shifted my perspective on what these models can achieve in theoretical physics."

Though the initial output contained some errors that required revision, the fundamental idea was correct. And in a total of just 40 prompts, the researchers had a verified publishable analytical solution. To their surprise, this solution was hidden directly within the equations themselves; they didn't need any external physical assumptions or deep connections between functions.

"It is entirely possible that a pure mathematician who works full time on such kind[s] of equations might have spotted the solution," Zamponi told Live Science. "But this is a particularly interesting point for us, as it highlights how Claude gave us instant access to a vast repository of mathematical training and formal skills that lay just outside our usual domain."

Whether Claude simply trawled the vast mathematical literature and used pattern matching to find a way to solve their problem or if it applied something akin to creativity is, for Zamponi, moot because they "could not see the path forward, and Claude did," he said. And although he admitted that interacting with AI forces him to reconsider his definitions of reasoning, intuition, and creativity, Zamponi will continue to collaborate with the technology to speed up mundane tasks and provide fresh perspectives on challenging problems.

Now, Zamponi is applying this collaborative approach to a problem involving the “random sequential addition of hard hyperspheres," he said. “It is another excellent case study because, while the AI drastically accelerates writing and optimizing code, I have had to provide the vast majority of the conceptual ideas, which suggests that human guidance remains indispensable, at least in this case."

It started with a plate of ginger chicken. In the late 1970s, physicist Richard Feynman — best known for his earlier work on the Manhattan Project — sat down for lunch with his friend Ralph Leighton at a restaurant in Glendale, California. Leighton was agonizing over ordering his usual favorite, or risking something new.

Feynman turned the choice into a math problem, and solved it on a piece of notebook paper. His equation showed exactly when Leighton — or any indecisive diner, for that matter — should stop taking risks and stick with what one knows is good.

For decades, Feynman’s notes on the "restaurant problem” were unreadable. But now, researchers reconstructed a decision-making problem from Richard Feynman's previously undeciphered notes and proved him to be right. The findings were published on June 1 in the journal Proceedings of the National Academy of Sciences.

The problem with picking lunch

Imagine you're visiting a new city for a week. Each night, you can either try an unknown restaurant or return to the best one you've already found. You want to maximize your total dining experience over the whole trip.

That kind of problem has a name in mathematics: an "optimal stopping problem." The same logic shows up in apartment hunting and job searching. But Feynman argued you can always go back to a previous restaurant. The goal is to maximize your cumulative enjoyment, not just find the single best spot.

Feynman's notes showed that the optimal strategy involves a quality threshold — a minimum score you require before committing — that starts high and drops as your trip runs out.

Brian Christian, a computer scientist and cognitive scientist at University of Oxford, began working on the problem about 13 years ago alongside his collaborator Tom Griffiths. They tracked down Feynman's original notes through the Feynman Lectures website.

The team proved that Feynman's solution was indeed optimal, then extended it to other versions of the problem: do people actually solve the problem this way?

They recruited 2,520 participants online and presented them with a digital version of the scenario: a grid of restaurants in a virtual city, each with a hidden quality score revealed only on the first visit. Participants aimed to maximize their total score over a fixed number of nights. Each person played just once.

"We wanted to really capture people's gut intuitions," Christian told Live Science. "When you just get thrown into this situation, what do you do?"

The answer: People don't follow Feynman's optimal curve in reality. Instead of the precise mathematical threshold, participants used a much simpler rule. Their quality bar started high and dropped by the same fixed amount each night regardless of how long the trip was or what the restaurant landscape looked like.

The simple strategy captured about 90% of the value that the optimal approach would yield.

"People are not doing the optimal thing. They're doing something radically simpler," Christian said. "And still the simple strategy is being tailored in a way that feels very situationally appropriate."

The slope of people's declining threshold was identical across every condition — a week-long trip or a month-long one, restaurants distributed evenly in quality or skewed toward extremes. What did shift was where people set their starting bar, adjusting it appropriately based on the landscape they'd seen.

In other words, people used a universal rule for how fast to lower their standards, but calibrated how high to set them in the first place.

An order of redemption

The results fit into an emerging framework in cognitive science called "resource rationality." The idea that humans aren't perfectly rational, but make good use of the limited time and brainpower they have.

"People don't do the perfect thing, but they make nearly perfect use of their constrained resources," Christian said. "I think this is a little bit more of a redemptive story about the human mind than we are used to from the 20th century."

That's a shift from the long tradition in behavioral economics emphasizing human irrationality and cognitive bias.

Christian says the findings also have implications for AI. Most AI systems assume people behave as perfectly rational agents. This study suggests that AI designed around how humans actually think — imperfectly — might work better.

Feynman died in 1988, never having published his restaurant analysis. But more than four decades after he scrawled those notes over lunch, the puzzle he left behind has finally been solved — and it turns out to say as much about the human mind as it does about what to eat.

A new theoretical study adds fresh support to the idea that a mathematical pattern of ripples in space-time geometry could give rise to naked singularities and microscopic black holes. The new finding advances research into a subject that has vexed physicists for decades.

In 1997, Stephen Hawking famously conceded defeat on a 1991 bet with fellow theoretical physicists Kip Thorne and John Preskill about the possible existence of naked singularities: objects like black holes but without an event horizon (a point beyond which light, and all other matter, cannot escape), making them observable. Hawking eventually admitted that such objects could exist. Thorne and Preskill’s prize? T-shirts to cover their "nakedness."

The evidence that swayed Hawking came from physicist Matthew Choptuik. In 1993, Choptuik studied a specific set of solutions to Albert Einstein's general relativity equations. When solved numerically, on what was then considered a supercomputer, he showed how naked singularities could hypothetically occur under very specific conditions.

Choptuik found that by modeling the gravitational collapse of a simple form of matter, such as a field, and fine-tuning the initial conditions, an unstable state can be constructed. This theoretical state later became known as a space-time crystal — a self-organized repetitive mathematical pattern of ripples in space-time geometry — containing a singularity with infinite curvature (a naked singularity). Because such a singularity wouldn’t form inside a black hole, it could theoretically be observable.

But much like the phase transition from liquid water to ice, this state is delicate, with the field teetering on the edge between dissipating to become empty space or forming a microscopic black hole.

However, significant doubt remained about such a state's existence, even theoretically.

"Whenever you formulate a system in numerical code, you always have a problem because you can only represent a finite number of digits on a computer," study co-author Christian Ecker, an astrophysicist at Goethe University in Germany, told Live Science. "The historic computer simulations could only go so far before inaccuracies became unavoidable."

Though more recent numerical methods offer much higher accuracy, they are not exact and can never provide deep understanding of the phenomenon that traditional analytical methods (such as manipulating equations using algebra and calculus) offer.

In the new study published May 12 in the journal Physical Review Letters, the researchers mathematically described the formation of space-time crystals, naked singularities and microscopic black holes precisely.

A pen and paper solution

They succeeded using just pen and paper, and some mathematical sleight of hand. "Whenever physicists find a small parameter, they are happy because they can first solve the equations when this parameter is zero, then add small corrections to it with standard perturbation theory," co-author Daniel Grumiller, an astrophysicist at the Institute for Theoretical Physics, Vienna University of Technology, told Live Science. "General relativity by itself doesn’t have a small parameter, but if you inject a small parameter [one over the number of dimensions and let this number be huge]… then you can use these perturbative tools and get a handle on otherwise very tough equations."

When taking the number of dimensions to be infinite, the team's exact solution could fit on just a few lines. This solution is unrealistic given we are most certainly not living in an infinite dimensional universe. However, as they brought the number of dimensions down to more realistic numbers, the solution required additional terms that made the expressions ever more complicated.

"The lowest dimension that we can consistently connect with so far is 52, but the numerical data extends only up to dimension 14 — so there's a gap," Grumiller said, referring to the fact that neither pen-and-paper nor numerical techniques are accurate enough to cross paths yet.

"In the future, we plan to extend the numerics to higher dimensions, so that we can actually connect the two," Grumiller added.

Doing so would provide a compelling case that space-time crystals, naked singularities and microscopic black holes are mathematically possible in a universe like ours — however, this would still not prove they actually exist in reality. In the end, Hawking may have awarded those T-shirts too soon.

See how much you know about black holes with our black hole quiz!

Recently, a cheerful 100-year-old message in a bottle was found on the south-west coast of Australia. In it, a world war one soldier proclaimed to be "as happy as Larry."

If you're a betting person, you probably wouldn't expect great odds of this happening. A bottle cast into the ocean could end up absolutely anywhere.

If it floats to a remote location, there is little chance of somebody stumbling upon it. And if it lands somewhere more favorable where people could potentially find it, there are other issues. The message itself will deteriorate over time as light degrades it. If the bottle fills with water, it will sink and almost certainly never be found.

So, what are the chances of a message in a bottle being found and it being over 100? And what are your chances of finding this bottle?

Despite these many possibilities during a bottle's lifetime, the probability we are after is a straightforward calculation. Just count up the number of bottles with messages that have been found and are over 100 years old, and divide by the number of messages that have been sent this way (assuming we know how many are sent):

Our diagram below shows a hypothetical situation where 20 bottles are sent in total, of which six are found (indicated in gold) and one of these is over 100 years old (indicated by the "100" stamp). So, one in 20 bottles are found and over 100 years old. (Note: This is only a hypothetical calculation, not the real data.)

Instead of calculating the probability directly, another way to do it is by breaking the problem into two parts: (A) a bottle with a message is found, and (B) the found bottle is over 100. These two probabilities can be calculated separately and multiplied together to get what we want:

This is known as the "multiplication rule" of probability, and we confirm from our hypothetical numbers that (6/20)×(1/6) = 1/20, as before.

Both approaches to calculating this probability are simple. However, the direct calculation requires knowing the total number of bottles sent out, which is very difficult to know in the real world.

The multiplication rule has the advantage that it breaks the calculation into two parts. We can tackle each separately, then bring the two results together to get the probability we want. This is useful in the real-world situation where we can draw information from different sources.

First, we'll deal with the probability that a bottle with a message is found, irrespective of its age.

Experts from the Federal Maritime and Hydrographic Agency of Germany suggest a 1 in 10 chance that a message in a bottle will be found. This aligns broadly with various historical "drift bottle" experiments, where oceanographers released large numbers of bottles to understand ocean currents.

For example, studies from the 1960s and '70s in the North Atlantic Ocean led to recovery rates of 14% from the Gulf of Mexico, 8% from the Caribbean Sea and 7% from the northern Brazilian coast. A more recent and more northerly study (between Canada and Greenland) from the 2000s led to a 5% recovery rate.

We would expect the results to vary naturally from different experiments in different parts of the world. But to keep things simple, we will stick with 1/10 as the probability that a bottle with a message is found.

Now for the second piece of the calculation: of the bottles that are found, what proportion are over 100 years old?

The table below summarises data from news articles collected on Wikipedia about very old bottles with messages that have been found. However, only data on bottles over 25 years old has been collected, presumably because older bottles are more newsworthy.

So, we needed to estimate the number of 0- to 25-year-old bottles with messages ourselves — here’s how we did this.

The table shows that fewer bottles with messages are found as they get older. Messages in bottles degrade over time, which means the bottles have an increased chance of breaking and sinking, or just getting covered in layers of sediment. Plotting this data in the graph below helped us see the trend in the ages of found bottles more clearly.

We drew a line to match this observed trend in the ages of found bottles. This red line in the graph corresponds to the equation:

This equation provides an estimate of how many bottles have been found for any specific age range (where 25 = 0-to-25, 50 = 25-to-50 and so on). We are interested in the the 0- to 25-year-old bottles, so the equation suggests 46 bottles have been found in this range.

Adding up this and all of the numbers in the table gives a total of 106 bottles found, of which 12 are over 100 years old, and 12/106 is about one in ten.

Recapping the above, we have that: (A) one in ten bottles with messages are found, of which (B) one in ten are over 100 years old. Bringing these results together using the multiplication rule, we estimate the chance of a message in a bottle being found and it being over 100 years old to be (1/10)×(1/10) = 1/100.

So, if there are 100,000 bottles with messages floating around the oceans waiting to be found, we’d expect 1,000 of these to be found and be 100 or more years old. Assuming anybody in the world is equally likely to find one of these, with 8 billion people currently, that’s about a 1 in 8 million chance of you finding one – pretty unlikely.

However, some people are more persistent at message-in-a-bottle hunting than others. Following the paths of ocean currents (known as gyres) could provide clues on where to look.

Specifically, peninsulas or islands intersecting with these gyres could be good spots. For this reason, it has been suggested the Caribbean islands are ideally placed for finding bottles as they lie on the path of the North Atlantic Gyre. Which seems like a great reason to travel to the Carribean!

But let's also spare a thought for the poor soul stranded on their desert island, who surely won't appreciate the low odds of their SOS being found.

This edited article is republished from The Conversation under a Creative Commons license. Read the original article.

As an irrational number, pi has no end — but that has not stopped computer engineers from chasing its eternal string of decimal places deeper into the unknown. Recently, technology media company StorageReview achieved a staggering new record, calculating 314 trillion digits of pi on a single Dell PowerEdge R7725 server that ran constantly for nearly four months.

The result shows that in modern pi calculations, the real battle is no longer just about processor speed but also storage space and efficiency. StorageReview's Dell PowerEdge R7725 server had 1.5 terabytes of memory to get the job done.

"The storage layer, specifically, is where this record was actually won," StorageReview representatives wrote in a December 2025 statement.

An irrational arms race

Previous pi records have jumped fast in the past few years, from Google Cloud's 100 trillion-digit run in 2022 to StorageReview's own 105 trillion-digit and 202 trillion-digit marks in 2024. In April 2025, Linus Media Group and Kioxia stole the crown by calculating pi to 300 trillion digits — but StorageReview reclaimed the record in November 2025.

The results were announced in time for Pi Day, March 14 (or 3/14) — a nod to the number's famous first three digits (3.14). The day has become a lighthearted tribute to math, marked by pie jokes, slices of pie, classroom contests and a public fascination with a number that never ends.

Why pi is important

Pi is a key constant in mathematics, linking every circle's circumference to its diameter. It appears in geometry, physics, engineering and statistics, showing up in everything from waves and orbits to bridges, buildings and computer models. Most people encounter pi in school as a simplified number used to find the area or circumference of a circle. But for engineers and scientists, it's a building block that helps describe how the physical world works.

Pi is considered an irrational number because it cannot be written as a simple fraction of two whole numbers. Its decimal form never ends and never settles into a repeating pattern. Mathematician Johann Lambert was the first to prove pi was irrational in 1761, showing that no fraction can exactly equal the ratio of a circle's circumference to its diameter. So, even though pi is a precise number, its decimal expansion is endless.

All those digits aren't strictly necessary for accurate calculations; NASA typically stops at 16 digits in its most precise calculations about the universe. Still, researchers challenge themselves to calculate pi to ever-more decimal places for multiple reasons. It's a way to test the limits of computers, storage and software, as a huge pi run can expose weaknesses in hardware better than many standard benchmarks. Calculating pi also helps researchers refine algorithms for handling other large calculations.

Then, of course, there's the fame of being the one to calculate pi out to the most decimal places yet. To achieve the mind-boggling result of 314 trillion digits of pi, StorageReview delivered around 280 GB/s bandwidth on its Dell server to handle the huge stream of intermediate calculations required for such a large run.

"If someone wants to take the record, we would like to see them take the whole thing: more digits, less power, shorter wall time, and the same zero-downtime reliability," the company said in the statement. "Until then, this is the benchmark for efficiency."

Pi quiz: How much do you know about this irrational number?

Earlier this month an artificial intelligence (AI) startup announced that their AI agent had confirmed a proof of two cases of the devilishly challenging "higher dimensional sphere-packing problem." In 2022, the proofs earned Ukrainian mathematician Maryna Viazovska a Fields Medal, one of the most prestigious prizes in math.

This was a giant step forward, and speaks to the emergence of a quiet revolution in the field.

On the surface, it may not seem so extraordinary. After all, mathematicians have long used tools to extend their abilities — abacuses, slide rules, calculators and, eventually, computers. Yet none of these tools ever replaced mathematicians; they just allowed us to refocus our attention on more interesting problems. The arrival of AI in mathematics might feel like another step in that same process. But there's a crucial difference: This time, the tools aren't just helping us calculate; they're helping us reason, or at least perform many of the routines that sit underneath human reasoning.

This represents a seismic shift in what it means to do mathematics. Instead of working unassisted, struggling at the boundaries of our own cognitive limits, we are starting to build and tune the instruments that will allow us to extend these limits, pairing human intuition with machine-level discipline. This might mean that our most sophisticated proofs won't be works a single mind can grasp; rather, they will be fully understood only in a collective mind that relies heavily on AI tools. It also means the scope of the math we can tackle will increase dramatically.

The change has been coming for a while. For years, our biggest proofs have not been the endeavours of single mathematicians. Many modern research articles in pure mathematics now rely on huge conceptual frameworks, long dependency chains, and catalogs of results that no single person can fully internalize. Computers have played a role in large proofs before, like the four-color theorem and the Kepler conjecture. But what's changing now is the level of autonomy and reliability we can expect from AI systems working alongside formal proof assistants — programs designed to check mathematical arguments.

These formal verification languages express mathematical arguments in a way a computer can check step by step, guaranteeing that every part of the proof is logically sound. Take the language Lean, for example. Unlike ordinary mathematical writing, Lean requires every definition and inference to be made explicit, and it checks each step mechanically and methodically. It's unforgiving, but in a productive way: If the argument is passed by Lean, that, in theory, means the proof doesn't have hidden assumptions or leaps of faith. Over the past few years, Lean has become a proving ground for research‑level mathematics, and mathematicians have been building "libraries" to support increasingly complex problems.

These libraries are huge collections of definitions and already‑verified theorems that have been painstakingly programmed, allowing researchers to prove new results in the language. But until recently, turning cutting‑edge proofs into machine‑checkable form required specialists to devote months or years to the work.

That's the context in which the recent formal verification of Viazovska's higher-dimensional sphere‑packing results should be understood. The sphere‑packing problem asks how tightly identical spheres can be packed together in spaces of any dimension, not just the 3D world we live in. Before Viazovska's breakthrough, the sphere‑packing problem had only been fully solved in dimensions one, two and three, with all higher‑dimensional cases remaining open. Viazovska's proofs of the eight- and 24‑dimensional sphere-packing problem, are profound pieces of mathematical insight that solve problems previously thought out of reach.

Fields Medal-level advancements

The recent important step forward is that a human-AI collaboration has now translated those arguments into fully verified Lean code, which then checked every step. The sheer scale of that achievement is astonishing; these are recent Fields Medal‑level results, and they have now been certified at a level of detail and certainty that would be impossible for individual referees, or even large human specialist teams, to reproduce unaided.

A key ingredient was Math, Inc.'s AI reasoning agent Gauss which had played a vital role in helping to turn human mathematical arguments into Lean proofs. The AI system wasn't working entirely unaided; mathematicians still had to set out the blueprint, shape the overall structure, and ensure the right concepts were in place. But once that scaffolding existed, the system could fill in the missing pieces at extraordinary speed. In the eight‑dimensional case, it completed work that the human contributors had estimated would take them months, and it did so in days. The 24‑dimensional case, which is even more intricate, followed soon after.

This is more than a technical accomplishment. It points toward a shift in the way mathematicians might organize their work. When I talked to UCLA mathematician and Fields Medalist Terence Tao, he suggested that the immediate value of AI might come not from cracking our hardest problems outright but from relieving us of the drudgery — the thousand small cases that are conceptually straightforward but too time‑consuming for any one person to tackle by hand.

Some AI systems, he argued, are already surprisingly good at handling these tasks, letting mathematicians devote their attention to strategy rather than bookkeeping. Tools like Lean matter because they give us a way to separate the creativity of generating ideas from the rigor of checking them.

AI proof expert Kevin Buzzard, of Imperial College London, expressed a complementary view. He worries, rightly, about the dangers of relying on large language models that sound authoritative without guaranteeing correctness. But he also argues that formalization offers a way through this. In Lean, if the program accepts all the steps, then it's a valid proof. This doesn't mean the computer has necessarily done something "intelligent" but rather that the formal verification language leaves no room for hidden steps or suggestive-but-incomplete arguments. The challenge, as he sees it, is that most of modern mathematics still hasn't been translated into formal libraries, so the systems don't yet have the concepts they need.

This latest step forward suggests the gap is beginning to close. The sphere‑packing project is probably the clearest demonstration yet of what is becoming possible.

None of this means mathematicians are on the brink of extinction. In fact, I suspect the opposite is true. As the space of verifiable mathematics expands, so too does the need for people who can pose good questions, create new definitions, and recognize when an argument is genuinely insightful. But we are going to have to adapt. We may find ourselves acting more like scientific-instrument builders and less like lone theorists, weaving together human intuition and AI tenacity to produce machine‑verified certainty.

Mathematics has always moved forward by partnering with assistive tools. AI doesn't change that practice; it just takes it to the next level. Mathematical concepts won't get easier to prove, but our capacity to test, verify and build upon them will surely increase.

Opinion on Live Science gives you insight on the most important issues in science that affect you and the world around you today, written by experts and leading scientists in their field.

Like physics, math has its own set of "fundamental particles" — the prime numbers, which can't be broken down into smaller natural numbers. They can only be divided by themselves and 1.

And in a new development, it turns out these mathematical "particles" are offering new ways to tackle some of physics' deepest mysteries. Over the past year, researchers have found that formulas based on the prime numbers can describe features of black holes. Number theorists have spent hundreds of years deriving theorems and conjectures based on the primes. These new connections suggest that the mathematical truths that govern prime numbers may also govern some fundamental laws of the universe. So can physics be expressed in terms of primes?

Black holes are the sites of the universe's most crushing gravitational force. At their centers lie single points called singularities, where classical physics predicts that gravity must be infinite, causing our understanding of space and time to break down. But in the 1960s, physicists found that, immediately surrounding the singularity, a type of chaos emerges — and it looks remarkably similar to a kind of chaos recently found in the primes.

Physicists hope to make use of the connection. "I'd say many high-energy physicists don't actually know much about that side of number theory," says Eric Perlmutter of the Institute of Theoretical Physics, Saclay.

