In the third century BCE, Apollonius of Perga asked how many circles one could draw that would touch three given circles at exactly one point each. It would take 1,800 years to prove the answer: eight. Such questions, which ask for the number of solutions that satisfy a set of geometric conditions, were a favorite of the ancient Greeks. And they’ve continued to entrance mathematicians for…
These days, large language models such as ChatGPT are omnipresent. Yet their inner workings remain deeply mysterious. To Naomi Saphra, that’s an unsatisfying state of affairs. “We don’t know what makes a language model tick,” she said. “If we have these models everywhere, we should understand what they’re doing.” Saphra, a research fellow at Harvard University’s Kempner Institute who will start…
Nearly 170 years ago, a scientist named Eunice Foote discovered a fundamental truth about the gases that surround us. In her home laboratory in New York, she filled one glass cylinder with carbon dioxide and another with regular air, placed a thermometer in each and left them out in the sun. Less than 20 minutes later, the carbon dioxide–filled cylinder was much warmer — and much harder to cool…
The Earth’s atmosphere is nothing but freely roaming molecules. Left alone, they would drift and collide, and eventually even out into a mixture that’s dynamic, yet stable and broadly unchanging. The sun’s rays complicate things. Energy enters the Earth system in daily cycles, the bulk of it going to whichever half of the planet is tilted toward the sun (and experiencing summer).
As we listen to a piece of music, our ears perform a calculation. The high-pitched flutter of the flute, the middle tones of the violin, and the low hum of the double bass fill the air with pressure waves of many different frequencies. When the combined sound wave descends through the ear canal and into the spiral-shaped cochlea, hairs of different lengths resonate to the different pitches…
Water is the most fundamental need for all life on Earth. Not every organism needs oxygen, and many make their own food. But for all creatures, from deep-sea microbes and slime molds to trees and humans, water is nonnegotiable. “The first act of life was the capture of water within a cell membrane,” a pair of neurobiologists wrote in a recent review. Ever since, cells have had to stay wet enough…
In 1971, Fan Chung, then in her second year of graduate school at the University of Pennsylvania, received an assignment. Her thesis adviser, Herbert Wilf, asked her to read the proof of a problem in Ramsey theory, an area of mathematics that explores the inevitable emergence of patterns in networks of vertices and edges called graphs. They planned to discuss it again the following week.
Hard problems are usually not a welcome sight. But cryptographers love them. That’s because certain hard math problems underpin the security of modern encryption. Any clever trick for solving them will doom most forms of cryptography. Several years ago, researchers found a radically new approach to encryption that lacks this potential weak spot. The approach exploits the peculiar features of…
Most cosmologists agree that our universe had a beginning. But the finer details about the Big Bang remain a mystery. A history of everything would explain all, or so theoretical physicists hoped. In his final years, Stephen Hawking working with Thomas Hertog proposed a striking idea: The laws of physics were not precisely determined before the Big Bang; they evolved as the universe evolved.
Take a deep breath. A flow of air has rushed into your lungs, where the oxygen moves into your bloodstream, fueling metabolic fires in cells throughout your body. You, being an aerobic organism, use oxygen as the cellular spark that frees molecular energy from the food you eat. But not all organisms on the planet live or breathe this way. Instead of using oxygen to harvest energy…
Randomness is a source of power. From the coin toss that decides which team gets the ball to the random keys that secure online interactions, randomness lets us make choices that are fair and impossible to predict. But in many computing applications, suitable randomness can be hard to generate. So instead, programmers often rely on things called hash functions, which swirl data around and…
When Thomas Hummel gets a whiff of an unripe, green tomato, he finds himself in his childhood home in Bavaria. Under the tilted ceilings of the bedroom that he shared with his two older brothers, there were three beds, a simple table and a cupboard. “My mother put those green tomatoes on the cupboard for them to ripen,” said Hummel, an olfaction researcher at the Carl Gustav Carus University…
We were once promised self-driving cars and robot maids. Instead, we’ve seen the rise of artificial intelligence systems that can beat us in chess, analyze huge reams of text and compose sonnets. This has been one of the great surprises of the modern era: physical tasks that are easy for humans turn out to be very difficult for robots, while algorithms are increasingly able to mimic our intellect.
Born in the 18th century when Leonhard Euler solved the puzzle of the seven bridges of Königsberg, graph theory has become a foundational tool in mathematics. It studies relationships through nodes (vertices) and the links (edges) that connect them, transforming the complexity of systems — from friendship networks to airline routes — into elegant abstractions that reveal underlying structure and…
Beneath the richness of our world lies a pristine simplicity. Everything is made of a set of just 17 fundamental particles, and those particles, though they may differ by mass or charge, come in just two basic types. Each is either a “boson” or a “fermion.” The physicist Paul Dirac coined both terms in a speech in 1945, naming the two particle kingdoms after physicists who helped elucidate their…
Last winter, at a meeting in the Finnish wilderness high above the Arctic Circle, a group of mathematicians gathered to contemplate the fate of a mathematical universe. It was minus 20 degrees Celsius, and while some went cross-country skiing, Juan Aguilera, a set theorist at the Vienna University of Technology, preferred to linger in the cafeteria, tearing pieces of pulla pastry and debating…
You’ve just gotten home from an exhausting day. All you want to do is put your feet up and zone out to whatever is on television. Though the inactivity may feel like a well-earned rest, your brain is not just chilling. In fact, it is using nearly as much energy as it did during your stressful activity, according to recent research. Sharna Jamadar, a neuroscientist at Monash University in…
For decades, string theory has been hailed as the leading candidate for the theory of everything in our universe. Yet despite its mathematical elegance, the theory still lacks empirical evidence. One of its most intriguing, yet vexing, implications is that if all matter and forces are composed of vibrations of tiny strands, then this allows for a vast landscape of possible universes with…
Inside Manu Prakash are two scientists. A bioengineer at Stanford University, he spends half his time studying urgent health issues with global impact and the rest pursuing questions “of no use to anyone,” he said. To him, though, there is no conflict between these pursuits. Together, they represent something of a philosophy for life. Prakash is widely recognized for his pioneering low-cost…
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