Number theory's foundational conjecture on primes is the 1859 Riemann hypothesis. In a hand-written paper, German mathematician Bernhard Riemann provided a formula with two main terms. The first offered a startlingly close estimate for how many prime numbers exist that are smaller than a given number. The second term is the zeta function, whose zeros (the places where the function is equal to zero) tune up the original estimate. The mysterious way in which the zeta zeros always improve the estimate is the subject of the Riemann hypothesis. The hypothesis is so crucial to number theory that anyone who can prove it will earn a $1-million Clay Mathematics Institute prize.

In the late 1980s physicists started to wonder if there was a physical system whose energy levels might be based on the prime numbers. Physicist Bernard Julia of the École Normale Supérieure in France was challenged by a colleague to find a physics analogue described by the zeta function. His solution was to propose a hypothetical kind of particle with energy levels given by the logarithms of prime numbers. Julia called these particles "primons" and a group of them a "primon gas." The partition function — a census of a system’s possible states — of this gas is exactly the Riemann zeta function.

At the time, Julia's concept was a thought experiment — most scientists doubted that primons actually existed. But deep inside black holes, a mathematical link awaited discovery. A little more than two decades later, physicists Yan Fyodorov of King's College London, Ghaith Hiary of Ohio State University and Jon Keating of the University of Oxford saw hints that fractal chaos emerges from the fluctuations of the zeta function's zeros, an idea that was conclusively proven in 2025.

Einstein's general theory of relativity shows that the same chaos also arises near a singularity.

In a February 2025 preprint, University of Cambridge physicist Sean Hartnoll and graduate student Ming Yang brought Julia's work into the real world. Inside the chaos close to a singularity, they found that a "conformal" symmetry emerges. Hartnoll likens conformal symmetry to Dutch artist M. C. Escher's famous drawings of bats — the same structure repeats on different scales. This scaling symmetry, together with a bit of math, revealed a quantum system near the singularity whose spectrum organizes into prime numbers — a conformal primon gas cloud.

Five months later, they uploaded a preprint with a new twist. The team, which now included University of Cambridge University physicist Marine De Clerck, expanded their analysis to a five-dimensional universe instead of the usual four. They found that the extra dimension forced a new feature: keeping track of the singularity's dynamics now required a "complex" prime number, known as a Gaussian prime, that includes an imaginary component (a number multiplied by the square root of –1). Gaussian primes can't be divided any further by other complex numbers. The authors dubbed this system a "complex primon gas."

"We don't know yet whether the appearance of prime number randomness close to a singularity has a deeper meaning," Hartnoll says. "However, to my mind, it is very intriguing that the connection extends to higher dimensional theories of gravity," including some candidates for a fully quantum mechanical theory of gravity.

And in a late 2025 preprint, Perlmutter proposed a new framework involving the zeta zeros. He relaxed the restrictions on the zeta function so it could rely not just on integers but on all real numbers, including irrationals. Doing so opened up even more powerful zeta function techniques to understand quantum gravity. Physicist Jon Keating of the University of Oxford, who was not involved in the new research, says that broader perspectives such as this can reveal new ways to tackle long-standing problems. "It's only when you step back and look at the whole mountain that you think, 'Ah, there's a much better way to get up over there,'" he says.

Perlmutter cautiously hopes the flurry of prime physics will hasten new discoveries, but the approach is one of many fighting for acceptance. "The kinds of things we're trying to understand, black holes in quantum gravity, are surely governed by some beautiful structures," he says. "And number theory seems to be a natural language."

This article was first published at Scientific American. © ScientificAmerican.com. All rights reserved. Follow on TikTok and Instagram, X and Facebook.

Albert Einstein quiz: What do you know about the life of the famous theoretical physicist?

At a secret meeting in 2025, some of the world's leading mathematicians gathered to test OpenAI's newest large language model, o4-mini.

Experts at the meeting were amazed by how much the model's responses sounded like a real mathematician when delivering a complex proof.

"I've never seen that kind of reasoning before in models," Ken Ono, a professor of number theory at the University of Virginia said at the time. "That's what a scientist does."

But was the artificial intelligence (AI) model being given more credit than it deserved? And do we run the risk of accepting AI-derived proofs without fully understanding them?

Ono acknowledged that the model might be giving convincing — but potentially incorrect — answers.

"If you say something with enough authority, people just get scared," Ono said. "I think o4-mini has mastered proof by intimidation; it says everything with so much confidence."

In the past, confidence and the appearance of a good argument were good signs because only the best mathematicians could make convincing arguments, and their reasoning was usually sound. That has changed.

"If you were a terrible mathematician, you would also be a terrible mathematical writer, and you would emphasize the wrong things," Terry Tao, a mathematician at UCLA and the 2006 winner of the prestigious Fields Medal, told Live Science. "But AI has broken that signal."

Naturally, mathematicians are beginning to worry that AI will spam them with convincing-looking proofs that actually contain flaws that are difficult for humans to detect.

Tao warned that AI-generated arguments might be incorrectly accepted because they look rigorous.

"Unfortunately, the AI is much better at sounding like they have the right answer than actually getting it … right or wrong; they will always look convincing," Tao said.

He urged caution on the acceptance of AI '"proofs." "One thing we've learned from using AIs is that if you give them a goal, they will cheat like crazy to achieve the goal," Tao said.

While it may seem largely abstract to ask whether we can truly "prove" highly technical mathematical conjectures if we can't understand the proofs, the answers can have significant implications. After all, if we can't trust a proof, we can't develop further mathematical tools or techniques from that foundation.

For instance, one of the major outstanding problems in computational math, dubbed P vs. NP, asks, in essence, whether problems whose solutions are easy to check are also easy to find in the first place. If we can prove that, we could transform scheduling and routing, streamline supply chains, accelerate chip design, and even speed up drug discovery. The flip side is that a verifiable proof might also compromise the security of most current cryptographic systems. Far from being arcane, there is real jeopardy in the answers to these questions.

Proof is a social construct

It might shock non-mathematicians to learn that, to some extent, human-derived mathematical proofs have always been social constructs — about convincing other people in the field that the arguments are right. After all, a mathematical proof is often accepted as true when other mathematicians analyze it and deem it correct. That means a widely accepted proof doesn't guarantee a statement is irrefutably true. Andrew Granville, a mathematician at the University of Montreal, suspects there are issues even with some of the better-known and more scrutinized human-made mathematical proofs.

There's some evidence for that claim. "There have been some famous papers that are wrong because of little linguistic issues," Granville told Live Science.

Perhaps the best-known example is Andrew Wiles' proof of Fermat's last theorem. The theorem states that although there are whole numbers where one square plus another square equals a third square (like 3²+4²=5²), there are no whole numbers that make the same true for cubes, fourth powers, or any other higher powers.

Wiles famously spent seven years working in almost complete isolation and, in 1993, presented his proof as a lecture series in Cambridge, to great fanfare. When Wiles finished his last lecture with the immortal line "I think I'll stop there," the audience broke into thunderous applause and Champagne was uncorked to celebrate the achievement. Newspapers around the world proclaimed the mathematician's victory over the 350-year-old problem.

During the peer-review process, however, a reviewer spotted a significant flaw in Wiles' proof. He spent another year working on the problem and eventually fixed the issue.

But for a short time, the world believed the proof was solved, when, in fact, it hadn't been.

Mathematical verification systems

To prevent this sort of problem—where a proof is accepted without actually being correct—there's a move to shore up proofs with what mathematicians call formal verification languages.

These computer programs, the best known example of which is called Lean, require mathematicians to translate their proofs into a very precise format. The computer then goes through every step, applying rigorous mathematical logic to confirm the argument is 100% correct. If the computer comes across a step in the proof it doesn't like, it flags it and doesn't let go. This encoded formalization leaves no room for the linguistic misunderstandings that Granville worries have plagued previous proofs.

Kevin Buzzard, a mathematician at Imperial College London, is one of the leading proponents of the formal verification. "I started in this business because I was worried that human proofs were incomplete and incorrect and that we humans were doing a poor job documenting our arguments," Buzzard told Live Science.

In addition to verifying existing human proofs, AI, working in conjunction with programs like Lean, could be game-changing, mathematicians said.

"If we force AI output to produce things in a formally verified language, then this, in principle, solves most of the problem," of AI coming up with convincing-looking, but ultimately incorrect proofs, Tao said.

Buzzard agreed. "You would like to think that maybe we can get the system to not just write the model output, but translate it into Lean, run it through Lean," he said. He imagined a back-and-forth interaction between Lean and the AI in which Lean would point out errors and the AI would attempt to correct them.

If AI models can be made to work with formal verification languages, AI could then tackle some of the most difficult problems in mathematics by finding connections beyond the scope of human creativity, experts told Live Science.

"AI is very good at finding links between areas of mathematics that we wouldn't necessarily think to connect," Marc Lackenby, a mathematician at the University of Oxford, told Live Science.

A proof that no one understands?

Taking the idea of formally verified AI proofs to its logical extreme, there is a realistic future in which AI will develop "objectively correct" proofs that are so complicated that no human can understand them.

This is troubling for mathematicians in an altogether different way. It poses fundamental questions about the purpose of undertaking mathematics as a discipline. What is ultimately the point of proving something that no one understands? And if we do, can we be said to have added to the state of human knowledge?

Of course, the notion of a proof so long and complicated that no one on Earth understands it is not new to mathematics, Buzzard said.

"There are papers in mathematics where nobody understands the whole paper. You know, there's a paper with 20 authors and each author understands their bit," Buzzard told Live Science. "Nobody understands the whole thing. And that's fine. That's just how it works."

Buzzard also pointed out that proofs that rely on computers to fill in gaps are nothing new. "We've had computer-assisted proofs for decades," Buzzard said. For instance, the four-color theorem states that if you have a map divided into countries or regions, you'll never need more than four distinct colors to shade the map such that neighboring regions are never the same colors.

Almost 50 years ago, in 1976, mathematicians broke the problem into thousands of small, checkable cases and wrote computer programs to verify each one. As long as the mathematicians were convinced there weren't any problems with the code they'd written, they were reassured the proof was correct. The first computer-assisted proof of the four-color theorem was published in 1977. Confidence in the proof built gradually over the years and was reinforced to the point of almost universal acceptance when a simpler, but still compute-aided, proof was produced in 1997 and a formally verified machine-checked proof was published in 2005.

"The four-color theorem was proved with a computer," Buzzard noted. "People were very upset about that. But now it's just accepted. It's in textbooks."

Uncharted territory

But these examples of computer-assisted proofs and mathematical teamwork feel fundamentally different from AI proposing, adapting and verifying a proof all on its own — a proof, perhaps, that no human or team of humans could ever hope to understand.

Regardless of whether mathematicians welcome it, AI is already reshaping the very nature of proofs. For centuries, the act of proof generation and verification have been human endeavors — arguments crafted to persuade other human mathematicians. We're approaching a situation in which machines may produce airtight logic, verified by formal systems, that even the best mathematicians will fail to follow.

In that future scenario — if it comes to pass — the AI will do every step, from proposing, to testing, to verifying proofs, "and then you've won," Lackenby said. "You've proved something."

However, this approach raises a profound philosophical question: If a proof becomes something only a computer can comprehend, does mathematics remain a human endeavor, or does it evolve into something else entirely? And that makes one wonder what the point is, Lackenby noted.

In January 1816, the secretary general of the Paris Academy of Sciences sent Marie-Sophie Germain a strange letter.

In it, he acknowledged that she had won the institute's prestigious "Grand Mathematics Prize" for her mathematical work describing how sound waves travel across 2D surfaces. And yet, the letter offered no congratulations, noted condescendingly that she was the only entrant, and admitted that she had not received tickets to attend the prize ceremony set to occur two days later. He grudgingly acknowledged that, if needed, handwritten tickets could be hastily produced.

Germain did not attend the ceremony.

"The class of mathematical and physical sciences of the Institute held its public session today, a very large assembly that attracted without doubt those desiring to see virtuoso of a new kind, Miss Sophie Germain, to whom the prize for elastic membranes was to be awarded. The expectation of the public was disappointed: the young lady did not go to take the trophy that no one of her gender has ever received in France," the newspaper Journal des Débats reported about the event that day.

The award was the culmination of a decade of work by Germain, a self-taught polymath. Born to a wealthy merchant's family, she became interested in math while reading books in her father's library during a period of seclusion during the French revolution.

Her parents were not pleased with her "unladylike" pursuit. They banked the fires that kept the house toasty and took away her warm clothes, hoping that she'd be too cold and uncomfortable to study. But when they went to sleep, she'd grab candles and cover herself in quilts to continue her math research. She taught herself number theory and calculus that way.

When the École Polytechnique opened in 1794, women were barred from attending, but the notes from lectures were publicly available. She began reading those notes and submitting answers to problems from the lectures under the pseudonym "Antoine August LeBlanc." Under her pseudonym, Germain also began corresponding with some of the leading mathematicians of her day, including Carl Friedrich Gauss and Joseph-Louis Lagrange.

Around 1806, she became intrigued by the physics behind a perplexing experiment. In his 1787 book, physicist and musician Ernst Chladni, often called the "father of acoustics," described a phenomenon in which a person can sprinkle sand across a glass plate and then drag a violin bow across various surfaces and edges. Not only could the plate be played like a violin, but varied geometric patterns formed in the sand depending on how the plates were bowed.

The French institute had offered a prize three years running to mathematically describe the "Chladni figures" that formed. No one else bothered to attempt a solution, with most believing the existing math of the day insufficient to explain the phenomenon.

Germain, however, submitted her proposed solutions all three years. Her third proposal, submitted in 1816, was titled "Research on the Vibrations of Elastic Plates." Though "awkward and clumsy" given the math available at the time, it was still a brilliant insight into the subject of 2D harmonic oscillation, or stably moving waves.

Germain ultimately decided to skip the ceremony because she felt the committee didn't sufficiently respect her work. For instance, her leading rival, Siméon Poisson, was part of the award committee and refused to discuss the problem with her or talk with her in public. Not all of Germain's contemporaries were so dismissive, however; Lagrange and Gauss strongly supported her work.

"But when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarizing herself with their knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius," Gauss wrote when he discovered her gender.

Germain would continue with her solitary math research for decades.

Her work with French mathematician Adrien-Marie Legendre was a major advance in the proof of Fermat's Last Theorem, which states that no three positive integers (a, b, c) can satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than 2.

Germain showed that Fermat's Last Theorem held for a special class of prime numbers, now called Germain primes, in which both p and 2p+1 are prime. Her work formed the foundation for the eventual, complete solution produced by Andrew Wiles in 1994. Nonetheless, Germain's theorem was mentioned only in a footnote in Legendre's work.

In 1831, her longtime correspondent and mentor Gauss pushed for the University of Göttingen to give Germain an honorary degree. She died of breast cancer a few weeks before she could be given the award.

The human body is extraordinarily complex, with several openings and a few exits. But exactly how many holes does each person have?

It sounds like a simple enough question to answer — list the openings and add them up. But it's not quite that easy once you start considering questions like: "What exactly is a hole?" "Does any opening count?" And "why don't mathematicians know the difference between a straw and a doughnut?"

Before we start counting, we need to agree on what constitutes a "hole." Katie Steckles, a lecturer in mathematics at Manchester Metropolitan University in the U.K. and a freelance mathematics communicator, told Live Science that mathematicians "use the term 'hole' to mean one like the hole in a donut: one that goes all the way through a shape and out the other side."

But if you dig a "hole" at the beach, your aim is probably not to dig right through to the other side of the world. Many people would think of a hole as a depression in a solid object. But "this isn't a true hole, as it has an end," Steckles said.

Similarly, mathematical communicator James Arthur, who is based in the U.K., told Live Science that "in topology, a 'hole' is a through hole, that is you can put your finger through the object."

When digging a tunnel under the sea, like the Channel Tunnel that connects the U.K. and France, engineers started off by digging two openings. But as soon as those two digging projects joined up, the Channel Tunnel became a fundamentally different object (what Arthur and engineers would call a "through hole") — like a straw, or a tube with an opening at either end.

And if you ask people how many holes a straw has you will get a range of different answers: one, two and even zero. This is a result of our colloquial understanding of what constitutes a hole.

To find a consistent answer, we can turn to mathematics. And the problem of classifying how many holes there are in an object falls squarely within the realm of topology.

To a topologist, the actual shapes of objects are not important. Instead, "topology is more concerned with the fundamental properties of shapes and how things connect together in space," Steckles said.

In topology, objects can be grouped together by the number of holes they possess. For example, a topologist sees no difference between a golf ball, a baseball or even a Frisbee. If they were all made of plasticine, or putty, they could theoretically be squashed, stretched or otherwise manipulated to look like each other without making or closing any holes in the plasticine or sticking different parts together, Steckles argued.

However, to a topologist, these objects are fundamentally different to a bagel, a doughnut or a basketball hoop, which each have a hole through the middle of them. A figure of eight with two holes and a pretzel with three are different topological objects again.

A useful way to get into the mathematicians' way of thinking about the straw problem is to "imagine our straw is made of play dough," Arthur said. "Let's take this straw and slowly squish the top down and down and down towards the bottom, making sure the hole in the middle stays open. We will squish it until we are in a shape that looks like a doughnut." Mathematicians, Arthur said, would say that "the straw is homeomorphic to a doughnut."

The long, thin aspect ratio of the straw, and the fact that the two openings are relatively far apart, are perhaps what gives rise to the suggestion of two holes. But to a topologist, bagels, basketball hoops and doughnuts are all topologically equivalent to a straw with a single hole. "The hole in a straw goes all the way through it, and the opening at the other end is just the back of that same hole," Steckles said.

Back to the human body

Armed with the topologists' definition of a hole, we can tackle the original question: How many holes does the human body have? Let's first try to list all the openings we have. The obvious ones are probably our mouths, our urethras (the ones we pee out of) and our anuses, as well as the openings in our nostrils and our ears. For some of us, there are also milk ducts in nipples and vaginas.

There are also four less-obvious openings that we all have in the corners of eyelids closest to our nose — the four lacrimal puncta, which drain tears from our eyes into our nasal cavities. At an even smaller scale there are the pores that enable sweat to escape our bodies and sebum to lubricate our skin. In total there are potentially millions of these openings in our bodies, but do they all count as holes?

To make the question interesting, think about whether we could pass a very thin string into one hole and out of another. If we set the size of this string to be about 60 microns (60 millionths of a meter) then it's possible that the string could enter an opening as small as a pore. However — and this is key — it wouldn't be able to leave. It wouldn't be able to come out the other end. It would be blocked by the cells at the bottom of the pore — too thick to pass through into the vasculature that supplies the pore.

"They're not actually holes in the topological sense, as they don't go all the way through," Steckles said. "They're just blind pits."

By this definition we can rule out all the pores, milk ducts and urethras. We couldn't thread a string in one of these openings and out of another. Even the ears canals have to go as they are separated from the rest of the sinuses by the ear drums.

"We have our mouth, our anus, and then our nostrils. They are four of the … openings that form a hole," Arthur said. "But we actually have eight. The remaining four come from the tear ducts, we each have two in each eye, an upper and a lower."

But this doesn't mean eight holes. Steckles pointed out ."When the holes that pass through a shape connect together inside the shape, it makes it harder to count how many there are."

Looking at underwear

A pair of underwear, for example, has three openings (one for the waist and one for each of the two legs), but it's not immediately clear how many holes a topologist would say it has. "A useful trick is to think about flattening it out," Steckles said. — "If we were to stretch the waistband of the pants out onto a big hula hoop, we'd see the two trouser legs sticking down, each being one hole."

So despite having three openings, the pair of underwear has only two holes. "So when the holes connect together in the middle, there's one fewer hole than there are openings," Steckles argued. Correspondingly, topology tells us that, despite eight interconnected openings, the human body has seven different holes.

But there might be one more. Although often counted as a blind hole, the vagina leads to the uterus, which then leads to one of two fallopian tubes. These tubes are open at the far end and lead to the peritoneal cavity near the ovary. It is the job of the finger-like projections of the funnel-shaped infundibulum at the end of the fallopian tube to catch the egg when it is released from the nearest ovary. However, it has been demonstrated that eggs released from one ovary can be captured by the fallopian tube on the other side, so that passage between the two open ends of the fallopian tubes is possible. Our tiny string could therefore be threaded all the way through the female reproductive tract and back out, counting as one more hole.

So the mathematician's answer is that humans have either seven or eight holes.

In the end, the question is not just about counting openings but about understanding connections. Topologically speaking, our bodies are less like Swiss cheese and more like a carefully constructed onesie for an octopus.

Human skeleton quiz: What do you know about the bones in your body?

In October 2024, news broke that Facebook parent company Meta had cracked an "impossible" problem that had stymied mathematicians for a century.

In this case, the solvers weren't human.

An artificial intelligence (AI) model developed by Meta determined whether solutions of the equations governing certain dynamically changing systems — like the swing of a pendulum or the oscillation of a spring — would remain stable, and thus predictable forever.

The key to the problem was finding Lyapunov functions, which determine the long-term stability of these systems.

Meta's work made headlines and raised a possibility once considered pure fantasy: that AI could soon outperform the world's best mathematicians by cracking math's marquee "unsolvable" problems en masse.

After looking under the hood, however, mathematicians were less impressed. The AI found Lyapunov functions for 10.1% of randomly generated problems posed to it. This was a substantial improvement over the 2.1% solved by previous algorithms, but it was by no means a quantum leap forward. And the model needed lots of hand-holding by humans to come up with the right solutions.

A similar scenario played out earlier this year, when Google announced its AI research lab DeepMind had discovered new solutions to the Navier-Stokes equations of fluid dynamics. The solutions were impressive, but AI was still some distance from solving the more general problem associated with the equations, which would garner its solvers the $1 million Millennium Prize.

Beyond the hype, just how close is AI to replacing the world's best mathematicians? To find out Live Science asked some of the world's best mathematicians.

While some experts were dubious about AI’s problem solving abilities in the short term, most noted that the technology is developing frighteningly fast. And some speculated that not so far into the future, AI may be able to solve hard conjectures — unproven mathematical hypotheses — at a massive scale, invent new fields of study, and tackle problems we never even considered.

"I think what's going to happen very soon — actually, in the next few years — is that AIs become capable enough that they can sweep through the literature at the scale of thousands — well, maybe hundreds, tens of thousands of conjectures," UCLA mathematician Terence Tao, who won the Fields Medal (one of mathematics' most prestigious medals) for his deep contributions to an extraordinary range of different mathematical problems, told Live Science. "And so we will see what will initially seem quite impressive, with thousands of conjectures suddenly being solved. And a few of them may actually be quite high-profile ones."

From games to abstract reasoning

To understand where we are in the field of AI-driven mathematics, it helps to look at how AI progressed in related fields. Math requires abstract thinking and complex multistep reasoning. Tech companies made early inroads into such thinking by looking at complex, multistep logical games.

In the 1980s, IBM algorithms began making progress in games like chess. It's been decades since IBM's Deep Blue beat what was then the world's best chess player, Garry Kasparov, and about a decade since Alphabet's DeepMind defeated the period's best Go player, Lee Sedol. Now AI systems are so good at such mathematical games that there's no point to these competitions because AI can beat us every time.

But pure math is different from chess and Go in a fundamental way: Whereas the two board games are very large but ultimately constrained (or, as mathematicians would say, "finite") problems, there are no limits to the range, depth and variety of problems mathematics can reveal.

In many ways, AI math-solving models are where chess-playing algorithms were a few decades ago. "They're doing things that humans know how to do already," said Kevin Buzzard, a mathematician at Imperial College London.

"The chess computers got good, and then they got better and then they got better," Buzzard told Live Science. "But then, at some point, they beat the best human. Deep Blue beat Garry Kasparov. And at that moment, you can kind of say, 'OK, now something interesting has happened.'"

That breakthrough hasn't happened yet for math, Buzzard argued.

"In mathematics we still haven't had that moment when the computer says, 'Oh, here's a proof of a theorem that no human can prove,'" Buzzard said.

Mathematical genius?

Yet many mathematicians are excited and impressed by AI's mathematical prowess. Ken Ono, a mathematician at the University of Virginia, attended this year's "FrontierMath' meeting organized by OpenAI. Ono and around 30 of the world's other leading mathematicians were charged with developing problems for o4-mini — a reasoning large language model from OpenAI — and evaluating its solutions.

After witnessing the heavily human-trained chatbot in action, Ono said, "I've never seen that kind of reasoning before in models. That's what a scientist does. That's frightening." He argued that he wasn't alone in his high praise of the AI, adding that he has "colleagues who literally said these models are approaching mathematical genius."

To Buzzard, these claims seem far-fetched. "The bottom line is, have any of these systems ever told us something interesting that we didn't know already?" Buzzard asked. "And the answer is no."

Rather, Buzzard argues, AI's math ability seems solidly in the realm of the ordinary, if mathematically talented, human. This summer and last, several tech companies' specially trained AI models attempted to answer the questions from the International Mathematical Olympiad (IMO), the most prestigious tournament for high school "mathletes" around the world. In 2024, Deepmind's AlphaProof and AlphaGeometry 2 systems combined to solve four of the six problems, scoring a total of 28 points — the equivalent of an IMO silver medal. But the AI first required humans to translate the problems into a special computer language before it could begin work. It then took several days of computing time to solve the problems — well outside the 4.5-hour time limit imposed on human participants.

This year's tournament witnessed a significant leap forward. Google's Gemini Deep Think solved five of the six problems well within the time limit, scoring a total of 35 points. This is the sort of performance that, in a human, would have been worthy of a gold medal — a feat achieved by less than 10% of the world's best math students.

Research-level problems

Although the most recent IMO results are impressive, it's debatable whether matching the performance of the top high school math students qualifies as "genius-level."

Another challenge in determining AI's mathematical prowess is that many of the companies developing these algorithms don't always show their work.

"AI companies are sort of shut. When it comes to results, they tend to write the blog post, try and go viral and they never write the paper anymore," Buzzard, whose own research lies at the interface of math and AI, told Live Science.

However, there's no doubt that AI can be useful in research-level mathematics.

In December 2021, University of Oxford mathematician Marc Lackenby's research with DeepMind was on the cover of the journal Nature.

Lackenby's research is in the area of topology which is sometimes referred to as geometry (the maths of shapes) with play dough. Topology asks which objects (like knots, linked rings, pretzels or doughnuts) keep the same properties when twisted, stretched or bent. (The classic math joke is that topologists consider a doughnut and a coffee cup to be the same because both have one hole.)

Lackenby and his colleagues used AI to generate conjectures connecting two different areas of topology, which he and his colleagues then went on to try to prove. The experience was enlightening.

It turned out that the conjecture was wrong and that an extra quantity was needed in the conjecture to make it right, Lackenby told Live Science.

Yet the AI had already seen that, and the team "had just ignored it as a bit of noise," Lackenby said.

Can we trust AI at the frontier of math?

Lackenby's mistake had been not to trust the AI enough. But his experience speaks to one of the current limitations of AI in the realm of research mathematics: that its outputs still need human interpretation and can't always be trusted.

"One of the problems with AI is that it doesn't tell you what that connection is," Lackenby said. "So we have to spend quite a long time and use various methods to get a little bit under the hood."

Ultimately, AI isn't designed to get the "right" answer; it's trained to find the most probable one, said Neil Saunders, a mathematician who studies geometric representation theory at City St George's, University of London and the author of the forthcoming book "AI (r)Evolution" (Chapman and Hall, 2026), told Live Science.

"That most probable answer doesn't necessarily mean it's the right answer," Saunders said.

AI's unreliability means it wouldn't be wise to rely on it to prove theorems in which every step of the proof must be correct, rather than just reasonable.

"You wouldn't want to use it in writing a proof, for the same reason you wouldn't want ChatGPT writing your life insurance contract," Saunders said.

Despite these potential limitations, Lackenby sees AI's promise in mathematical hypothesis generation. "So many different areas of mathematics are connected to each other, but spotting new connections is really of interest and this process is a good way of seeing new connections that you couldn't see before," he said.

The future of mathematics?

Lackenby's work demonstrates that AI can be helpful in suggesting conjectures that mathematicians can then go on to prove. And despite Saunders' reservations, Tao thinks AI could be useful in proving existing conjectures.

The most immediate payoff might not be in tackling the hardest problems but in picking off the lowest-hanging fruit, Tao said.

The highest-profile math problems, which "dozens of mathematicians have already spent a long time working on — they're probably not amenable to any of the standard counterexamples or proof techniques," Tao said. "But there will be a lot that are."

Tao believes AI might transform the nature of what it means to be a mathematician.

"In 20 or 30 years, a typical paper that you would see today might indeed be something that you could automatically do by sending it to an AI," he said. "Instead of studying one problem at a time for months, which is the norm, we're going to be studying 10,000 problems a year … and do things that you just can't dream of doing today."

Rather than AI posing an existential threat to mathematicians, however, he thinks mathematicians will evolve to work with AI.

"We've had situations in the past where entire fields of mathematics became basically solvable by computer," Tao said. At one point, we even had a human profession called a "computer," he added. That job has disappeared, but humans just moved on to harder problems. "It didn't mean mathematics died," Tao said.

Andrew Granville, a professor of number theory at the University of Montreal, is more circumspect about the future of the field. "My feeling is that it's very unclear where we're going," Granville told Live Science. "What is clear is that things are not going to be the same. What that means in the long term for us depends on our adaptability to new circumstances."

Lackenby similarly doesn't think human mathematicians are headed for extinction.

While the precise degree to which AI will infiltrate the subject remains uncertain, he's convinced that the future of mathematics is intertwined with the rise of AI.

"I think we live in interesting times," Lackenby said. "I think it's clear that AI will have an increasing role in mathematics."

A dropped vase, a crushed sugar cube and an exploding bubble all have something in common: They break apart in similar ways, a new mathematical equation reveals.

A French scientist recently discovered the mathematical equation, which describes the size distribution of fragments that form when something shatters. The equation applies to a variety of materials, including solids, liquids and gas bubbles, according to a new study, published Nov. 26 in the journal Physical Review Letters.

Though cracks spread through an object in often unpredictable ways, research has shown that the size distribution of the resulting fragments seems to be consistent, no matter what they're made of — you can always expect a certain ratio of larger fragments to smaller ones. Scientists suspected that this consistency pointed to something universal about the process of fragmenting.

Rather than focusing on how fragments form, Emmanuel Villermaux, a physicist at Aix-Marseille University in France, studied the fragments themselves. In the new study, Villermaux argued that fragmenting objects follow the principle of "maximal randomness." This principle suggests that the most likely fragmentation pattern is the messiest one — the one that maximizes entropy, or disorder.

But that randomness has to obey certain limits. To account for this, Villermaux introduced a conservation law that he and his colleagues discovered in 2015. This law adds physical constraints on the density of fragments in space when an object shatters.

By combining the two principles, Villermaux derived a mathematical equation that describes the pattern of fragment sizes from a shattered object. He then validated the equation by comparing the equation's predictions to years' worth of fragmentation data collected on various objects, including glass, spaghetti, liquid droplets, gas bubbles, plastic fragments in the ocean, and even flakes from early stone tools. All matched the predicted size distribution.

Villermaux also tested the equation by dropping heavy objects onto sugar cubes and observing how they fragmented. "That was a summer project with my daughters," Villermaux told New Scientist. "I did this a long time ago when my children were still young and then came back to the data, because they were illustrating my point well."

However, the newly discovered law doesn't always apply: It doesn't apply in situations with no randomness, such as a smooth stream of liquid breaking into droplets of equal size; and it doesn't cover conditions where the fragments interact with each other, such as in certain plastics.

Ferenc Kun, a physicist at the University of Debrecen in Hungary, told New Scientist that understanding fragmentation could help scientists determine how energy is spent on shattering ore in industrial mining or how to prepare for rockfalls.

Future work could involve determining the smallest possible size a fragment could have, Villermaux told New Scientist.

It's also possible that the shapes of different fragments could follow a similar relationship, Kun wrote in an accompanying viewpoint article.

On a cold day in November, a man living quietly in Russia posted a paper to a public server.

Published by "Grisha Perelman" and titled "The entropy formula for the Ricci flow and its geometric applications," it was the foundation for one of the most important math proofs.

The paper was the first of three published over the next year solving the long-standing Poincaré conjecture, a hypothesis posed nearly a century earlier by Henri Poincaré.

In simple terms, Poincaré hypothesized that if you were to take any kind of 3D space — from a cat to the Empire State Building — and draw a 2D loop on it, if you can shrink that loop down to a point without breaking either the loop or the shape, then the space is mathematically equivalent to a sphere.

Proving this conjecture was crucial to topology, the mathematical study of shapes. Mathematician Stephen Smale had solved the conjecture in five dimensions in 1961, earning math's prestigious Fields Medal in the process. But the 3D case proved the most intractable.

In the 1980s, Richard Hamilton, a mathematician at Columbia University, proposed solving the conjecture using a math technique called Ricci flow, which had been useful for Einstein's theory of general relativity, as well as string theory.

In 2006, New York Times reporter Dennis Overbye likened the Ricci flow technique to using heat from a hair dryer to smooth out shrink-wrap. Similarly, the Ricci flow could smooth out wrinkles and curvature and reduce a complicated shape to a more fundamental one.

Ricci flow worked to simplify roundish shapes to spheres, but singularities — points of infinite density — kept cropping up in more complicated shapes. Topologists can perform a kind of "surgery" to excise these singularities, but there was still a possibility that the singularities would keep emerging forever. Researchers were stuck.

Perelman's work solved the singularity problem. Perelman (whose first name is Grigori, also spelled Grigory; Grisha was a nickname) had spent the prior decade doing postdoctoral research in the U.S. at several institutions. In the mid-1990s, he turned down very prestigious math fellowships in the U.S. and Europe, returned to St. Petersburg, and took a position at the Steklov Institute of Mathematics.

The friendly-but-shy and "unworldly" mathematician "looked like Rasputin, with long hair and fingernails," and he told colleagues he enjoyed hiking in the woods around St. Petersburg, hunting for mushrooms, Robert Greene, a mathematician at UCLA, told Overbye in 2006. He seemed completely uninterested in wealth or material success, his colleagues reported.

Perelman receded into obscurity after he returned to Russia in the mid- to late 1990s, and many of his colleagues thought he had left mathematics altogether.

Then Perelman published his 2002 paper. Over the next year, he published two more papers and gave a series of talks at several East Coast colleges, explaining his process. Then, he receded into the background once more.

Perelman's work showed that all of the singularities actually reduced to simple shapes, like spheres or tubes, and that if you could follow the Ricci process to its end, you would find the 3D shape reduced to a sphere. He had proved the Poincaré conjecture, but it would take another few years for mathematicians to wade through his brilliant, original and highly technical proofs and confirm that the great topographical problem had, indeed, been solved.

In 2006, mathematicians John Morgan and Gang Tian published a 473-page paper showing that Perelman's work, building on Hamilton's, did in fact prove the elusive conjecture.

Perelman was offered the prestigious Fields Medal and the Clay Millennium math prize, which came with a $1 million award. He turned them down, reportedly due to objections about how credit was given for solving the problem.

Perelman resigned from his position at the Steklov Institute in 2005 and has since ferociously avoided the limelight. It's unclear whether he is still working on math in his St. Petersburg apartment, where as of the early 2010s, his neighbors said he cared for his elderly mom.

When a reporter tried to contact him in 2010, he rejected an interview, saying, "You are disturbing me. I am picking mushrooms."

Rogue waves have captivated the attention of both seafarers and scientists for decades. These are giant, isolated waves that appear suddenly in the open ocean.

These puzzling giants are brief, typically lasting less than a minute before disappearing. They can reach heights of 65 feet (20 meters) or greater and often more than twice the height of surrounding waves. Once a nautical myth, rogue waves have now been observed around the world. Because they're so tall and powerful, they can pose a danger to ships and offshore structures.

To rethink what rogue waves are and what causes them, I gathered an international team of researchers. Our study, published in Nature Scientific Reports, sheds light on these oceanic giants using the most comprehensive dataset of its kind.

By analyzing 18 years of high-frequency laser measurements from the Ekofisk oil platform in the central North Sea, we reached the surprising conclusion that rogue waves aren't just freak occurrences. They arise under the natural laws of the sea. They are not mysterious, but somewhat simple.

27,500 sea states

We analyzed nearly 27,500 half-hour wave records, or sea states, collected between 2003 and 2020 in the central North Sea. These records, taken every 30 minutes, describe how elevated the sea surface was compared to the average sea level. They include major storms, such as the Andrea wave event in 2007.

Under normal conditions, waves arise from wind blowing over the sea surface. It's like when you blow over your cup of coffee and form small ripples on the surface. At sea, with enough time and space, those ripples can turn into large waves.

We focused on understanding what causes waves to suddenly go rogue and rise far above their neighboring waves. One proposed theory is based on modulational instability, a phenomenon described by complex mathematical models. I've revised these models in the past, as my work suggests that this theory doesn't fully explain what causes rogue waves in the open ocean.

Related: 4-story rogue wave that randomly appeared in the Pacific Ocean is the 'most extreme' ever detected

When waves are trapped within a narrow channel, the modulational instability theory describes their rippling movement well. However, it starts to fall apart when you look at the real ocean. In open environments such as the North Sea, waves are free to propagate from multiple directions.

To understand the difference, imagine a crowd of spectators leaving a stadium after a football game. If the exit is a long, narrow hallway with tall walls, people are forced to move in a single direction. Those at the back push forward, and some may even climb over others, piling up between the confining walls. This catastrophic pileup would resemble a rogue wave, caused by their confinement.

In contrast, if the stadium's exit opens onto a wide field, spectators can disperse freely in all directions. They don't push on each other, and they avoid pileups.

Similarly, researchers can generate rogue waves in a confined channel in the lab, where they obey modulational instability. But without the confinement of a channel, rogue waves usually won't follow those physics or form the same way in the open sea.

Our team knew we had to study the open sea directly to figure out what was really going on. The real-world data my team examined from the North Sea doesn't line up with modulational instability — it tells a different story.

It's just a bad day at sea

We analyzed the sea state records using statistical techniques to uncover patterns behind these rare events. Our findings show that instead of modulational instability, the extreme waves observed more likely formed through a process called constructive interference.

Constructive interference happens when two or more waves line up and combine into one big wave. This effect is amplified by the natural asymmetry of sea waves — their crests are typically sharper and steeper than their flatter troughs.

Rogue waves form when lots of smaller waves line up and their steeper crests begin to stack, building up into a single, massive wave that briefly rises far above its surroundings. All it takes for a peaceful boat ride to turn into a bad day at sea is a moment when many ordinary waves converge and stack.

These rogue waves rise and fall in less than a minute, following what's called a quasi-deterministic pattern in space and time. This type of pattern is recognizable and repeatable, but with touches of randomness. In an idealized ocean, that randomness would almost vanish, allowing rogue waves to grow to nearly infinite heights. But it would also take an eternity to witness one of these waves, since so many would have to line up perfectly. Like waiting for Fortuna, the goddess of chance, to roll a trillion dice and have nearly all of them land on the same number.

In the real ocean, nature limits how large a rogue wave can grow thanks to wave breaking. As the wave rises in height and energy, it can't hold itself beyond a certain point of no return. The tip of the wave spills over and breaks into foam, or whitecap, releasing the excess energy.

The quasi-deterministic pattern behind rogue waves

Rogue waves aren't limited to the sea. Constructive interference can happen to many types of waves. A general theory called the quasi-determinism of waves, developed by oceanographer Paolo Boccotti, explains how rogue waves form, both in the ocean and in other wave systems.

For example, for turbulent water flowing through a confined channel, a rogue wave manifests in the form of an intense, short-lived spike in vortices — patterns of spinning swirls in the water that momentarily grow larger as they move downstream.

While ocean waves seem unpredictable, Boccotti's theory shows that extreme waves are not completely random. When a really big wave forms, the waves in the sea around it follow a recognizable pattern formed through constructive interference.

We applied Boccotti's theory to identify and characterize these patterns in the measured North Sea wave records.

The giant waves observed in these records carry a kind of signature or fingerprint, in the form of a wave group, which can reveal how the rogue wave came to life. Think of a wave group like a small package of waves moving together. They rise, peak and then fade away through constructive interference. Tracking these wave groups allows researchers to understand the bigger picture of a rogue event as it unfolds.

As one example, a powerful storm hit the North Sea on Nov. 24, 2023. A camera at the Ekofisk platform captured a massive 55 foot (17 meter) rogue wave. I applied the theory of quasi-determinism and an AI model to investigate the origin of this extreme wave. My analysis revealed that the rogue event followed these theories — quasi-determinism and constructive interference — and came from multiple smaller waves repeatedly stacking together.

Recognizing how rogue waves form can help engineers and designers build safer ships and offshore platforms — and better predict risks.

This edited article is republished from The Conversation under a Creative Commons license. Read the original article.

Do you think you've got decent science knowledge? It's time to put your gray matter to the test with our weekly, free science crossword puzzle.

We've spent hours carefully writing our puzzles to make sure they are challenging but accessible to people of all ages and scientific backgrounds, with answers ranging from unusual animals and ancient rulers, to fundamental theories and Nobel Prize-winning scientists. Don't expect it to be easy, but it will be fun!

All you have to do to play is register once and then you should be logged in for next time, and if you need a hint tap the question mark next to the clue to reveal a letter.

Be sure to share this with your friends, and if you want more fun and games, you can also try out one of our amazing science quizzes on more than 50 different topics.

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Previous science crosswords

Want to try luck with our previous crossword puzzles? The most recent ones can be found below, but you can access the full list of science crosswords here.

—#52: The moon's other name — 2 down

—#51: Largest rodent on Earth — 4 down

—#50: Longest-serving president in US history — 1 across

—#49: 'Short' tempered French emperor — 13 across

—#48: Largest fish on Earth — 6 across

The idea of an animal that can count or do math might sound like something out of a viral news story or TikTok video. But a sense of quantity, sometimes called "numerosity," appears across a surprising range of species.

So, which animals can count and understand simple math?

"Many species, including insects, mollusks, lizards, birds and many types of mammals (land living and sea living) can discriminate between quantities of things," Michael Beran, a professor of psychology at Georgia State University, told Live Science in an email. This ability has the evolutionary benefit of helping animals find more food, thus helping them stay alive and pass on their genes.

For instance, research has shown that honeybees (Apis mellifera) count landmarks while flying toward nectar-rich flowers. Golden orb weaver spiders (Nephila clavipes) keep track of how many insects are caught in their webs.

Túngara frogs (Physalaemus pustulosus) even stage numerical duels as part of their mating ritual: One male makes a whining call that ends in a strange, brief sound called a "chuck," and another male frog replies with an extra chuck. This competition goes on, with an increasing number of chucks, until they run out of breath.

Lionesses (Panthera leo) weigh their odds in battle by counting the number of roars from an approaching rival pride before deciding whether to attack or retreat.

And, in 2024, researchers discovered that carrion crows (Corvus corone) are capable of vocalizing a precise number of caws in response to visual or auditory cues, allowing them to count out loud between one and four.

But it's likely that these species "cannot count in the way that we mean this in humans," Beran pointed out.

Instead, many animals have a cognitive tool that scientists call the approximate number system (ANS), or "number sense," Giorgio Vallortigara, a professor of neuroscience at the University of Trento in Italy, told Live Science. The ANS seems to rely on "number neurons," nerve cells that show a peak response to specific quantities, he said. "Interestingly, we found these neurons even in newly hatched chicks, suggesting that the ANS could be innate."

Related: When was math invented?

This "number sense" doesn't work like counting on fingers — it's more about making quick comparisons. Its two defining traits are the distance effect and the size effect. The distance effect is the idea that it's easier to distinguish numbers that are farther apart, such as 8 and 4, compared with 8 and 6, and the size effect is the idea that it's easier to compare smaller numbers than larger ones, even if the difference is the same. For example, it's easier to compare 2 and 4 than it is to compare 12 and 14, Vallortigara said.

The ANS follows Weber's law, which states that animals perceive differences in quantities based on ratios rather than absolute amounts, Irene Pepperberg, an adjunct research professor at Boston University who famously worked with Alex the parrot, told Live Science.

Pepperberg and Beran said the ability to estimate quantities using the ANS is quite different from the human ability to count, which involves knowing that a numeral like "4" means exactly four things, regardless if they are corks, keys or marbles. True counting, Pepperberg explained, also involves learning symbols, understanding the value represented by each numeral and knowing their order. Human children need years of learning to fully grasp these ideas, and "only a very few nonhumans" — Alex the parrot and two chimps called Sheba and Ai — "have gotten anywhere close to true counting," Pepperberg said.

Alex the parrot could identify and correctly order Arabic numerals from one to eight and even add two sets of objects, like crackers or jelly beans.

Can animals do math?

For many researchers, counting is seen as a precursor and foundation of math, but not actually math itself. So although many animals seem capable of basic counting — at least when it comes to noticing changes in quantity, the vast majority are not actually doing math, which at a basic level involves arithmetic: numbers and symbols that are used to do addition, subtraction, multiplication and division.

"Formal arithmetic of the sort our children learn at school is clearly a quite recent cultural invention," Vallortigara said. Some traditional societies, like the Himba tribe in Namibia, still rely on the ANS for estimating quantities, he pointed out. With the rise of agriculture and livestock farming, humans needed more precise calculations, likely giving rise to formal arithmetic.

However, scientists have devised clever experiments to show that a few select species might be able to handle simple math, such as addition and subtraction, Beran said.

When trained to associate certain colors or symbols with arithmetic operations, several animal species — including African grey par­rots, pigeons, cer­tain primates, honeybees, stingrays and cichlids — have demonstrated the ability to perform basic addition and subtraction with small numbers. In these experiments, animals learned to interpret visual cues (like a blue dot for "add one") and then apply those rules to solve simple math problems.

"And, perhaps, with creative designs, even some forms of multiplication and division, which are really just special cases of addition/subtraction [might be possible]," Beran said.

But if math involves larger numbers — like solving 12 + 22 — or more complex formulas, like those used in algebra, "then the case is much, much weaker for nonhuman animal capacities for arithmetic," Beran said.

Pi quiz: How much do you know about this irrational number?

For centuries, prime numbers have captured the imaginations of mathematicians, who continue to search for new patterns that help identify them and the way they're distributed among other numbers. Primes are whole numbers that are greater than 1 and are divisible by only 1 and themselves. The three smallest prime numbers are 2, 3 and 5. It's easy to find out if small numbers are prime — one simply needs to check what numbers can factor them. When mathematicians consider large numbers, however, the task of discerning which ones are prime quickly mushrooms in difficulty. Although it might be practical to check if, say, the numbers 10 or 1,000 have more than two factors, that strategy is unfavorable or even untenable for checking if gigantic numbers are prime or composite. For instance, the largest known prime number, which is 2^136279841 − 1, is 41,024,320 digits long. At first, that number may seem mind-bogglingly large. Given that there are infinitely many positive integers of all different sizes, however, this number is minuscule compared with even larger primes.

Furthermore, mathematicians want to do more than just tediously attempt to factor numbers one by one to determine if any given integer is prime. "We're interested in the prime numbers because there are infinitely many of them, but it's very difficult to identify any patterns in them," says Ken Ono, a mathematician at the University of Virginia. Still, one main goal is to determine how prime numbers are distributed within larger sets of numbers.

Recently, Ono and two of his colleagues — William Craig, a mathematician at the U.S. Naval Academy, and Jan-Willem van Ittersum, a mathematician at the University of Cologne in Germany — identified a whole new approach for finding prime numbers. "We have described infinitely many new kinds of criteria for exactly determining the set of prime numbers, all of which are very different from 'If you can't factor it, it must be prime,'" Ono says. He and his colleagues' paper, published in the Proceedings of the National Academy of Sciences USA, was runner-up for a physical science prize that recognizes scientific excellence and originality. In some sense, the finding offers an infinite number of new definitions for what it means for numbers to be prime, Ono notes.

At the heart of the team's strategy is a notion called integer partitions. "The theory of partitions is very old," Ono says. It dates back to the 18th-century Swiss mathematician Leonhard Euler, and it has continued to be expanded and refined by mathematicians over time. "Partitions, at first glance, seem to be the stuff of child's play," Ono says. "How many ways can you add up numbers to get other numbers?" For instance, the number 5 has seven partitions: 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1 and 1 + 1 + 1 + 1 + 1.

Yet the concept turns out to be powerful as a hidden key that unlocks new ways of detecting primes. "It is remarkable that such a classical combinatorial object — the partition function — can be used to detect primes in this novel way," says Kathrin Bringmann, a mathematician at the University of Cologne. (Bringmann has worked with Ono and Craig before, and she's currently van Ittersum's postdoctoral adviser, but she wasn't involved with this research.) Ono notes that the idea for this approach originated in a question posed by one of his former students, Robert Schneider, who's now a mathematician at Michigan Technological University.

Ono, Craig and van Ittersum proved that prime numbers are the solutions of an infinite number of a particular type of polynomial equation in partition functions. Named Diophantine equations after third-century mathematician Diophantus of Alexandria (and studied long before him), these expressions can have integer solutions or rational ones (meaning they can be written as a fraction). In other words, the finding shows that "integer partitions detect the primes in infinitely many natural ways," the researchers wrote in their PNAS paper.

George Andrews, a mathematician at Pennsylvania State University, who edited the PNAS paper but wasn't involved with the research, describes the finding as "something that's brand new" and "not something that was anticipated," making it difficult to predict "where it will lead."

Related: What is the largest known prime number?

The discovery goes beyond probing the distribution of prime numbers. "We're actually nailing all the prime numbers on the nose," Ono says. In this method, you can plug an integer that is 2 or larger into particular equations, and if they are true, then the integer is prime. One such equation is (3n³ − 13n² + 18n − 8)M₁(n) + (12n² − 120n + 212)M₂(n) − 960M₃(n) = 0, where M₁(n), M₂(n) and M₃(n) are well-studied partition functions. "More generally," for a particular type of partition function, "we prove that there are infinitely many such prime detecting equations with constant coefficients," the researchers wrote in their PNAS paper. Put more simply, "it's almost like our work gives you infinitely many new definitions for prime," Ono says. "That's kind of mind-blowing."

The team's findings could lead to many new discoveries, Bringmann notes. "Beyond its intrinsic mathematical interest, this work may inspire further investigations into the surprising algebraic or analytic properties hidden in combinatorial functions," she says. In combinatorics — the mathematics of counting — combinatorial functions are used to describe the number of ways that items in sets can be chosen or arranged. "More broadly, it shows the richness of connections in mathematics," she adds. "These kinds of results often stimulate fresh thinking across subfields."

Bringmann suggests some potential ways that mathematicians could build on the research. For instance, they could explore what other types of mathematical structures could be found using partition functions or look for ways that the main result could be expanded to study different types of numbers. "Are there generalizations of the main result to other sequences, such as composite numbers or values of arithmetic functions?" she asks.

"Ken Ono is, in my opinion, one of the most exciting mathematicians around today," Andrews says. "This isn't the first time that he has seen into a classic problem and brought really new things to light."

There remains a glut of open questions about prime numbers, many of which are long-standing. Two examples are the twin prime conjecture and Goldbach's conjecture. The twin prime conjecture states that there are infinitely many twin primes — prime numbers that are separated by a value of two. The numbers 5 and 7 are twin primes, as are 11 and 13. Goldbach's conjecture states that "every even number bigger than 2 is a sum of two primes in at least one way," Ono says. But no one has proven this conjecture to be true.

"Problems like that have befuddled mathematicians and number theorists for generations, almost throughout the entire history of number theory," Ono says. Although his team's recent finding doesn't solve those problems, he says, it's a profound example of how mathematicians are pushing boundaries to better understand the mysterious nature of prime numbers.

This article was first published at Scientific American. © ScientificAmerican.com. All rights reserved. Follow on TikTok and Instagram, X and Facebook.

Imagine this scenario: Scientists have intercepted a transmission from an alien race. It's clear that the message comes from an intelligent being, but everything about it — the syntax, the grammar, the context — is unintelligible to us Earthlings.

That's how most mathematicians feel about the Inter-universal Teichmüller Theory (IUT), a proof introduced by mathematician Shinichi Mochizuki over a decade ago in an attempt to solve the famous ABC conjecture, one of the most famous unsolved problems in number theory, which deals with the sum of prime numbers and has implications on many other conjectures.

IUT bears so little resemblance to other branches of math that it's been nicknamed the "alien's language." Only about 20 people in the world have managed to comprehend it to any extent. But now, a 28-year-old engineer named Zhou Zhongpeng has made significant progress in demystifying IUT.

Mochizuki developed IUT in the early 2000s and published it across a series of four preprints in 2012. The proof is over 2,000 pages long, and Mochizuki claims it offers a solution to the ABC conjecture. If proven, the conjecture could help clarify other major mathematical enigmas, such as Fermat's Last Theorem — a nearly 400-year-old theorem that states no three positive integers a, b, and c satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than 2. The theorem was first solved by mathematician Andrew Wiles in 1995, but Zhou's new framework could prove the theorem in much fewer steps.

However, IUT employs concepts and symbols that are wholly unique in the world of math. In other words, Mochizuki essentially created his own mathematical language — and it confounded many of the world's leading mathematicians. A few brave souls, including mathematician Ivan Fesenko, have chipped away at parts of it and lent some credence to Mochizuki's claims. Yet despite existing for over a decade, IUT has not been fully verified by peer review because it is so difficult to understand.

Related: 14-year-old known as 'the human calculator' breaks 6 math world records in 1 day

Enter Zhou. He has a background in mathematics, having studied graph theory as a doctoral candidate, but he ultimately left before completing his degree to work as a software engineer. However, this didn't diminish his interest in pure math. He became obsessed with IUT, studying the theory in his spare time despite a busy workweek. Over the course of five months, he detailed several refinements and new applications in a paper, which he sent to both Mochizuki and Fesenko. The work, if correct, proves the majority of cases of generalized Fermat's Last Theorem, using principles from IUT.

The mathematicians were impressed; Fesenko even offered to fly him out to Westlake University in China, where he works. Zhou accepted the offer and is currently working under Fesenko's tutelage on furthering the proof. There are myriad potential applications of this work, ranging from cryptography and quantum computing to a better understanding of space-time — but only if they're comprehensible to other researchers.

And parts of IUT still remain inscrutable. It will likely be years before someone cracks it fully, if at all. "Those papers are based on the research of predecessors; my work has only made some minor innovations and explorations, and I hope to contribute a modest amount to the relevant field," Zhou said in a social media post.

Editor's Note: This story was corrected at 5:00 p.m. EDT on June 4 to correct the formatting of Fermat's Last Theorem, and to clarify that Zhou's work could prove the theorem in fewer steps than Wiles' version.

Mathematics is the basis of all science and has come a long way since humans started counting. But when did people start doing math? The answer is complicated because abstract mathematics is thought to be different from counting — although counting is the foundation of math — and because many advanced types of mathematics, such as calculus, were developed only within the past few hundred years.

The origins of counting

Humans couldn't have mastered complex and abstract math without figuring out how to count first, and evidence suggests our species was counting tens of thousands of years ago.

The Ishango bone from Africa's Congo region indicates that Homo sapiens have been making "tallies" — a kind of counting — for at least 20,000 years. The 4-inch-long (10 centimeters) bone, probably from a baboon or a bobcat, was found in the 1950s. Researchers think the dozens of parallel notches cut into its surface were a "tally" — a recorded count of some unknown item — and in 1970, archaeologist Alexander Marshack argued it was a six-month lunar calendar.

There's also the Lebombo bone, which was unearthed in southern Africa in the 1970s and was made about 43,000 years ago. It, too, is covered with cut notches and may have been a tally for the 29 days of a lunar month or for a human menstrual cycle.

Danish historian of mathematics Jens Høyrup told Live Science that the very ancient origins of counting could never be known but that it might have been inspired by observations of the night sky by early Homo sapiens, before our species left Africa.

"There was no artificial light then, only the fires within caves," he said. "And when you have no light pollution, the moon and the stars are a wonder to look at."

Related: When did humans discover how to use fire?

Sumerian advances

The next major step in mathematics came with the ancient Sumerians, who are also credited — perhaps coincidently — with inventing cuneiform, the earliest known type of writing.

The Sumerians were one of the first Mesopotamian civilizations, and their city-states thrived in what's now southern Iraq from about 4500 to 1900 B.C. Among their key contributions were numerals that could be written on clay tablets in cuneiform's wedge-shaped marks, and the sexagesimal number system, which is the traditional base-60 system still used today for trigonometry, navigation and timekeeping.

Mathematics, as opposed to simple counting, is the study of patterns and relationships using logical reasoning and abstract concepts. The ancient Sumerians developed the concepts of arithmetic — including tables for multiplication and division — and algebra, where unknown quantities were represented by symbols. They also developed formulas to calculate the areas of triangles, rectangles and irregular shapes, with which they measured land and designed irrigation systems.

St. Lawrence University mathematician Duncan Melville told Live Science these developments were driven by the growing Sumerian bureaucracy.

"Record-keepers needed to know not just what came into or left their stores, but how much or how many," he said in an email. Different mathematical notations were used depending on what was measured, and Sumerian scribes converted between these systems in tasks such as finding the area of a field from its measurements. "In this way we see the beginnings of arithmetic and computational geometry," he said.

Modern math

In addition to the developments of the Sumerians and their Mesopotamian successors, especially the Babylonians, early mathematical expertise and innovations came from ancient Egypt, Greece, India and China, and later from the Islamic civilization.

Mathematics flourished in early modern Europe, where two scientists both claimed to have invented calculus — a way to determine the geometric area enclosed by any curve and an important advance in mathematics that underpins much of modern engineering and science.

One was Isaac Newton, who said he'd invented calculus for his 1687 work "Principia Mathematica" (although he called his calculus "the method of fluxions"), and the other was the German polymath Gottfried Wilhelm Leibniz, who had published a mathematical system of differentials and integrals a few years earlier. (His notation is still used today.)

The two men and their supporters engaged in a bitter dispute about who deserved recognition for the invention, which included allegations that Leibniz had snuck a look at Newton's unpublished manuscript. But historians now think Newton and Leibniz developed calculus independently of each other.

Pi quiz: How much do you know about this irrational number?

Polynomial equations are a cornerstone of modern science, providing a mathematical basis for celestial mechanics, computer graphics, market growth predictions and much more. But although most high schoolers know how to solve simple polynomial equations, the solutions to higher-order polynomials have eluded even seasoned mathematicians.

Now, University of New South Wales mathematician Norman Wildberger and independent computer scientist Dean Rubine have found the first general method for solving these devilishly difficult equations. They detailed their approach April 8 in the journal The American Mathematical Monthly.

A polynomial is a type of algebraic equation that involves variables raised to a non-negative power — for example, x² + 5x + 6 = 0. It is among the oldest mathematical concepts, tracing its roots back to ancient Egypt and Babylon.

Mathematicians have long known how to solve simple polynomials. However, higher-order polynomials, where x is raised to a power greater than four, have proved trickier. The approach most often used to solve two-, three- and four-degree polynomials relies on using the roots of exponential numbers, called radicals. The problem is that radicals often represent irrational numbers — decimals that keep going to infinity, like pi.

Related: Mathematicians just solved a 125-year-old problem, uniting 3 theories in physics

Although mathematicians can use radicals to find approximate solutions to individual higher-order polynomials, they have struggled to find a general formula that works for all of them. That's because irrational numbers can never fully resolve. "You would need an infinite amount of work and a hard drive larger than the universe," Wildberger said in a statement.

In their new method, Wildberger and his colleagues avoided radicals and irrational numbers entirely. Instead, they employed polynomial extensions known as power series. These are hypothetically infinite strings of terms with the powers of x, commonly used to solve geometric problems. They belong to a sub branch of mathematics known as combinatorics.

The mathematicians based their approach on the Catalan numbers, a sequence that can be used to describe the number of ways to break down a polygon into triangles. This sequence was first delineated by Mongolian mathematician Mingantu around 1730 and was independently discovered by Leonhard Euler in 1751. Wildberger and Rubine realized that they could look to higher analogues of the Catalan numbers to solve higher-order polynomial equations. They called this extension "the Geode."

The Geode has numerous potential applications for future research, especially in computer science and graphics. "This is a dramatic revision of a basic chapter in algebra," Wildberger said.

When the greatest mathematician alive unveils a vision for the next century of research, the math world takes note. That's exactly what happened in 1900 at the International Congress of Mathematicians at Sorbonne University in Paris. Legendary mathematician David Hilbert presented 10 unsolved problems as ambitious guideposts for the 20th century. He later expanded his list to include 23 problems, and their influence on mathematical thought over the past 125 years cannot be overstated.

Hilbert's sixth problem was one of the loftiest. He called for "axiomatizing" physics, or determining the bare minimum of mathematical assumptions behind all its theories. Broadly construed, it's not clear that mathematical physicists could ever know if they had resolved this challenge. Hilbert mentioned some specific subgoals, however, and researchers have since refined his vision into concrete steps toward its solution.

In March mathematicians Yu Deng of the University of Chicago and Zaher Hani and Xiao Ma of the University of Michigan posted a new paper to the preprint server arXiv.org that claims to have cracked one of these goals. If their work withstands scrutiny, it will mark a major stride toward grounding physics in math and may open the door to analogous breakthroughs in other areas of physics.

In the paper, the researchers suggest they have figured out how to unify three physical theories that explain the motion of fluids. These theories govern a range of engineering applications from aircraft design to weather prediction — but until now, they rested on assumptions that hadn't been rigorously proven. This breakthrough won't change the theories themselves, but it mathematically justifies them and strengthens our confidence that the equations work in the way we think they do.

Related: 'Can you predict the future? Yes, of course you can.': Inside the 1 equation that can predict the weather, sporting events, and more

Each theory differs in how much it zooms in on a flowing liquid or gas. At the microscopic level, fluids are composed of particles — little billiard balls bopping around and occasionally colliding — and Newton's laws of motion work well to describe their trajectories.

But when you zoom out to consider the collective behavior of vast numbers of particles, the so-called mesoscopic level, it's no longer convenient to model each one individually. In 1872 Austrian theoretical physicist Ludwig Boltzmann addressed this when he developed what became known as the Boltzmann equation. Instead of tracking the behavior of every particle, the equation considers the likely behavior of a typical particle. This statistical perspective smooths over the low-level details in favor of higher-level trends. The equation allows physicists to calculate how quantities such as momentum and thermal conductivity in the fluid evolve without painstakingly considering every microscopic collision.

Zoom out further, and you find yourself in the macroscopic world. Here we view fluids not as a collection of discrete particles but as a single continuous substance. At this level of analysis, a different suite of equations — the Euler and Navier-Stokes equations — accurately describe how fluids move and how their physical properties interrelate without recourse to particles at all.

The three levels of analysis each describe the same underlying reality — how fluids flow. In principle, each theory should build on the theory below it in the hierarchy: the Euler and Navier-Stokes equations at the macroscopic level should follow logically from the Boltzmann equation at the mesoscopic level, which in turn should follow logically from Newton's laws of motion at the microscopic level. This is the kind of "axiomatization" that Hilbert called for in his sixth problem, and he explicitly referenced Boltzmann's work on gases in his write-up of the problem. We expect complete theories of physics to follow mathematical rules that explain the phenomenon from the microscopic to the macroscopic levels. If scientists fail to bridge that gap, then it might suggest a misunderstanding in our existing theories.

Unifying the three perspectives on fluid dynamics has posed a stubborn challenge for the field, but Deng, Hani and Ma may have just done it. Their achievement builds on decades of incremental progress. Prior advancements all came with some sort of asterisk, though; for example, the derivations involved only worked on short timescales, in a vacuum or under other simplifying conditions.

The new proof broadly consists of three steps: derive the macroscopic theory from the mesoscopic one; derive the mesoscopic theory from the microscopic one; and then stitch them together in a single derivation of the macroscopic laws all the way from the microscopic ones.

The first step was previously understood, and even Hilbert himself contributed to it. Deriving the mesoscopic from the microscopic, on the other hand, has been much more mathematically challenging. Remember, the mesoscopic setting is about the collective behavior of vast numbers of particles. So Deng, Hani and Ma looked at what happens to Newton's equations as the number of individual particles colliding and ricocheting grows to infinity and their size shrinks to zero. They proved that when you stretch Newton's equations to these extremes, the statistical behavior of the system — or the likely behavior of a "typical" particle in the fluid — converges to the solution of the Boltzmann equation. This step forms a bridge by deriving the mesoscopic math from the extremal behavior of the microscopic math.

The major hurdle in this step concerned the length of time that the equations were modeling. It was already known how to derive the Boltzmann equation from Newton's laws on very short timescales, but that doesn't suffice for Hilbert's program, because real-world fluids can flow for any stretch of time. With longer timescales comes more complexity: more collisions take place, and the whole history of a particle's interactions might bear on its current behavior. The authors overcame this by doing careful accounting of just how much a particle's history affects its present and leveraging new mathematical techniques to argue that the cumulative effects of prior collisions remain small.

Gluing together their long-timescale breakthrough with previous work on deriving the Euler and Navier-Stokes equations from the Boltzmann equation unifies three theories of fluid dynamics. The finding justifies taking different perspectives on fluids based on what's most useful in context because mathematically they converge on one ultimate theory describing one reality. Assuming that the proof is correct, it breaks new ground in Hilbert's program. We can only hope that with just such fresh approaches, the dam will burst on Hilbert's challenges and more physics will flow downstream.

This article was first published at Scientific American. © ScientificAmerican.com. All rights reserved. Follow on TikTok and Instagram, X and Facebook.

The mysterious Antikythera Mechanism may not have been a cryptic celestial measuring device, but just a toy prone to constant jamming. And the secret to its true purpose, according to new research, is its triangle-shaped teeth.

Discovered in a shipwreck in 1901, the Antikythera Mechanism has remained an enigma for more than a century. Several years ago, CT scans suggested that the 2,000-year-old device was an astronomical tool.

Consisting of a hand crank, many interlocking gears and various indicators, the Mechanism could seemingly perform many tasks. These include giving the date according to the Egyptian and Greek calendars, displaying the positions of the sun, moon and planets within the zodiac, and predicting future lunar and solar eclipses.

But was celestial measurement really the purpose of this fascinating device? Was it meant to be a powerful computer, or a simple toy? And was it a one-off, made for a wealthy patron, or just one example of a mass-produced tool?

For decades, researchers have attempted to understand how accurate the Mechanism is, as that would help resolve its intended purpose. If it wasn't very accurate, for example, then it might have been a toy or an educational model. But if it was incredibly precise, it might have been used by court astrologers to make forecasts and horoscopes.

Related: Antikythera mechanism, world's oldest computer, followed Greek lunar calendar

Unfortunately, the Antikythera Mechanism spent more than two millennia buried at the bottom of the sea, and an unknown amount of time functioning before that. Its gears are highly corroded, and many parts are missing.

Triangles point the way

In the new study, submitted April 1 to the preprint server arXiv, Argentinian scientists created a computer simulation that replicated the Antikythera Mechanism's movements. This simulation incorporated errors from the imprecise nature of its manufacture, where the gears didn't have exact spacing between them.

Crucially, unlike previous efforts to recreate the Mechanism, the researchers also included an accurate model of the Mechanism's triangle-shaped gear teeth, which affect how well gears interlock with one another, and how well the indicators point to the intended astronomical target.

From this model, the researchers found that the Mechanism wasn't very useful at all. It could only be cranked to about four months into the future before it inevitably jammed, or its gears simply disengaged. The user would then have had to reset everything to get it going again — similar to trying to fix a modern printer. Considering that the indicators marking the date cover an entire year, this jamming problem seems unfortunate.

One possibility is that the Antikythera Mechanism was a fancy toy that was never intended to be fully accurate, or that it came with an instruction manual that required users to reset it after a few turns — much like a mechanical watch whose mainspring must be occasionally adjusted by hand.

But given the obvious craftmanship that went into creating such a complex device, the researchers don't believe that the Mechanism was just a janky toy. After all, if it was never intended to be accurate, detailed or forward-looking, why bother putting in all that hard work in the first place?

Another possibility, which the researchers think is more likely, is that current measurements of the gears and teeth are off. CT scans can only provide a certain level of resolution, and two thousand years of corrosion may have warped or distorted the components far beyond their original state. The Mechanism's original creators may have made it precise enough to avoid jamming and still provide reliable predictions for years into the future.

Either way, the Antikythera Mechanism represents the apex of ancient engineering expertise. And despite modern technology and tools such as CT scans, its ultimate purpose may stay forever mysterious.

Navigating a busy crowd is often an awkward experience, but sometimes, it feels much easier than others. In a crowded hallway, people seem to spontaneously organize themselves into lanes, while in an open city square, people travel in every direction, darting from one side to the other.

But what determines the way people move in busy spaces?

Karol Bacik, a mathematician at MIT, and colleagues have developed a mathematical theory that accurately predicts pedestrian flow and the point where it changes from organized lanes to an entangled crowd. The work, which they reported in the journal PNAS March 24, could help architects and city planners design safer and more efficient public spaces that promote ordered crowds.

The team started by creating a mathematical simulation of a moving crowd in different spaces, using fluid dynamics equations to analyze the motion of pedestrians across various scenarios.

"If you think about the whole crowd flowing, rather than individuals, you can use fluid-like descriptions," Bacik said in a statement. "If you only care about the global characteristics like, are there lanes or not, then you can make predictions without detailed knowledge of everyone in the crowd."

Crowd math

Both the width of the space and the angles at which people moved across it heavily influenced the overall order of the crowd. Bacik's team identified "angular spread" — the number of people walking in different directions — as the key factor in whether people self-organized into lanes.

Related: 14-year-old known as 'the human calculator' breaks 6 math world records in 1 day

Where the spread of people walking in different directions is relatively small — such as in a narrow corridor or on pavement — pedestrians tend to form lanes and meet oncoming traffic head-on. However, a broader range of individual travel directions — for example, in an open square or airport concourse — dramatically increases the likelihood of disorder as pedestrians dodge and weave around one another to reach their separate destinations.

The tipping point, according to this theoretical analysis, was an angular spread of around 13 degrees, meaning ordered lanes could descend into disordered flow once pedestrians start traveling at more extreme angles.

"This is all very common sense," Bacik said. "[But] now we have a way to quantify when to expect lanes — this spontaneous, organized, safe flow — versus disordered, less efficient, potentially more dangerous flow."

However, the researchers were keen to investigate whether the reality of a human crowd bears out this theory, so they devised an experiment to simulate a busy road crossing. Volunteers, each wearing a paper hat labeled with a unique barcode, were assigned various start and end positions and were asked to walk between opposite sides of a gymnasium without bumping into other participants. An overhead camera recorded each scenario, tracking both the movement of individual pedestrians and the overall motion of the crowd.

Subsequent analysis of the 45 trials confirmed the importance of angular spread, showing a transition from ordered lanes to disordered movement at angles close to the theoretically predicted 13 degrees. Furthermore, as disorder increased, pedestrians were forced to move more slowly to avoid collisions, with a roughly 30% speed reduction for random crowds versus ordered lanes, the team found.

Bacik's team is now looking to test these predictions in real-world scenarios, and they hope the work will ultimately help improve crowded environments.

"We would like to analyze footage and compare that with our theory," he said. "We can imagine that, for anyone designing a public space, if they want to have a safe and efficient pedestrian flow, our work could provide a simpler guideline, or some rules of thumb."

Originally defined as the ratio between the circumference of a circle and its diameter, pi — written as the Greek letter π — appears throughout mathematics, including in areas that are completely unconnected to circles such as chemistry, physical sciences and medicine.

Pi belongs to a huge mathematical group called irrational numbers, which go on forever and cannot be written as fractions. Scientists have calculated pi to 105 trillion digits, although most of us are more familiar with the approximation 3.14. But how do we know that pi is an irrational number?

Rational numbers, which make up the majority of numbers we use in day-to-day life (although less than half of all possible numbers), can be written in the form of one whole number divided by another. Pi, with its complicated string of decimals, certainly doesn't appear to be part of this group at first glance.

"Rationality is the practical property of having access to the number explicitly, i.e. without any approximation … so being able to write the number in a finite amount of symbols," Wadim Zudilin, a mathematician at Radboud University in the Netherlands, told Live Science.

Related: What is the largest known prime number?

However, actually proving that you can't write pi as a fraction is a surprisingly knotty issue. Mathematicians don't have a universal method to show that a particular number is irrational, so they must develop a different proof for each case, explained Keith Conrad, a mathematician at the University of Connecticut. "How do you know a number is not a fraction?" he said. "You're trying to verify a negative property."

Despite this difficulty, over the past 300 years, mathematicians have established different proofs of pi's irrationality, using techniques from across mathematics. Each of these arguments begins with the assumption that pi is rational, written in the form of an equation. Through a series of manipulations and deductions about the properties of the unknown values in this equation, it subsequently becomes clear that the math contradicts this original assertion, leading to the conclusion that pi must be irrational.

The specific math involved is often incredibly complex, typically requiring a university-level understanding of calculus, trigonometry and infinite series. However, each approach relies on this central idea of proof by contradiction.

"There are proofs using calculus and trigonometric functions," Conrad said. "In some of them, π is singled out as the first positive solution to sin(x) = 0. The first proof by Lambert in the 1760s used a piece of mathematics called infinite continued fractions — it's a kind of infinitely nested fraction."

However, rather than proving pi is irrational directly, it's also possible to confirm irrationality using a different property of the number. Pi belongs to another numerical group called transcendental numbers, which are not algebraic and, importantly, cannot be written as the root of a polynomial equation. Because every transcendental number is irrational, any proof showing that pi is transcendental also proves that pi is irrational.

"Using calculus with complex numbers, you can prove π is transcendental," Conrad said. "The proof uses the very famous equation called Euler's identity: e^iπ +1 = 0."

Although pi's universal importance may arise from this intangible irrationality, seven or eight decimal places is usually more than sufficient for any real-world applications. Even NASA uses only 16 digits of pi for its calculations.

"We approximate the value for practical purposes, 3.1415926 — that's already a lot of information!" Zudilin said. "But of course in mathematics, it's not satisfactory. We care about the nature of the numbers."

Pi Day quiz: How much do you know about this irrational number?

As one of the most famous scientists of the 20th century, Albert Einstein has become synonymous with genius. He taught himself geometry and calculus while living in Germany in his teens before training in Switzerland to become a math teacher. Unable to find a teaching position, Einstein bounced around until he completed a doctorate in physics in 1905 and began publishing groundbreaking scientific papers that ultimately changed the world.

But Einstein had other interests that are less well known — he was a staunch pacifist, invented an early refrigerator and called himself a "deeply religious non-believer." What do you know about the man, the myth, the genius Einstein? Time to take a quantum leap, because this quiz will test what you know about the famous physicist!

Related: 32 fun and random facts about Albert Einstein

Remember to log in to put your name on the leaderboard; hints are available if you click the yellow button. Consider the gravity of the situation and be sure to ace this quiz at the speed of light!

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On March 14 mathematicians everywhere enjoy a slice of pie as they recite the digits of math's most famous constant. Pi, defined as the ratio between the circumference of a circle and its diameter, is what's known as an irrational number. Despite its ubiquity, not only across math but also in fields as varied as medicine, space exploration, and geography, this number can't be written as a fraction and is instead expressed as a decimal with digits going on forever.

For most of us, this fundamental constant is usually approximated to 3.14, but math enthusiasts are constantly trying to push the boundaries and calculate pi to as many digits as possible.

Pi Day is celebrated on March 14th (3/14 — see what they did there?) and is the perfect occasion for mathematicians to celebrate their favourite number. Sharing math puzzles, strange pi facts, and new records is all part of the fun. So to celebrate Pi Day — or just to celebrate Pi — here's a quick quiz to test your knowledge. If you need a clue, press the yellow button.

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A teenager has set multiple math world records by performing calculations in his head. Fourteen-year-old Aaryan Shukla from Maharashtra, India, recently took down no less than six mental math world records in a single day — proving he's earned his nickname, "the Human Calculator." His times were certified by Guinness World Records.

Shulka's feats include setting the fastest known time to mentally add 100 four-digit numbers (30.9 seconds), 200 four-digit numbers (1 minute, 9.68 seconds), and 50 five-digit numbers (18.71 seconds), as well as the fastest time to multiply two five-digit number sets of 10 (51.69 seconds) and two eight-digit number sets of 10 (2 minutes, 35.41 seconds), and the fastest time to divide a set of 10 20-digit numbers by a set of 10-digit numbers (5 minutes, 42 seconds). He crunched these numbers quicker than most people can punch the digits into a calculator.

Shulka set these new records at an event hosted by Guinness in Dubai. Shulka's successful record-setting attempt was captured on video, which is available to watch for free on the Guinness World Records website, and in the player below.

Related: Infamous 'sofa problem' that boggled mathematicians for decades may finally have a solution

A 'one in a billion kind of person'

In addition to these six most recent records, Shulka already holds the record for the quickest time to add 50 five-digit numbers, which he set a year ago. He credits his numerical prowess to his yoga practice, which "helps me keep calm and focused," he told People Magazine in an interview. Shulka also practices math for five or six hours a day, in-between more typical teenage hobbies like reading and playing video games.

In an interview with Guinness, Shulka's father insisted that extraordinary math skills don't run in the family. "We are a normal family," he said. "Aaryan is a one in a billion kind of person, but I don't think that we are a family of mental calculators."

Shulka is not the only teen earning attention for tackling complicated and impressive mathematical problems. In 2022, American high school seniors Ne'Kiya Jackson and Calcea Johnson discovered a new "impossible" proof for the Pythagorean theorem — the 2,000-year-old theorem that describes the relationships between a right triangle's three sides. The teens used trigonometry to prove the theorem, a feat that was previously considered unworkable by mathematicians. Their work was published in the peer-reviewed journal American Mathematical Monthly in 2024, in a paper that included nine more trigonometry-based proofs that no one had ever come up with before.

Twenty-five years too late to help Ross get his new couch into his apartment in "Friends," a mathematician has finally solved the pesky "sofa problem."

The math problem delineates the largest-size sofa that can fit around a corner of a given width — exactly the problem faced by the characters in an episode of "Friends" that aired in 1999. Ross' pleas of "Pivot!" could have been avoided, it turns out, if he'd only considered a Gerver's sofa with 18 curve sections and a maximum area of 2.2195 units. (Okay, so maybe it wouldn't have been that helpful.)

The solution to the sofa problem is a first for mathematics. The problem was posited by Austrian-Canadian mathematician Leo Moser in 1966. Moser asked for the largest possible area of a single shape in one plane that could move around a right-angled corner of a hallway with a unit width of one. While this might seem simple, the math is quite complicated, as the problem involves both area maximization and movement of the shape.

Now, Jineon Baek, a postdoctoral researcher in mathematics at Yonsei University in South Korea, has arrived at an answer. Baek posted his solution on Dec. 2 on the preprint website ArXiv. In just over 100 pages of mathematical proofs, Baek found that for a hallway with a width of 1 unit, the imaginary sofa's maximum area can be 2.2195 units — narrowing the answer down with precision from the previously known range of between 2.2195 and 2.37 units. The proof has not yet been published in a peer-reviewed journal and will need to be worked through by other mathematicians to determine that it is, indeed, optimal.

Related: High school students who came up with 'impossible' proof of Pythagorean theorem discover 9 more solutions to the problem

The "Gerver" of Gerver's sofa is mathematician Joseph Gerver, an emeritus professor at Rutgers University who posited the lower bound of 2.2195 in 1992. But there had been debate over whether the sofa could be larger, with a team in 2018 using a computer-assisted proof to suggest that 2.37 was actually the upper bound.

Gerver's sofa is a broad U-shaped couch with a curved "seat" that can squeeze around the corner without getting hung up. The question was whether this painstakingly designed sofa — made of 18 separate curves put together — was really the largest, most optimal shape that could make the turn. Baek worked through the geometry of the shape and its movement and found that Gerver's solution was, in fact, correct.

The proof created a ripple of interest on social media.

"This is the optimal sofa," user @morallawwithin wrote on the social platform X on Dec. 6, posting a picture of the rather wide-armed sofa shape. "You may not like it, but this is what peak optimization looks like."

Name: Babylonian geometry homework

What it is: A cuneiform mathematical clay tablet with an incorrect answer

Where it is from: Tell Ingharra, Kish (Sumer), modern-day Iraq

When it was made: Old Babylonian period, between 1900 and 1600 B.C.

Related: Mask of Xiuhtecuhtli: A 600-year-old mask of the Aztec fire god taken as treasure by conquistadors

What it tells us about the past:

This round clay tablet, which is in the collection of the Ashmolean Museum at the University of Oxford, is one of two dozen examples of ancient Babylonian mathematics homework found at the archaeological site of Kish in 1931.

However, the student who used this tablet as a "scratch pad" to calculate the area of a triangle made a key mistake, and their error has been preserved for nearly 4,000 years.

The tiny tablet is just 3.2 inches (8.2 centimeters) in diameter and depicts a right triangle with three sets of cuneiform style numbers — one set along each of the two sides representing the length and height of the triangle, and one in the middle for its area.

Along the top line (height) of the triangle, the student has written 3.75, while the vertical line (base) is indicated as 1.875. These values mean the area of the triangle should be 3.5156. The student, though, has incorrectly calculated it as 3.1468.

Several of these ancient scratch pads have been found at Kish and nearby Babylon, both of which were major areas of early mathematics education. While this particular tablet is blank on the back, other examples include a teacher's composition on one side and the student's on the other.

The rise of mathematics education in ancient Babylon corresponded to the time when large empires began to develop. Higher math techniques like algebra and geometry were likely invented around 3000 B.C. in Sumer as the growing civilization needed ways to calculate taxes, tally up trade and commerce and establish calendars.

Babylonian mathematics had a base 60 number system, which we still use today to tell time — 60 seconds in a minute, 60 minutes in an hour. And these ancient people understood the Pythagorean theorem more than a millennium before the ancient Greek philosopher Pythagoras became famous for establishing that the sum of the squares of two sides of a right triangle equals the square of the hypotenuse.

So this student's messed-up math actually shows an important cultural development: The way people accumulated and passed on knowledge was switching from memorization to written information. This switch — which began around 3500 B.C. in Kish — was so dramatic that it is sometimes compared to the switch from paper to digital recordkeeping in the 20th century.

Mathematicians have stumped the most advanced generative artificial intelligence (AI) models with a series of mind-bending new math problems.

These problems typically require doctorate-level mathematicians hours to days to solve, according to the research institute Epoch AI. But in the new tests, the most advanced AI models on the market got correct answers on less than 2% of these problems.

In the past decade, a number of AI tests have been developed to determine whether the answers these models return are actually correct. In many cases, AI models now breeze through these benchmarks.

For example, in the commonly used Measuring Massive Multitask Language Understanding (MMLU) benchmark test, today's AI models answer 98% of math problems correctly.

Most of these benchmarks are geared toward testing AI's ability to do high-school and college-level math, Elliot Glazer, a mathematician at Epoch AI, and colleagues wrote in a new paper posted on the preprint database arXiv. (The paper has not yet been peer-reviewed or published in a scientific journal.)

Related: Scientists design new 'AGI benchmark' that indicates whether any future AI model could cause 'catastrophic harm'

The new set of benchmarks, called FrontierMath, aims for a higher level of reasoning. Epoch AI developed the questions with the help of mathematics professors, including some winners of the Fields Medal, perhaps the most prestigious prize in math. The problems cover a wide range of subfields, from number theory to algebraic geometry, and are available on Epoch AI's website.

"These are extremely challenging," 2006 Fields Medal winner Terence Tao, a mathematician at UCLA, wrote in a review of the problems for Epoch AI. "I think that in the near term basically the only way to solve them, short of having a real domain expert in the area, is by a combination of a semi-expert like a graduate student in a related field, maybe paired with some combination of a modern AI and lots of other algebra packages."

The problems were also unique — a step taken to ensure that none of the problems were already in the AI models' training data. When complex reasoning problems are included in the training data, the AI may appear to solve the problems, but in reality, it already has a "cheat sheet," since it has been trained on the answers.

The researchers tested six state-of-the-art AI models: Google's Gemini 1.5 Pro (002), Anthropic's Claude 3.5 Sonnet, OpenAI's o1-preview, o1-mini, and GPT4o and xAI's Grok-2 Beta. Gemini and Claude managed to solve 2%, which was just slightly better than the showings from o1-preview, o1-mini and GPT-4o's 1%. Grok-2 Beta failed to get any problems right.

However, these rankings are misleading because the low success rate means that a single right answer can have an outsize impact on each model's overall score, the researchers cautioned.

"[E]ven when a model obtained the correct answer, this does not mean that its reasoning was correct," the paper authors wrote. "For instance, on one of these problems running a few simple simulations was sufficient to make accurate guesses without any deeper mathematical understanding. However, models' low overall accuracy shows that such guessing strategies do not work on the overwhelming majority of FrontierMath problems."

The findings show that right now, AI models don't possess research-level math reasoning, Epoch AI's collaborators concluded. However, as AI models advance, these benchmark tests will provide a way to find out if their reasoning abilities are deepening.

"By regularly evaluating state-of-the-art models and collaborating with the AI research community," the team wrote in the statement, "we aim to deepen our understanding of AI’s capabilities and limitations."

Two students who discovered a seemingly impossible proof to the Pythagorean theorem in 2022 have wowed the math community again with nine completely new solutions to the problem.

While still in high school, Ne'Kiya Jackson and Calcea Johnson from Louisiana used trigonometry to prove the 2,000-year-old Pythagorean theorem, which states that the sum of the squares of a right triangle's two shorter sides are equal to the square of the triangle's longest side (the hypotenuse). Mathematicians had long thought that using trigonometry to prove the theorem was unworkable, given that the fundamental formulas for trigonometry are based on the assumption that the theorem is true.

Jackson and Johnson came up with their "impossible" proof in answer to a bonus question in a school math contest. They presented their work at an American Mathematical Society meeting in 2023, but the proof hadn't been thoroughly scrutinized at that point. Now, a new paper published Monday (Oct. 28) in the journal American Mathematical Monthly shows their solution held up to peer review. Not only that, but the two students also outlined nine more proofs to the Pythagorean theorem using trigonometry.

"To have a paper published at such a young age — it's really mind-blowing," Johnson, who is now studying environmental engineering at Louisiana State University, said in a statement emailed to Live Science. "I am very proud that we are both able to be such a positive influence in showing that young women and women of color can do these things."

Related: Largest known prime number, spanning 41 million digits, discovered by amateur mathematician using free software

By proving Pythagoras' theorem using trigonometry, but without using the theorem itself, the two young women overcame a failure of logic known as circular reasoning. Trigonometry is a branch of mathematics that lays out how the sides, lengths and angles in a triangle are related, and as such, the discipline often includes expressions of the Pythagorean theorem. But Jackson and Johnson managed to prove the theorem using a result of trigonometry called the Law of Sines, dodging circular reasoning.

In the new study, and on top of their initial proof, the young mathematicians described four new ways to prove Pythagoras' theorem using trigonometry, as well as a novel method that revealed five more proofs, totaling 10 proofs.

Jackson and Johnson are only the third and fourth people known to have proven the Pythagorean theorem using trigonometry and without resorting to circular reasoning. The two other people were professional mathematicians, according to the statement.

"I didn't think it would go this far," Jackson, who currently studies pharmacology at the Xavier University of Louisiana, said in the statement. "I was pretty surprised to be published."

In the paper, Jackson and Johnson say there are two ways to present trigonometry and its functions sine and cosine, but these versions are often conflated into one. Sine and cosine are ratios that are defined in the context of a triangle's right angle, and they can be presented according to either the trigonometric method or a method that uses polynomials of complex numbers, according to the paper.

The conflation means that "trying to make sense of trigonometry can be like trying to make sense of a picture where two different images have been printed on top of each other," Jackson and Johnson wrote.

By teasing the two methods apart, researchers can discover "a large collection of new proofs of the Pythagorean theorem," the young mathematicians added.

If you liked reading this story, here are some mathematics books you might also enjoy:

The largest known prime number has been discovered by an amateur researcher and former Nvidia employee.

The new number is 2^136,279,841 – 1, which beats the previous title holder (2^82,589,933 – 1) by more than 16 million digits.

Prime numbers, described by mathematicians as the "atoms of integers," are numbers that are divisible only by themselves and 1. The smallest prime numbers are 2, 3, 5, 7 and 11. Technically, prime numbers run to infinity, but finding them becomes significantly harder the bigger they get.

To find the new prime, Luke Durant used a free program called the Great Internet Mersenne Prime Search, or GIMPS, to sift through the possibilities with an algorithm. His efforts required the harnessing of thousands of graphics processing units (GPUs) across 24 data centers in 17 countries — a feat that "ends the 28-year reign of ordinary personal computers finding these huge prime numbers," according to a statement released on the GIMPS website.

The newly confirmed prime number contains 41,024,320 decimal digits, according to the statement.

Related: Pi calculated to 105 trillion digits, smashing world record

The new prime number is also the 52nd known Mersenne prime — a series named after Marin Mersenne, a French monk and polymath who devised a formula for finding prime numbers by subtracting 1 from powers of 2. (The smallest Mersenne prime is 3 — or 2 to the power of 2, minus 1.) Though far from being the only way to discover primes, the method is slightly easier than others.

As for the usefulness of the discovery, "At present there are few practical uses for these large Mersenne primes, prompting some to ask, 'Why search for these large primes?'" the GIMPS team wrote in the statement. "Those same doubts existed a few decades ago until important cryptography algorithms were developed based on prime numbers."

The discovery has netted Durant a $3,000 cash prize from GIMPS. Further prizes of $150,000 and $250,000 await those who discover the first hundred-million-digit prime and the first billion-digit prime, respectively.

On October 16 1843, the Irish mathematician William Rowan Hamilton had an epiphany during a walk alongside Dublin's Royal Canal. He was so excited he took out his penknife and carved his discovery right then and there on Broome Bridge.

It is the most famous graffiti in mathematical history, but it looks rather unassuming:

i ² = j ² = k ² = ijk = –1

Yet Hamilton's revelation changed the way mathematicians represent information. And this, in turn, made myriad technical applications simpler — from calculating forces when designing a bridge, an MRI machine or a wind turbine, to programming search engines and orienting a rover on Mars. So, what does this famous graffiti mean?

Rotating objects

The mathematical problem Hamilton was trying to solve was how to represent the relationship between different directions in three-dimensional space. Direction is important in describing forces and velocities, but Hamilton was also interested in 3D rotations.

Mathematicians already knew how to represent the position of an object with coordinates such as x, y and z, but figuring out what happened to these coordinates when you rotated the object required complicated spherical geometry. Hamilton wanted a simpler method.

He was inspired by a remarkable way of representing two-dimensional rotations. The trick was to use what are called "complex numbers", which have a "real" part and an "imaginary" part. The imaginary part is a multiple of the number i, "the square root of minus one", which is defined by the equation i ² = –1.

Related: Imaginary numbers could be needed to describe reality, new studies find

By the early 1800s several mathematicians, including Jean Argand and John Warren, had discovered that a complex number can be represented by a point on a plane. Warren had also shown it was mathematically quite simple to rotate a line through 90° in this new complex plane, like turning a clock hand back from 12.15pm to 12 noon. For this is what happens when you multiply a number by i.

Hamilton was mightily impressed by this connection between complex numbers and geometry, and set about trying to do it in three dimensions. He imagined a 3D complex plane, with a second imaginary axis in the direction of a second imaginary number j, perpendicular to the other two axes.

It took him many arduous months to realize that if he wanted to extend the 2D rotational wizardry of multiplication by i he needed four-dimensional complex numbers, with a third imaginary number, k.

In this 4D mathematical space, the k-axis would be perpendicular to the other three. Not only would k be defined by k ² = –1, its definition also needed k = ij = –ji. (Combining these two equations for k gives ijk = –1.)

Putting all this together gives i ² = j ² = k ² = ijk = –1, the revelation that hit Hamilton like a bolt of lightning at Broome Bridge.

Quaternions and vectors

Hamilton called his 4D numbers "quaternions", and he used them to calculate geometrical rotations in 3D space. This is the kind of rotation used today to move a robot, say, or orient a satellite.

But most of the practical magic comes into it when you consider just the imaginary part of a quaternion. For this is what Hamilton named a "vector".

A vector encodes two kinds of information at once, most famously the magnitude and direction of a spatial quantity such as force, velocity or relative position. For instance, to represent an object's position (x, y, z) relative to the "origin" (the zero point of the position axes), Hamilton visualised an arrow pointing from the origin to the object's location. The arrow represents the "position vector" x i + y j + z k.

This vector's "components" are the numbers x, y and z — the distance the arrow extends along each of the three axes. (Other vectors would have different components, depending on their magnitudes and units.)

Half a century later, the eccentric English telegrapher Oliver Heaviside helped inaugurate modern vector analysis by replacing Hamilton's imaginary framework i, j, k with real unit vectors, i, j, k. But either way, the vector's components stay the same — and therefore the arrow, and the basic rules for multiplying vectors, remain the same, too.

Hamilton defined two ways to multiply vectors together. One produces a number (this is today called the scalar or dot product), and the other produces a vector (known as the vector or cross product). These multiplications crop up today in a multitude of applications, such as the formula for the electromagnetic force that underpins all our electronic devices.

A single mathematical object

Unbeknown to Hamilton, the French mathematician Olinde Rodrigues had come up with a version of these products just three years earlier, in his own work on rotations. But to call Rodrigues' multiplications the products of vectors is hindsight. It is Hamilton who linked the separate components into a single quantity, the vector.

Everyone else, from Isaac Newton to Rodrigues, had no concept of a single mathematical object unifying the components of a position or a force. (Actually, there was one person who had a similar idea: a self-taught German mathematician named Hermann Grassmann, who independently invented a less transparent vectorial system at the same time as Hamilton.)

Hamilton also developed a compact notation to make his equations concise and elegant. He used a Greek letter to denote a quaternion or vector, but today, following Heaviside, it is common to use a boldface Latin letter.

This compact notation changed the way mathematicians represent physical quantities in 3D space.

Take, for example, one of Maxwell's equations relating the electric and magnetic fields:

∇ × E = –∂B/∂t

With just a handful of symbols (we won't get into the physical meanings of ∂/∂t and ∇ ×), this shows how an electric field vector (E) spreads through space in response to changes in a magnetic field vector (B).

Without vector notation, this would be written as three separate equations (one for each component of B and E) — each one a tangle of coordinates, multiplications and subtractions.

The power of perseverance

I chose one of Maxwell's equations as an example because the quirky Scot James Clerk Maxwell was the first major physicist to recognise the power of compact vector symbolism. Unfortunately, Hamilton didn't live to see Maxwell's endorsement. But he never gave up his belief in his new way of representing physical quantities.

Hamilton's perseverance in the face of mainstream rejection really moved me, when I was researching my book on vectors. He hoped that one day — "never mind when" — he might be thanked for his discovery, but this was not vanity. It was excitement at the possible applications he envisaged.

He would be over the moon that vectors are so widely used today, and that they can represent digital as well as physical information. But he'd be especially pleased that in programming rotations, quaternions are still often the best choice — as NASA and computer graphics programmers know.

In recognition of Hamilton's achievements, maths buffs retrace his famous walk every October 16 to celebrate Hamilton Day. But we all use the technological fruits of that unassuming graffiti every single day.

This edited article is republished from The Conversation under a Creative Commons license. Read the original article.

Whether the sight of an equation makes you jump for joy or run to the hills, there is no doubting that so much of science is guided by the principles laid out in these beautiful collections of symbols and numbers. But from medical testing to artificial intelligence, one mathematical rule guides much of the modern world — Bayes' theorem.

To this day, the seemingly simple equation, developed by an 18th-century Presbyterian minister and amateur mathematician, is used in modeling and forecasting to help us predict everything from future weather events, fluctuations in the stock market to the winners of sporting events.

The book "Everything is Predictable," by award-winning science writer Tom Chivers is a captivating tour of this curious theorem and how it impacts modern life, and has been shortlisted for the prestigious 2024 Royal Society Trivedi Science Book Prize. Below is a short excerpt from the book's introduction, which explores to what extent we can predict the future.

Related: 32 sci-fi technology predictions that came true

Can you predict the future? Yes, of course you can.

You can predict with near-certain accuracy that in the next few seconds, you'll take a breath, and let it out again. Your heart will beat, somewhere between one and three times a second. Tomorrow morning, the sun will come up, at a particular time which depends upon your latitude and the time of year but which nonetheless you can find out with great accuracy. All of these events you can predict with confidence.

You can also predict that the train will arrive at a certain time, or that your friend will arrive on time at the restaurant at which you've arranged to meet her. Though, depending on the rail company, or your friend, you might be less confident in that.

And you can predict that the world's population will continue to grow until around the middle of the century, and then start to fall again. You can predict that global average surface temperatures in the year 2030 will be higher than they were in the year 1930.

The future isn't opaque. You can see into it. Some parts are more predictable than others – the Newtonian dance of the planets we can predict out for thousands of years; the Lorenzian chaos of the weather, really only a few days. But you can peer through the murk, after a fashion.

That's not what people normally mean when they say, "I can predict the future." They are referring to something mystical, some psychic or magical vision. We probably can't do that. (You'll read about a scientist in this book who thinks we can, and you’ll also read about why he's almost certainly wrong.) But we don’t need to. All that we do, all the time, is predict the future. We couldn’t function if we couldn’t. We make very basic predictions, like "the air will continue to be breathable," implicitly, with every breath we take. We make more complex predictions, like "The corner shop will have Alpen [a breakfast cereal] when I get there," each time we make a decision. We're not basing them on mystical visions, but on information we have gathered in the past.

The thing with all these predictions is that they are uncertain. The universe may or may not be deterministic; perhaps if we had perfect, God-like knowledge of the position, movement and qualities of every particle in the universe, we could perfectly predict everything, the fall of every sparrow. But we don't. Instead, we have partial information. We can see bits of the universe, imperfectly, through our imperfect senses. We have best guesses for the way those bits move — we know the human-shaped bits tend to seek food and company; we know the rock-shaped bits tend to sit still. We can make messy, imperfect predictions with that information.

Life isn’t chess, a game of perfect information, one that can in theory be "solved." It's poker, a game where you're trying to make the best decisions using the limited information you have. This book is about the equation that lets you do that.

Excerpted from "Everything is predictable: How Bayesian Statistics Explain our World." Copyright © 2024 by Tim Chivers.

Physicists may have created the world's most difficult maze using a chess sequence, and it could help them understand the properties of otherworldly quasicrystals. 

The maze is an example of a Hamiltonian cycle — a path that visits all of the points on a graph at least once. The researchers were inspired by the movement of a knight around a chessboard. The result is an infinitely expandable fractal maze that describes the structure of quasicrystals. The researchers published their findings May 1 in the journal Physical Review X.

"When we looked at the shapes of the lines we constructed, we noticed they formed incredibly intricate mazes," study lead author Felix Flicker, a physicist at the University of Bristol in the U.K., said in a statement. "The sizes of subsequent mazes grow exponentially — and there are an infinite number of them."

First represented in the irregular, non-repeating tile patterns of early Islamic art, quasicrystals are very rare crystals whose atoms fit into an ordered arrangement and yet never repeat. They are crystals, but they stubbornly break the rules of symmetry that scientists once used to divide traditional crystals from more chaotically structured solids.

Related: Pi calculated to 105 trillion digits, smashing world record

First theorized in 1981 and discovered in 1982, the once-controversial structures got Dan Shechtman, the scientist who found them, booted from his lab for defending his discovery — before later earning him the 2011 Nobel Prize in chemistry. Since then, quasicrystals have been synthesized in labs, discovered in meteorites and fossilized lightning, and found to have formed in the wake of the Trinity bomb test in 1945.

To recreate the quasicrystals' bizarre structure, the researchers in the new study used a 2D version of Ammann-Beenker tiling, a type of aperiodic tiling similar to Penrose tiles. The researchers created an algorithm to find a Hamiltonian cycle over these tiles, enabling them to mathematically represent each atom inside a quasicrystal from beginning to end.

The result is an infinitely scalable fractal maze, but modeling quasicrystals has much deeper applications than a mind-boggling pattern. The researchers said their Hamiltonian cycle offers the fastest way for scanning tunneling microscopes to scan an object. It also provides insight into how complex proteins fold and offers suggestions for how to efficiently capture carbon dioxide molecules from the atmosphere.

"We show that certain quasicrystals provide a special case in which the problem is unexpectedly simple," Flicker said. "In this setting, we therefore render some seemingly-impossible problems tractable."

For many, the idea of math will bring back endless hours of formulas and equations at school. So it may seem hard to imagine, but there once was a time when arithmetic didn't exist. Of course, there was still the need to use complex computations to solve real-world problems, but it wasn't until Muhammad ibn-­Mūsā al-­Khwārizmī, the so-called "father of algebra," established the fundamentals for solving equations that we began to set the foundations for modern mathematics.

In this excerpt from her new book "Vector: A Surprising Story of Space, Time, and Mathematical Transformation", mathematician Robyn Arianrhod explores the 4,000-year evolution of the language of mathematics — from complex descriptions to the symbolic form we know today.

Learning to think symbolically

Algebra has been part of mathematics since records began nearly 4,000 years ago, but not always in the symbolic form we learn today. In fact, for most of those four millennia it was written entirely in words and numerals — although works such as Euclid’s famous 300 B.C.E. textbook "Elements" also included geometric diagrams, to help prove such things as Pythagoras's theorem, and to show how to expand squares that we would write today as (a+b)².

So "algebra" was communicated in cumbersome word problems or increasingly complicated diagrams — although geometry did have its advantages. For instance, it's the easiest way to prove Pythagoras's theorem. In figure 1.1, I've given an algebraic adaptation of such a proof, although the ancients simply rearranged the diagram to show visually that the shaded area is equal to the sum of the areas of the squares on the adjacent sides of the triangle — a pretty clever approach!

It took a long time for algebra to emerge from arithmetic and geometry as a separate subject. It didn't even get its name until medieval times, and that was thanks to the ninth-­century Persian mathematician Muhammed ibn-­Mūsā (al-­)Khwārizmī… He studied at Caliph al-­Ma’mūn’s pioneering Baghdad-­based university, or "House of Wisdom," when the great Arabic translation movement was at its height: Greek, Indian, and other ancient manuscripts were being collected from all corners of the burgeoning Islamic empire and translated into Arabic.

Imperialism is rarely ethical and often violent, but it can ultimately lead to cultural cross-­fertilization, and in this case the visionary translation movement was so important that by the 12th century, Europeans were learning Arabic in order to translate these manuscripts into Latin — including Ptolemy’s "Almagest" and Euclid’s "Elements," along with new Arabic works such as those of al-­Khwārizmī. The name "algebra” famously comes from the first word in the title of his book "Al-Jabr wa’l muqābalah" — which means something like "The Compendious Book on Calculation by Completion and Balancing."

Judging from the problems al-­Khwārizmī included, an example of what he meant by "Completion" is "completing the square," the method you might have learned in school to solve quadratic equations...

Al-­Khwārizmī didn't write equations in the symbolic form we use today, either. In fact, to modern eyes his book is more arithmetical than algebraic, and one of its important impacts in Europe, when it was translated into Latin, was the popularization of the Hindu-­Arabic decimal system of numeration that eventually evolved into our modern one.

Yet Al-­Khwārizmī is often called the "father of algebra." He may have used words rather than symbols, and the problems he included may have been simple — his purpose, he tells us, was to teach students how to solve basic problems in "cases of inheritance, legacies, partitions, lawsuits and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned."

But he systematically set out word-­form linear and quadratic equations, with algorithmic methods for solving them — that is, for finding the "unknown numbers," our modern x's and y's. In fact, the English word "algorithm” — meaning a set of rules for performing a calculation or other operation — comes from "algorismi,” an early Latinized attempt at Al-­Khwārizmī.

…

The beauty of symbolic equations is that it's much easier to see these general patterns when you can see a problem at a glance. Compare this:

Take the square of the unknown number,

then add the unknown number to itself

and take the sum away from the square;

now let the total be eight.

with this:

x²–2x=8

And there's more: The earliest mathematicians solved each equation separately, but it's easier if you can see that whatever method works for the equation x²–2x=8 will also work for any equation of the same form, x²–ax=b. Eventually, ancient mathematicians did begin to recognize this, but progress was relatively slow because they had to keep all these patterns in their heads, or in long, convoluted sentences, and it was easy to lose track.

The first to publish any equation in a transparent, recognizably modern symbolic form were [Thomas] Harriot's executors in 1631, and then [René] Descartes in an appendix to his 1637 "Discourse on Method." (There were a few earlier attempts, but the symbolism — more properly called abbreviation — was tortured and idiosyncratic.) Even the +, −, =, and × signs we take for granted only came into widespread use in the 17th century. Which means that the earlier algebraists we know of — the ancient Mesopotamians, Egyptians, Chinese, and Greeks, the medieval Indians, Persians, and Arabs, as well as the early modern Europeans — all had expressed their equations mostly in words or pictorial word images.

Related: 9 equations that changed the world

It is a singular skill to think symbolically, as this long history shows. Take the word problem I gave above: it is an example of algorithmic thinking. But symbolic thinking is algorithmic and more, for its symbols sometimes contain the seeds of a new kind of creativity — a new kind of far-­reaching yet economical thought.

A classic case is Albert Einstein’s E=mc². Einstein did not set out to find the connection between energy and matter. Rather, he simply wanted to calculate the kinetic energy of a moving electron according to his new theory of relativity, so that his theoretical prediction could be tested experimentally.

A few months later, however, 26-­year-­old Einstein began to realize the significance of his equation. He wrote it up in his fifth groundbreaking paper of 1905, his annus mirabilis, but it would take him two more years to tease out the full, dramatic implications of this symbolic relationship. To realize that this wasn’t just a calculation about a particular form of energy and a particular type of matter, it was general: if a body gains (or loses) energy, it also gains (or loses) mass. This bizarre idea is alien to all our commonsense experience — but there it was, hidden in the symbols of his equation. It took experimental physicists decades to experimentally confirm this astonishing mathematical prediction.

A much simpler and earlier example is the sequence of powers x, x², x³ and so on. The first "power" is 1, so x is really x¹ , where the 1 was traditionally linked geometrically to a 1-­D line. The next two, x² and x³, are pronounced "x squared" and "x cubed" by analogy with the area of a square and the volume of a cube. These names highlight the way that early mathematicians thought geometrically rather than algebraically, because of the tangible nature of geometry. By contrast, symbolic algebra is abstract: you have to give it meaning, even if it is simply the display of an interesting pattern such as x, x², x³, x⁴,... But this flexibility is algebra's great strength. You can write down as many (finite) higher powers as you like, without having to visualize them as physical objects.

This may sound obvious today, but it took three and a half thousand years for mathematicians to move from solving quadratic equations — "quadratic" derives from the Latin for "square," so quadratic equations are those whose highest power is x² (the unknown multiplied by itself, as the ancients put it) — to solving "cubic" and higher equations. These higher-degree equations are much more difficult, of course; but part of the reason solutions didn't come easily was that algebra was tied to words and concrete images for such a very long time.

For instance, I mentioned Al-­Khwārizmī’s "completing the square" in order to solve a quadratic equation. It's actually a 4,000-­year-­old problem, dating back (as far as the historical record shows) to cuneiform tablets made by mathematicians living, like Al-Khwārizmī, in the region of modern-­day Iraq. These ancient Mesopotamians solved quadratic equations by literally completing a square.

Here is a typical teaching problem of the time: "Add 20 of my length to the area of my square, [to get] 21. How square is my square?" This type of problem, and the algorithm for solving it, is similar to those taught today — except that four millennia ago, the method was worked out entirely geometrically. First, draw a square of arbitrary side x (in modern notation); then add to it a rectangle of dimensions 20 [by] x. Now split this additional rectangle into two equal smaller ones and arrange them beside and below the original square. Finally, complete this new, larger square, as in figure 1.2.

The Mesopotamians had practical problems in mind when they developed this method, at least initially. Living in a land where water was at a premium, their tablets contain many problems relating to canal and reservoir excavations, the capacity of cisterns, the construction and repair of dams and levees, and administrative accounts relating to these tasks — and to solve these problems, these ancient mathematicians had to solve equations relating to areas and volumes.

Nearly 3,000 years later, Al-­Khwārizmī, too, focused on similar practical problems, and he used a similar geometrical method of completing the square — and so did other mathematicians right up to the 17th century.

This excerpt has been edited for style and length. Reprinted with permission from "Vector: A Surprising Story of Space, Time, and Mathematical Transformation" by Robyn Arianrhod, published by The University of Chicago Press. © 2024 by Robyn Arianrhod. All rights reserved.

The 2023 Turing Award has been given to Avi Wigderson, a mathematician who discovered the strange connection between computation and randomness. 

Wigderson was announced the winner of the Association for Computing Machinery (ACM) A.M. Turing Award, often called the "Nobel Prize of Computing," on April 10, 2024.

The award, given with a prize of $1 million, comes just three years after Wigderson, a professor of mathematics at the Institute for Advanced Study in Princeton, New Jersey, won the 2021 Abel Award for his contributions to computer science. Wigderson's theoretical work has been key to the development of numerous advances in computing, from cloud networks to cryptography methods that underpin cryptocurrencies.

"Wigderson is a towering intellectual force in theoretical computer science, an exciting discipline that attracts some of the most promising young researchers to work on the most difficult challenges," Yannis Ioannidis, president of the ACM, said in a statement. "This year's Turing Award recognizes Wigderson's specific work on randomness, as well as the indirect but substantial impact he has had on the entire field of theoretical computer science."

Related: Scientists uncover hidden math that governs genetic mutations

Computer algorithms are deterministic by nature, which enables them to make predictions but also limits their grasp of the messy randomness found in the real world. In fact, many problems are considered computationally “hard”, and deterministic algorithms struggle to solve them efficiently.

But Wigderson and his colleague Richard Karp, a computer scientist at the University of California, Berkeley, found a way to tame computational hardness. After inserting randomness into their algorithms, they found that they made some problems much easier to solve.

Wigderson chased this observation, proving in later work that the reverse also applied: Randomness could always be stripped from probabilistic algorithms to transform them into deterministic ones. His findings illuminated the connection between computational hardness and randomness in ways that reshaped computer science.

"From the earliest days of computer science, researchers have recognized that incorporating randomness was a way to design faster algorithms for a wide range of applications," Jeff Dean, chief scientist at Google Research and Google DeepMind, said in the statement. "Efforts to better understand randomness continue to yield important benefits to our field, and Wigderson has opened new horizons in this area."

A data storage company has decoded more than 100 trillion digits of pi — smashing the world record for calculating the never-ending number. Unraveling this hefty slice of pi required the equivalent computing power of hundreds of thousands of smartphones.   

Pi — often abbreviated as 3.14 — is an irrational number, meaning it has infinite nonrepeating decimal places. The value of pi is equal to the circumference of a circle (the distance around its edge) divided by its diameter (the distance between two directly opposite points). It means you can figure out the circumference of any circle if you know its diameter or radius (half the diameter) or vice versa because we know the value of pi.

Unraveling the hidden decimal places of pi has no real impact on mathematics because calculations rarely require more than a few dozen digits. For example, NASA scientists only need to know the first 15 decimal places of pi to understand most of the universe. Instead, calculating the number to its most exact value has long been used as a benchmark for testing new computer programs and data storage systems.

On Pi Day (March 14), Solidigm — a U.S. computer storage company based in California — revealed in a statement that it has calculated pi to approximately 105 trillion decimal places. 

To put that into context, if you typed out this number on paper using a 10-point font in one continuous line, the number would be around 2.3 billion miles (3.7 billion kilometers) long, meaning it could stretch from Earth to somewhere between Uranus and Neptune. And in case you were wondering, the 105 trillionth digit of pi is 6.

Related: 12 numbers that are cooler than pi

The calculation, which took around 75 days to complete, was carried out with 36 of the company's proprietary solid-state drives (SSDs) — a storage medium fitted into many of the newest laptops — that stored altogether around 1 petabyte (1 million gigabytes) of data. 

Processors are also needed to perform the number-crunching — with more powerful components reducing the time it takes to perform the necessary calculations. However, reliable and large-capacity storage is arguably more important because you need to store a massive amount of data in such a process.  

The achievement "was no small feat," Solidigm owner Brian Beeler said in the statement. "It involved meticulous planning, optimization, and execution."

In April 2023, Solidigm matched the record of 100 trillion digits of pi, which was calculated by Google Cloud in 2022.

Before that, the record was 62.8 trillion digits, which were calculated over 108 days by a supercomputer at the University of Applied Sciences of the Grisons in Switzerland in 2021. Going back even further, the record was set at 50 trillion digits in 2020 by Timothy Mullican of Huntsville, Alabama, using his personal computer. 

Using the human computer (the brain), the current world record for the most digits of pi memorized by a person is 70,000, which was achieved by Rajveer Meena at the VIT University in India, on March 21, 2015, according to Guinness World Records.

As computers continue to get more powerful in the future, we will inevitably start to uncover larger and larger slices of pi. However, no matter how powerful computers get, we will never be able to unravel the entire number due to its infinite nature. 

Happy Pi Day 2024!

Have you ever wondered what it would be like to solve problems for NASA to help with the exploration of other planets in the solar system?

In celebration of Pi Day 2024, you can do just that and take the annual NASA Pi Day Challenge. This is a fun way to put on your scientist and engineer thinking cap and try your best at a series of questions all surrounding the mathematical constant, pi. 

What is pi? If you recall from mathematics class back in grade school, it's approximately 3.14159 and can be used to figure out the circumference of a circle of the volume of a square.

While there are many uses for it in different STEM jobs and fields, it's also very important for engineers and scientists at NASA to help study not just our planet but others across the solar system and even other galaxies.

Related: Pi Day 2024: Why NASA uses only 16 of the 62 trillion digits of pi we know

This challenge is a tradition that has been on-going for the last decade put on by NASA's Jet Propulsion Laboratory's Education Office and features numerous math problems you have to solve using pi.

Some of the questions you can answer this year pertain to missions including the Deep Space Optical Communications technology on NASA's Psyche spacecraft, the Double Asteroid Redirection Test (DART) spacecraft, Earth-orbiting satellites, rovers on the Moon, and even the Hubble Space Telescope and James Webb Space Telescope.

So, let's get solving! You can find each of the problems with an accompanying worksheet you can do all your work online and the answers will be posted by NASA so you can check your work!

There are nearly four dozen different questions you can figure out, so try a few or do them all to "cook up" a unique way to get space-y and celebrate Pi Day 2024!

Originally posted on Space.com.

Math nerds everywhere are digging into a slice of pie today to celebrate their most iconic irrational number: pi. After all, March 14, or 3/14, is the perfect time to honor the essential mathematical constant, whose first digits are 3.14.

Pi, or π, is the ratio of a circle's circumference to its diameter. Because it is irrational, it can't be written as a fraction. Instead, it is an infinitely long, nonrepeating number.

But how was this irrational number discovered, and after thousands of years of being studied, does this number still have any secrets? From the number's ancient origins to its murky future, here are some of the most surprising facts about pi.

Related: The 9 most massive numbers in existence

Memorizing pi

The record for the most digits of pi memorized belongs to Rajveer Meena of Vellore, India, who recited 70,000 decimal places of pi over the course of 10 hours on March 21, 2015, according to Guinness World Records. (As of March 2024, Meena still holds this record.) Previously, Chao Lu, of China, who recited pi from memory to 67,890 places in 2005, held the record, according to Guinness World Records.

The unofficial record holder is Akira Haraguchi, who videotaped a performance of his recitation of 100,000 decimal places of pi in 2005, and more recently topped 117,000 decimal places, the Guardian reported.

Number enthusiasts have memorized many digits of pi. Many people use memory aids, such as mnemonic techniques known as piphilology, to help them remember. Often, they use poems written in Pilish (in which the number of letters in each word corresponds to a digit of pi). Speaking of which...

There's a pi "language"

Literary nerds invented a dialect known as Pilish, in which the numbers of letters in successive words match the digits of pi. For example, Mike Keith wrote the book "Not A Wake" (Vinculum Press, 2010) entirely in Pilish:

Now I fall, a tired suburbian in liquid under the trees, Drifting alongside forests simmering red in the twilight over Europe.

("Now" has three letters, "I" has one letter, "fall" has four letters, and so on.)

Exponential increase

Because pi is an infinite number, humans will, by definition, never determine every single digit of pi. However, the number of decimal places calculated has grown exponentially since pi's first use. The Babylonians thought the fraction 3 1/8 was good enough in 2000 B.C., while the ancient Chinese and the writers of the Old Testament (Kings 7:23) seemed perfectly happy to use the integer 3. But by 1665, Sir Isaac Newton had calculated pi to 16 decimal places. By 1719, French mathematician Thomas Fantet de Lagny had calculated 127 decimal places, according to "A History of Pi" (St. Martin's Press, 1976).

The advent of computers radically improved humans' knowledge of pi. Between 1949 and 1967, the number of known decimal places of pi skyrocketed from 2,037 on the ENIAC computer to 500,000 on the CDC 6600 in Paris, according to "A History of Pi" (St. Martin's Press, 1976). 

Of course, the latest computations blow those early records out of the water. 

The most digits of pi ever calculated

In 2021, researchers in Switzerland used a supercomputer to smash the existing record for calculating pi.

Running the supercomupter for 108 days straight, the team calculated pi to more than 62.8 trillion decimal places, blowing past the previous computational record by more than 12 trillion decimal places. That's a lot of pi.

This level of pi precision is unprecedented. However, it may be wasted on one of the world's top science agencies...

NASA only uses 16 digits of pi

How many digits of pi does NASA need to make high-precision calculations about the universe? Way less than you'd think.

According to NASA officials, the space agency rarely needs to use more than 16 digits of pi (or 3.141592653589793) to make accurate calculations about our solar system and its cosmic neighborhood. Using any longer versions of pi start to give severely diminishing returns, according to NASA.

For example: Earth has a diameter of around 7,900 miles (12,700 kilometers), which means its circumference is around 24,900 miles (40,100 km). If you were to calculate this circumference with the first 16 digits of pi and a more accurate version of pi with hundreds of decimal places, the difference between the two answers would be around 300 times less than the width of a human hair, according to NASA. 

When looking at the scale of the entire universe, larger versions of pi can be useful, the agency added. But with most of the interesting space action happening within our solar system, NASA rarely needs to go bigger than 3.141592653589793.

Hand-calculating pi

Those who are hoping to calculate pi using an old-fashioned technique can accomplish the task using a ruler, a can and a piece of string, or a protractor and a pencil. The downside of the can method is that it requires a can that is actually round, and the accuracy is limited by how well a person can loop string around its circumference. Similarly, drawing a circle with a protractor and then measuring its diameter or radius with a ruler involves a fair amount of dexterity and precision.

A more precise option is to use geometry. Break up a circle into multiple segments (such as eight or 10 pizza slices). Then, calculate the length of a straight line that would turn the slice into an isosceles triangle, which has two sides of equal length. Adding up all the sides yields a rough approximation for pi. The more slices you create, the more accurate the approximation of pi will be.

Discovery of pi

The ancient Babylonians knew of pi's existence nearly 4,000 years ago. A Babylonian tablet from between 1900 B.C. and 1680 B.C. calculates pi as 3.125, and the Rhind Mathematical Papyrus of 1650 B.C., a famous Egyptian mathematical document, lists a value of 3.1605. The King James Bible (I Kings 7:23) gives an approximation of pi in cubits, an archaic unit of length corresponding to the length of the forearm from the elbow to the middle finger tip (estimated at about 18 inches, or 46 centimeters), according to the University of Wisconsin-Green Bay. The Greek mathematician Archimedes (287-212 B.C.) approximated pi using the Pythagorean theorem, a geometric relationship between the length of a triangle's sides and the area of the polygons inside and outside of circles.

Pi rebranded

Prior to the association of the symbol pi with the circle constant, mathematicians had to say a mouthful to even describe the number. One phrase found in the old math books was the Latin phrase "quantitas in quam cum multiflicetur diameter, proveniet circumferencia," which roughly translates to "the quantity which, when the diameter is multiplied by it, yields the circumference," according to History Today.

The irrational number rocketed to fame when Swiss polymath Leonhard Euler used it in 1737 in his disquisitions on trigonometry. But it didn't get its pithier, Greek-symbol name from Euler. The first mention of pi as such occurred in a book by a lesser-known mathematician, William Jones, who used it in 1706 in his book "Synopsis Palmariorum Matheseos." Jones likely used the symbol for pi to denote the periphery of a circle, according to the book "A History of Pi," (St. Martin's Press, 1976).

Is pi normal?

Pi is definitely weird, but is it normal? Though mathematicians have plumbed many of the mysteries of this irrational number, there are still some unanswered questions.

Mathematicians still don't know whether pi belongs in the club of so-called normal numbers — or numbers that have the same frequency of all the digits — meaning that 0 through 9 each occur 10 percent of the time, according to Trueb's website pi2e.ch. In a paper published Nov. 30, 2016, in the preprint journal arXiv, Trueb calculated that, at least based on the first 2.24 trillion digits, the frequency of the numbers 0 through 9 suggest pi is normal. Of course, given that pi has an infinite number of digits, the only way to show this for sure is to create an airtight math proof. So far, proofs for this most famous of irrational numbers has eluded scientists, though they have come up with some bounds on the properties and distribution of its digits.

Pi sounds divine

While scientists don't know whether pi is normal, they have a better understanding of its other traits. Eighteenth-century mathematician Johann Heinrich Lambert proved pi's irrationality by expressing the tangent of x using a continued fraction.

Later, mathematicians showed that pi was also transcendental. In math terminology, transcendental means the number can't be the solution to any polynomial that has rational number coefficients. In other words, there's no finite, root-finding formula that can be used to calculate pi using rational numbers.

Downgrading pi

While many mathletes are enamored with pi, there is a resistance movement growing. Some argue that pi is a derived quantity, and that the value tau (equal to twice pi) is a more intuitive irrational number.

Tau directly relates the circumference to the radius, which is a more mathematically consequential value, Michael Hartl, author of the "Tau Manifesto," previously told Live Science. Tau also works better in trigonometric calculations, so that tau/4 radians corresponds to an angle that sweeps a quarter of a circle, for instance.

Pi likes a party

There wasn’t always a Pi Day (filled with delectably filled pies, of course). In 1988, physicist Larry Shaw launched the pi-partying day at the San Francisco-based Exploratorium science museum. Every year, on March 14 (3/14) staff and visitors walk a circular parade (and yes the diameter of the circle is like pi times its circumference), each holding one of the infinite numbers of pi. But Pi Day didn’t become a national event until 2009, when the House of Representatives passed Resolution 224, "supporting the designation of Pi Day". The goal? An increased interest in math and science. Let the celebrations begin!

Editor's note: An earlier version of this article was updated on March 14, 2024 to include new updates and information. Live Science contributor Tanya Lewis and editor Brandon Specktor contributed to this article.

Pi is an irrational number, meaning it has an infinite number of nonrepeating decimal places. But it turns out, NASA scientists need only a small slice of pi — the first 15 decimal places — to solve most of their math problems. And even when working out problems on the most mind-bending cosmic scales, they never need more than a few dozen extra digits.    

You may remember pi from school. It's the ratio between a circle's diameter (the distance between opposite points) and circumference (the distance around the edge). Simply put, pi is equal to the circumference of a circle divided by its diameter, which means you can figure out the circumference of any circle if you know its diameter or radius (half the diameter) and pi, which is most commonly abbreviated as 3.14 — which is why Pi Day is celebrated on March 14.   

We currently know more than 105 trillion digits of pi, thanks to a computer company that crunched the number for 75 days. But after that, pi's decimal places are a complete mystery to us.

However, you need to memorize only the first few decimals of pi for most real-world applications. And even if you're a NASA scientist, you can get away with knowing just 15, especially if you're only studying our solar system. (With 15 decimal places, pi looks like this: 3.141592653589793.)

The reason scientists don't need to bother including any more digits in their calculations is that the numbers they are using, even at planetary or stellar scales, are too small for additional decimal places to have any real effect on the output value, according to NASA's Jet Propulsion Laboratory.

Related: The universe would not make sense without mathematics

Take our planet: Earth has a diameter of around 7,900 miles (12,700 kilometers), which means its circumference is around 24,900 miles (40,100 km). If you were to calculate this exact circumference with the first 16 digits of pi (the number three followed by 15 decimal places) and a more accurate version of pi with hundreds of decimal places, the difference between the two answers would be around 300 times less than the width of a human hair, according to NASA. 

The error gets greater the larger the number you use. But for most practical calculations NASA scientists need to make, the difference is still negligible. 

For example, the Voyager 1 probe, which is now in interstellar space, is currently more than 15 billion miles (24 billion km) from Earth. If you wanted to calculate the circumference of a circle with this distance as the radius, the difference between using the first 16 digits of pi compared with hundreds of digits would be less than the width of a little finger, according to NASA.  

However, some calculations still require more decimal places. 

For example, if you wanted to calculate the circumference of a circle that encapsulated the known universe, the radius of that circle would be around 46 billion light-years — the distance light has traveled since the Big Bang when factoring in the expansion of the universe. In this case, you would need 38 decimals of pi to get a value with the same level of accuracy with which we can currently measure the width of an atom, according to NASA.

Edit: This article was updated on March 15 after a new world record for the known digits of pi was achieved.

The decimal point is 150 years older than historians thought it was, newfound notes from 15th-century Italy reveal.

Decimal points are so simple, it seems like they should have existed forever. These handy mathematical tools break up whole numbers into tenths, hundredths and thousandths, making computation much simpler than with fractions. And some versions of decimals have been around since the 900s (in Damascus) or the 1200s (in China).

But a consistent system of decimals wasn't fully cemented until 1593, when German mathematician Christopher Clavius used decimals in an astronomical treatise. Now, new research suggests Clavius was playing with an older tradition, picking up the use of decimals from a 15th-century Venetian merchant named Giovanni Bianchini.

Related: The World's Most Beautiful Equations

Bianchini's work dates to between 1441 and 1450, making the decimal point a century and a half older than Clavius' use of it, according to the authors of the new research.

While teaching a math camp for middle schoolers, Glen Van Brummelen, a historian of mathematics at Trinity Western University in Canada, noticed the use of the decimal in one of Bianchini's treatises.

"I remember running up and down the hallways of the dorm with my computer trying to find anybody who was awake, shouting 'look at this, this guy is doing decimal points in the 1440s!'" Van Brummelen told Nature News.

The idea of breaking up whole numbers into pieces is very old, but most mathematicians prior to the Middle Ages used fractions. Astronomers did use decimals, but not in the familiar base-10 system that elementary schoolers learn today. Instead, they used base-60 decimals, created by dividing 360-degree circles into 60 minutes, which could then be subdivided into 60 seconds.

Occasionally, mathematicians did use notations that are reminiscent of today's decimal system, Van Brummelen wrote in a paper published online for the journal Historia Mathematica. But these ideas tended to fizzle instead of getting passed down from mathematician to mathematician.

"Thus, attempting to identify a 'first' among this host of different players may be a fool's errand, depending on one's criteria for the historical actor's level of appreciation of the power of operations with decimal fractions and the persistence of their systems," Van Brummelen wrote.

However, it's easier to pin down the history of the decimal point — a symbol that persists to this day, he wrote. And that notation first appears in Bianchini's "Tabulae primi mobilis B," a text on the calculation of stellar coordinates. Bianchini was a merchant who became an administrator to the ruling family of Venice at the time, the d'Estes. As part of this job, he was responsible for calculating horoscopes and astrology. In some of the tables in his text, he uses the decimal point just as mathematicians do today.

Though the notation was slow to catch on, Clavius would have known about Bianchini, José Chabás, a historian of astronomy at the Pompeu Fabra University in Barcelona, Spain, told Nature News. And writers inspired by Clavius picked up the decimal point and ran with it. Finally, Scottish mathematician John Napier, the inventor of logarithms, cemented the decimal point in mathematics in the early 1600s.

Feb. 29 is a date that happens only once every four years, but what are leap years? Why do we need them? And how did they come about?

Leap years are years with 366 calendar days instead of the normal 365. They happen every fourth year in the Gregorian calendar — the calendar used by the majority of the world. The extra day, known as a leap day, is Feb. 29, which does not exist in non-leap years. Every year that is divisible by four, such as 2020 and 2024, is a leap year except for some centenary years, or years that end in 00, such as 1900. (We'll explain why further down.)   

The name "leap" comes from the fact that from March onward, each date of a leap year moves forward by an extra day from the previous year. For example, March 1, 2027 will be a Monday but in the next leap year, 2028, it will fall on a Wednesday. (Normally, the same date only moves forward by a single day between consecutive years.) 

Related: Earth's core wobbles every 8.5 years, new study suggests

Other calendars, including the Hebrew calendar, Islamic calendar, Chinese calendar and Ethiopian calendar, also have versions of leap years, but these years don't all come every four years and often occur in different years than those in the Gregorian calendar. Some calendars also have multiple leap days or even shortened leap months.

In addition to leap years and leap days, the Gregorian calendar also has a handful of leap seconds, which have sporadically been added to certain years — most recently in 2012, 2015 and 2016. However, the International Bureau of Weights and Measures (IBWM), the organization responsible for global timekeeping, will abolish leap seconds from 2035 onward.

Why do we need leap years? 

On the face of it, all of this "leaping" may seem like a silly idea. But leap years are very important, and without them our years would eventually look very different. 

Leap years exist because a single year in the Gregorian calendar is slightly shorter than a solar, or tropical, year — the amount of time it takes for Earth to completely orbit the sun once. A calendar year is exactly 365 days long, but a solar year is roughly 365.24 days long, or 365 days, 5 hours, 48 minutes and 56 seconds. 

If we did not account for this difference, then for each year that passes the gap between the start of a calendar year and a solar year would widen by 5 hours, 48 minutes and 56 seconds. Over time, this would shift the timing of the seasons. For example, if we stopped using leap years, then in around 700 years the Northern Hemisphere's summer would begin in December instead of June, according to the National Air and Space Museum.

Related: How many times has Earth orbited the sun?

Adding leap days every fourth year largely removes this problem because an extra day is around the same length as the difference that accumulates during this time. 

However, the system is not perfect: We gain around 44 extra minutes every four years, or a day every 129 years. To solve this problem, we skip the leap years every centenary year except for those that are divisible by 400, such as 1600 and 2000. But even then, there is still a tiny difference between calendar years and solar years, which is why the IBWM have experimented with leap seconds. 

But overall, leap years mean that the Gregorian calendar stays in sync with our journey around the sun.

When was the last leap year? When is the next leap year?

The history of leap years 

The idea of leap years dates back to 45 B.C. when the Ancient Roman emperor Julius Caesar instituted the Julian calendar, which was made up of 365 days separated into the 12 months we still use in the Gregorian calendar. (July and August were originally named Quintilis and Sextilis respectively but were later renamed after Julius Caesar and his successor Augustus.)

The Julian calendar included leap years every four years without exception and was synced up to Earth's seasons thanks to the "final year of confusion" in 46 B.C., which included 15 months totaling 445 days, according to the University of Houston.

For centuries, it appeared that the Julian calendar worked perfectly. But by the mid-16th century, astronomers noticed that the seasons were beginning around 10 days earlier than expected when important holidays, such as Easter, no longer matched up with specific events, such as the vernal, or spring, equinox.

To remedy this, Pope Gregory XIII introduced the Gregorian calendar in 1582, which is the same as the Julian calendar but with the exclusion of leap years for most centenary years (as outlined above). 

For centuries, the Gregorian calendar was only used by Catholic countries, such as Italy and Spain, but it was eventually adopted by Protestant countries, such as Great Britain in 1752, when their years began to greatly deviate from Catholic countries. 

Because of the discrepancy between calendars, countries that later switched to the Gregorian calendar had to skip days to sync up with the rest of the world. For example, when Britain swapped calendars in 1752, Sept. 2 was followed by Sept. 14, according to the Royal Museums Greenwich. 

At some point in the distant future, the Gregorian calendar may have to be re-evaluated as it slips out of sync with solar years. But it will take thousands of years for this to happen. 

Why is leap day on Feb. 29?

In the eighth century B.C., the Roman calendar had just 10 months, beginning in March and ending in December. The cold winter season was ignored, with no months to signify it. But this calendar had only 304 days, so January and February were eventually added to the end of the religious year. As the last month, February had the fewest days. But Romans soon began associating these months with the start of the civil year, and by around 450 B.C., January was viewed as the first month of the new year.

When Pope Gregory XIII added the leap day to the Gregorian calendar in 1582, he chose February because it was the shortest month, making it one day longer on leap years.

Prime numbers have been investigated for more than 2,000 years, since at least the era of the ancient Greek mathematician Euclid. There are infinitely many, but what is the largest known prime number?

Prime numbers are those that can be evenly divided only by 1 and themselves, such as 3 and 7. They are key building blocks in math; per the fundamental theorem of arithmetic, every number greater than 1 is either a prime number or a multiple of a prime number, according to the University of Houston.

"Prime numbers are the 'atoms' of number theory," Thomas Kecker, a mathematician at the University of Portsmouth in England, told Live Science.

A major difference between real atoms and prime numbers is that the number of different types of stable atoms is finite. In contrast, "it is known at least since the times of Euclid in ancient Greece that there is an infinitude of prime numbers," Kecker said. "Finding larger and larger prime numbers therefore became a quest for many mathematicians."

Related: How many atoms are in the observable universe?

Currently, the largest known prime number is 2^136,279,841 – 1. To calculate this number, multiply 2 by itself 136,279,841 times, and then subtract 1. The result, also known as M136279841, possesses a whopping 41,024,320 digits, more than 16 million digits more than the previous record holder, called M82589933.

Both recent record holders are Mersenne primes, a kind of number named after the French monk Marin Mersenne, who investigated these numbers more than 350 years ago. To calculate a Mersenne prime, 2 is multiplied by itself a number of times, and then 1 is subtracted, according to the Great Internet Mersenne Prime Search (GIMPS).

GIMPS is a distributed computing project in which groups of volunteers run software in the background on their computers to collectively solve problems — in this case, finding Mersenne primes. Founded in 1996, GIMPS is the longest continuously running distributed computing project, according to the project website.

"This distributed computing approach to finding the largest known prime number has been very successful," Curtis Cooper, a mathematician retired from the University of Central Missouri who helped discover several of the previous largest primes, told Live Science. "Most of these were the largest known prime numbers at the time of their discovery."

The new largest prime was discovered by amateur researcher and former Nvidia employee Luke Durant, who ran GIMPS on a cloud-based computer network. His efforts required the harnessing of thousands of graphics processing units (GPUs) across 24 data centers in 17 countries — a feat that "ends the 28-year reign of ordinary personal computers finding these huge prime numbers," according to a statement released on the GIMPS website.

This was the first new Mersenne prime discovered since 2018.

"For a large whole number — say, with a few thousand digits — it becomes more and more time-consuming to check whether or not that number is prime," Kecker said. "Even with the most advanced algorithms and latest supercomputers to run them on, testing whether or not a number is prime could easily exceed a human lifespan."

However, over the years, mathematicians have discovered strategies for finding out if Mersenne numbers are prime, and these methods are far quicker than the techniques used for other kinds of prime numbers. Until 2018, GIMPS discovered a new Mersenne prime about every other year. "It is almost like waiting for a volcanic eruption after a long period of inactivity — although one expects the next one to happen any time, one never knows when it strikes again, if it ever strikes again," Kecker said.

Editor's note: This article was updated on Oct. 23, 2024 after a new largest Mersenne prime was discovered.

​​There are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute — so surely a second is just 1/(24 x 60 x 60), or 1/86400, of a day, right? Well, it turns out that defining time isn't that simple.

We're used to thinking of a second as a fixed increment of time, but this small unit has changed several times over the centuries.

"The second was originally based on the length of the day," Peter Whibberley, a senior scientist at the National Physical Laboratory in the U.K., told Live Science. "People observed the sun passing overhead and started measuring its movement using sundials. Devices like that give a time based directly on the position of the sun in the sky, which is called apparent solar time."

However, sundials have a few drawbacks. Aside from the obvious problem of not being able to read a sundial when the sun isn't visible, relying on Earth's daily rotation (also known as astronomical time) is surprisingly inaccurate.

"The rotation is not precisely constant," Whibberley said. "The Earth speeds up and slows down over time. There's a seasonal variation, big unpredictable variations from decade to decade due to changes in the molten core, and a longer-term slowing caused by the tides moving backwards and forwards."

So how can we precisely measure time if using the length of a day is so unreliable?

Related: Where does the concept of time travel come from?

In the 16th century, people turned to technological solutions to this problem, and the first recognizable mechanical clocks began to emerge.

"The heart of making a clock basically moved from keeping time by following the position of the sun, to making an oscillator and defining a fixed number of oscillations to be equivalent to one second," Sumit Sarkar, a physicist at the University of Amsterdam, told Live Science.

The earliest mechanical examples were pendulum clocks, which were designed to tick at a specific frequency, equivalent to an astronomical second, averaged over the course of a year. Over the next several hundred years, scientists worked on building better, more precise oscillators and developed myriad other timekeeping systems, including springs and gears.

By around 1940, quartz crystal clocks had become the new gold standard. "If you apply a voltage to a carefully shaped piece of quartz, it starts vibrating and you can tune the frequency of that oscillation very precisely," Sarkar said. "But while this precision is fine for general use, it's just not good enough for really technical applications, like the internet, GPS systems or studying fundamental research."

Problems arise because every piece of quartz is unique and resonates slightly differently depending on physical conditions such as temperature and pressure. To be truly accurate, clocks need to be set against some independent, unchanging reference. This is where atomic clocks come in.

"Atoms have natural fixed resonances. They exist only in particular energy states and can only change from one state to another by absorbing or emitting a fixed amount of energy," Whibberley explained. "That energy corresponds to a precise frequency, so you can use that frequency as a reference for time keeping."

The first practical atomic clock, unveiled in 1955, measured the number of these microwave-induced energy transitions in cesium atoms during a single astronomical second. In 1967, the global scientific community agreed to redefine the second according to this number, and the International System of Units and Measurements now defines a second as the duration of 9,192,631,770 energy oscillations in a cesium atom.

Since then, the astronomical second has continued to vary, while the atomic second has remained at precisely 9,192,631,770 oscillations. These variations in astronomical time actually mean that, every few years, scientists must add a leap second to allow Earth's slowing rotation to keep up with atomic time. This leap second is being abolished in 2035, but scientists and government agencies haven't yet figured out how to handle this tiny discrepancy, Whibberley said.

But scientists are not content to rest with this definition, which is accurate to 10^-15s or one-quadrillionth of a second. Across the world, research teams are working on even more precise optical atomic clocks, which use higher-energy visible light-induced atomic transitions in elements such as strontium and ytterbium to improve this accuracy more than 100-fold. In fact, scientists are discussing whether it's time to redefine the second again according to optical clock oscillations, using UV and visible light sources in place of microwaves.

But while several important questions still need to be answered before this happens, it's clear that the precise definition of a second is subject to change.

You might be surprised to learn that you can't comb the hairs flat on a coconut without creating a cowlick. Perhaps even more surprising, this silly claim with an even sillier name, "the hairy ball theorem," is a proud discovery from a branch of math called topology. Juvenile humor aside, the theorem has far-reaching consequences in meteorology, radio transmission and nuclear power.

Here, "cowlick" can mean either a bald spot or a tuft of hair sticking straight up, like the one the character Alfalfa sports in "The Little Rascals." Of course, mathematicians don't refer to coconuts or cowlicks in their framing of the problem. In more technical language, think of the coconut as a sphere and the hairs as vectors. A vector, often depicted as an arrow, is just something with a magnitude (or length) and a direction. Combing the hair flat against the sides of the coconut would form the equivalent of tangent vectors—those that touch the sphere at exactly one point along their length. Also, we want a smooth comb, so we don't allow the hair to be parted anywhere. In other words, the arrangement of vectors on the sphere must be continuous, meaning that nearby hairs should change direction only gradually, not sharply. If we stitch these criteria together, the theorem says that any way you try to assign vectors to each point on a sphere, something ugly is bound to happen: there will be a discontinuity (a part), a vector with zero length (a bald spot) or a vector that fails to be tangent to the sphere (Alfalfa). In full jargon: a continuous nonvanishing tangent vector field on a sphere can't exist.

This claim extends to all sorts of furry figures. In the field of topology, mathematicians study shapes, as they would in geometry, but they imagine these shapes are made from an ever elastic rubber. Although that rubber is capable of molding into other forms, it is incapable of tearing, fusing or passing through itself. If one shape can be smoothly deformed into another without doing these things, then those shapes are equivalent, as far as topologists are concerned. This means that the hairy ball theorem automatically applies to hairy cubes, hairy stuffed animals and hairy baseball bats, which are all topologically equivalent to spheres. (You could mold them all from a ball of Play-Doh without violating the rubbery rules.)

Something that is not equivalent to a sphere is your scalp. A scalp on its own can be flattened into a surface and combed in one direction like the fibers on a shag carpet. So sadly, math can't excuse your bedhead. Doughnuts are also distinct from spheres, so a hairy doughnut—an unappetizing image, no doubt—can be combed smoothly.

Here's a curious consequence of the hairy ball theorem: there will always be at least one point on Earth where the wind isn't blowing across the surface. The wind flows in a continuous circulation around the planet, and its direction and magnitude at every location on the surface can be modeled by vectors tangent to the globe. (Vector magnitudes don't need to represent physical lengths, such as those of hairs.) This meets the premises of the theorem, which implies that the gusts must die somewhere (creating a cowlick). A cowlick could occur in the eye of a cyclone or eddy, or it could happen because the wind blows directly up toward the sky. This neat online tool depicts up-to-date wind currents on Earth, and you can clearly spot the swirly cowlicks.

To observe another weird ramification of the theorem, spin a basketball any which way you want. There will always be a point on the surface that has zero velocity. Again, we associate a tangent vector with each point based on the direction and speed at that point on the ball. Spinning is a continuous motion, so the hairy ball theorem applies and assures a point with no speed at all. Upon further reflection, this might seem obvious. A spinning ball rotates around an invisible axis, and the points on either end of that axis don’t move. What if we bored a tiny hole through the ball exactly along that axis to remove the stationary points? It seems then that every point would be moving. Does this violate the hairy ball theorem? No, because drilling a hole transformed the ball into a doughnut! Even doughnuts with unusually long, narrow holes flout the rules of the theorem—contradiction averted.

Moving on from toy scenarios—the hairy ball theorem actually imposes tangible limitations on radio engineers. Antennas broadcast radio waves in different directions depending on design choices. Some target their signals in a specific direction, while others beam more broadly. One might be tempted to simplify matters and build only antennas that send equal-strength signals in every direction at once, which are called isotropic antennas. There's just one problem: a certain hirsute fact from topology mandates that isotropic antennas can’t exist. Picture an orb of waves emanating from a central source. Sufficiently far away from the source, radio waves exhibit an electric field perpendicular to the direction they're traveling, meaning the field is tangent to the sphere of waves. The hairy ball theorem insists that this field must drop to zero somewhere, which implies a disturbance in the antenna's signal. Isotropic antennas serve merely as theoretical ideals against which we compare real antenna performance. Interestingly, sound transmits a different kind of wave without the perpendicular property of radio waves, so loudspeakers that emanate equal-intensity sound in every direction are possible.

Perhaps the coolest application of the hairy ball theorem concerns nuclear fusion power. Fusion power carries immense promise to—perhaps someday—help ease the energy crisis. It has the potential to generate vast quantities of energy without the environmental concerns that plague fossil fuels and with far fewer of the radioactive risks associated with traditional nuclear fission reactors. In a nutshell, fusion reactors begin by taking a fuel such as hydrogen and subjecting it to intense heat and pressure, which rips it into its constituent parts to form plasma. Plasma is a cloud of electrons and other charged particles that bop around and occasionally fuse together to form new particles, releasing energy in the process.

There's a fundamental engineering hurdle when building fusion reactors: How do you contain plasma that’s 10 times hotter than the sun's core? No material can withstand that temperature without disintegrating into plasma itself. So scientists have devised a clever solution: they exploit plasma’s magnetic properties to confine it within a strong magnetic field. The most natural container designs (think boxes or canisters) are all topologically equivalent to spheres. A magnetic field around any of these structures would form a continuous tangent vector field, and at this point we know what befalls such hairy constructions. A zero in the magnetic field means a leak in the container, which spells disaster for the whole reactor. This is why the leading design for fusion reactors, the tokamak, has a doughnut-shaped chamber. The International Thermonuclear Experimental Reactor (ITER) megaproject plans to finish construction of a new tokamak in France by 2025, and those involved claim their magnetic confinement system will be "the largest and most integrated superconducting magnet system ever built." That's topology playing its part in our clean energy future.

This article was first published at ScientificAmerican.com. © ScientificAmerican.com. All rights reserved. Follow Scientific American on Twitter @SciAm and @SciamBlogs. Visit ScientificAmerican.com for the latest in science, health and technology news.

Scientists have discovered that a key function from a "pure" branch of mathematics can predict how often genetic mutations lead to changes in function. 

These rules, laid out by the so-called sum-of-digits function, also govern some aspects of protein folding, computer coding and certain magnetic states in physics. 

"Part of what we're trying to do is find a universal explanation for a lot of these trends which have been observed in nature," said lead study author Vaibhav Mohanty, a theoretical physicist and doctoral and MD candidate at Harvard Medical School and the Massachusetts Institute of Technology. 

For every genotype — letters of DNA for a given gene — there is a phenotype, or end result: a new protein, or even a behavior in the case of a gene that regulates another set of genes. A given genotype can accrue a number of mutations before its phenotype changes; this accumulation of neutral mutations is a major way evolution proceeds. 

"We want to understand, how robust is the actual phenotype to mutations?" Mohanty said. "It turns out that that robustness has been observed to be pretty high." In other words, a lot of the "letters," or base pairs that make up the code of DNA, can change before the output does.

Because this robustness pops up not only in genetics but also in fields such as physics and computer science, Mohanty and his colleagues suspected its roots might lie in the fundamental mathematics of the possible sequences. They envisioned these possible sequences as a cube of many dimensions, known as a hybercube, with each point on this impossible-to-visualize cube as a possible genotype. Genotypes with the same phenotype should ultimately cluster together, Mohanty said. The question was, what shape would those clusters form? 

The answer turned out to be found in number theory, the area of mathematics concerned with the properties of positive integers. The average robustness of a phenotype to mutations turned out to be defined by what's called a sum-of-digits function. This means that by adding the digits representing each genotype on the cube, you can arrive at the average robustness of the genotype. 

"Let's say there are five genotypes that map to a particular phenotype," Mohanty said. So, for instance, five letter sequences of DNA, each with a different mutation, but which all still code for the same protein. 

Adding up the digits used to represent these five sequences gives you the average number of mutations those genotypes can take on before their phenotypes shift, the researchers found. 

This led to the second intriguing discovery: These sums of digits, plotted out on a graph, formed what's called a blancmange curve, a fractal curve named after a French dessert (which looks like a fancy molded pudding). 

In a fractal curve, "if you zoom into the curve it looks exactly the same as if you were zoomed out, and you can continue to zoom in infinitely and infinitely and infinitely and it would be the same," Mohanty said.

These findings revealed some interesting secrets about error correction, Mohanty said. For instance, the natural systems the researchers studied tended to handle errors differently than humans do when setting up data storage, like in digital messages or on CDs or DVDs. In these technological examples, all errors are treated equally, while biological systems tend to protect certain sequences more than others.

That's not surprising for genetic sequences, where there might be several linchpin sequences and then others that are more peripheral to the main gene function, Mohanty said. 

Understanding the dynamics of these neutral mutations could eventually be important for preventing disease, Mohanty said. Viruses and bacteria evolve rapidly, and they accumulate many neutral mutations in the process. If there were a way to prevent these pathogens from landing on the needle-in-the-haystack beneficial mutation among all the chaff, researchers might be able to stymie pathogens' ability to become more infectious or resistant to antibiotics, for example. 

The researchers published their findings July 26 in the Journal of the Royal Society Interface.

British theoretical physicist Paul Dirac was one of the most significant figures in the early days of quantum physics, who along with Erwin Schrödinger won the Nobel Prize for physics in 1933. But it was in 1927 that this quiet, but brilliant mind set to work looking for "pretty mathematics," and in doing so formulated what would become one of his greatest achievements — the Dirac equation.

In this extract from the Antimatter chapter of his book "'The One Thing You Need to Know'," author Marcus Chown explains how Dirac's unusual methods and mannerisms helped guide us towards understanding the fundamental physics that forms the world around us.

Related: What is antimatter?

Nature has chosen to double the number of its basic building blocks. For every subatomic particle, remarkably there exists an "'antiparticle"' with opposite properties such as electric charge. Before 1927, nobody had the slightest suspicion that such a world of "'antimatter"' existed. But that year, the British physicist Paul Dirac wrote down an equation that described an electron travelling at close to the speed of light and noticed that it contained something odd.

Dirac was one of the pioneers of quantum theory, the revolutionary description of the submicroscopic realm of atoms and their constituents. The theory reconciled two seemingly contradictory characteristics of the world revealed in experiments in the first quarter of the twentieth century: the ability of atoms and their like to behave both as localized particles and as spread-out waves. In 1926, the Austrian physicist Erwin Schrödinger encapsulated this in the Schrödinger equation, which describes quantum waves of probability spreading through space.

The problem with the Schrödinger equation is that it does not incorporate the other revolution of twentieth-century physics. In his special theory of relativity of 1905, Einstein showed that strange things happen to space and time as a body with mass approaches the speed of light. Although the Schrödinger equation works fine when describing an electron in a small atom, where the electric force of only a handful of protons in the nucleus causes it to orbit at much less than the speed of light, in heavier atoms, where there are lots of protons in the nucleus and an electron is whirled around at close to the cosmic speed limit, the equation breaks down. What was needed was an equation that was compatible with the special theory of relativity — relativistic — and that was what Dirac set out to find.

Dirac was a strange man who today would probably be diagnosed as being on the autism spectrum. Tall, gangly and reminiscent of a stick insect, his habit was to work hard all week and on Sundays take long walks in the countryside around Cambridge, where he would climb tall trees dressed in his suit and tie. Literal to the point of obtuseness, he was the Mr. Spock of physics. When a student put up their hand during one of his lectures and said, "'Professor Dirac, I don't understand the equation on the blackboard," he replied: "'That's a comment not a question'," and continued with his lecture.

Dirac's approach to physics was no less strange than his character. Whereas other physicists sought everyday analogues of the phenomena they wanted to describe, which they then tried to encapsulate in a mathematical equation, Dirac had the courage to simply sit with a pen and paper and guess at the form of an equation. "It's a peculiarity of myself that I like to play about with equations, just looking for beautiful mathematical relations which maybe don't have any physical meaning at all," said Dirac. 'Sometimes they do.'

It was while looking for "pretty mathematics" in his spartan rooms at St. John's College in late November 1927 that Dirac literally plucked from thin air what would become known as the Dirac equation. Today, it is one of two equations inscribed on flagstones on the floor of London's Westminster Abbey. The other is Stephen Hawking's equation for the temperature of a black hole. "Of all the equations of physics, perhaps the most magical is the Dirac equation," says American physicist Frank Wilczek (in the book "It Must Be Beautiful: Great Equations Of Modern Science" by Graham Farmelo (Granta, 2003)). "It is the most freely invented, the least conditioned by experiment, the one with the strangest and most startling consequences."

Dirac had found it impossible to describe a relativistic electron's properties, such as its energy, with a mere number, so instead had to use a two-by-two table of numbers known as a matrix. This "two-ness" explained a puzzling feature of the electron. Experiments had revealed that the particle behaved as if it was spinning in one of two ways: clockwise or anticlockwise. However, if an electron was really spinning, its behavior could be understood only if it was spinning faster than light, which, according to Einstein, was impossible. Physicists were forced to conclude that the "spin" of an electron was something entirely new. It was an intrinsic quantum property with no analog in the everyday world. And here it was, Dirac saw, just popping unbidden out of the formula he had written down. "My equation gave just the properties one needed for an electron," said Dirac. "That was really an unexpected bonus for me, completely unexpected." According to the American physicist John Hasbrouck Van Vleck, Dirac's explanation of an electron's spin was comparable to "a magician's extraction of rabbits from a silk hat."

Spin was strange. But another aspect that emerged from Dirac's equation was even stranger. When Dirac wrote down his equation, he noticed that its machinery was oddly duplicated. It appeared to describe not only a negatively charged electron, but also a particle with the same mass as an electron that had a positive charge. At the time, only three subatomic particles were known: the proton in the nucleus of the atom; the electron, which orbited the nucleus; and the photon, the particle of light. There appeared to be no need for another one. Even the great physicists of the day such as Werner Heisenberg and Wolfgang Pauli thought that the Dirac equation must be wrong. However, Dirac was right and they were wrong, as an experiment 8,000 kilometers away from Cambridge would later show.

In 1932, Carl Anderson, an American physicist at the California Institute of Technology in Pasadena, was trying to understand cosmic rays, extremely high-energy particles from space. He expected them to smash into atoms in the atmosphere, kicking out their electrons. If he could just measure the energy of such ejected electrons, he reasoned, he would have a handle on the energy of the cosmic rays. To this end he used an extremely strong magnetic field to bend the electrons, deducing that if they had high energy and so were moving fast, they would spend little time in the vicinity of his magnetic field and be bent less sharply than if they had low energy and spent more time there.

Anderson made his electrons visible by means of a "cloud chamber." Inside the device, tiny trails of water droplets formed along the tracks of electrons, and he could photograph these trails. On 2 August 1932, Anderson developed a photographic plate and was astonished to see a particle of the mass of an electron that was bent by the magnetic field in the opposite way to an electron. He knew nothing of Dirac's prediction. Nevertheless, he had stumbled on Dirac's positively charged electron, a particle which he immediately christened the "positron."

Excerpted from The One Thing You Need to Know. Copyright © 2023 by Marcus Chown.

Published by Michael O'Mara.