Math
135 articles
Math’s “Oldest Problem Ever” Gets a New Answer
A new proof significantly strengthens a decadesold result about the ubiquity of ways to represent whole numbers as sums of fractions.

Imaginary Numbers Are Reality
How the modern world arose from imaginary numbers.

An Ancient Geometry Problem Falls to New Mathematical Techniques
Three mathematicians show, for the first time, how to form a square with the same area as a circle by cutting them into interchangeable pieces that can be visualized.

Mathematicians Find Structure in Biased Polynomials
New work establishes a tighter connection between the rank of a polynomial and the extent to which it favors particular outputs.

In Topology, When Are Two Shapes the Same?
As topologists seek to classify shapes, the effort hinges on how to define a manifold and what it means for two of them to be equivalent.

Computer Scientists Discover Limits of Major Research Algorithm
The most widely used technique for finding the largest or smallest values of a math function turns out to be a fundamentally difficult computational problem.

Why You May Have More Friends Than Your Friends Do
There’s a rude charm to the title, “Why Your Friends Have More Friends Than You.” It’s catchy, like the title of an antagonistic explainer: Here are the causes of your lackluster social life. It sounds more like a New York Times oped than an academic paper. But in fact, “Why Your Friends Have More Friends […]

Pandemic Puts Mathematical Modeling Through Its Paces
Mathematical tools that proved essential during the pandemic were in many cases invented by mathematicians who had no particular goal in mind.

Pandemic Puts Mathematical Modeling Through Its Paces
Mathematical tools that proved essential during the pandemic were in many cases invented by mathematicians who had no particular goal in mind.

Mathematicians Answer Old Question About Odd Graphs
A pair of mathematicians solved a legendary question about the proportion of vertices in a graph with an odd number of connections.

How Mathematicians Use Homology to Make Sense of Topology
Originally devised as a rigorous means of counting holes, homology provides a scaffolding for mathematical ideas, allowing for a new way to analyze the shapes within data.

New Quantum Algorithms Finally Crack Nonlinear Equations
Two teams found different ways for quantum computers to process nonlinear systems by first disguising them as linear ones.

How the Slowest Computer Programs Illuminate Math’s Fundamental Limits
The goal of the “busy beaver” game is to find the longestrunning computer program. Its pursuit has surprising connections to some of the most profound questions and concepts in mathematics.

Undergraduate Math Student Pushes Frontier of Graph Theory
At 21, Ashwin Sah has produced a body of work that senior mathematicians say is nearly unprecedented for a college student.

Inside the Secret Math Society Known Simply as Nicolas Bourbaki
For almost a century, the anonymous members of Nicolas Bourbaki have written books intended as pure expressions of mathematical thought.

An Infinite Universe of Number Systems
The padics form an infinite collection of number systems based on prime numbers. They’re at the heart of modern number theory.

At the Math Olympiad, Computers Prepare to Go for the Gold
Computer scientists are trying to build an AI system that can win a gold medal at the world’s premier math competition.

At the Math Olympiad, Computers Prepare to Go for the Gold
Reprinted with permission from Quanta Magazine‘s Abstractions blog. The 61st International Mathematical Olympiad, or IMO, began yesterday. It may go down in history for at least two reasons: Due to the COVID19 pandemic it’s the first time the event has been held remotely, and it may also be the last time that artificial intelligence doesn’t compete. […]

A New Algorithm for Graph Crossings, Hiding in Plain Sight
Two computer scientists found—in the unlikeliest of places—just the idea they needed to make a big leap in graph theory.

Why Mathematicians Should Stop Naming Things After Each Other
A past generation’s glory can be the next generation’s headache.

Mathematicians Report New Discovery About the Dodecahedron
Three mathematicians have resolved a fundamental question about straight paths on the 12sided Platonic solid.

Computer Scientists Attempt to Corner the Collatz Conjecture
A powerful technique called SAT solving could work on the notorious Collatz conjecture. But it’s a long shot.

Math of the Penguins
Emperor penguins display rigorously geometric spacing and mathematical efficiency when they huddle together for warmth, which may reveal secrets to their overall health.

Mathematicians Will Never Stop Proving the Prime Number Theorem
Why do mathematicians enjoy proving the same results in different ways?

How Gödel’s Proof Works
His incompleteness theorems destroyed the search for a mathematical theory of everything. Nearly a century later, we’re still coming to grips with the consequences.

The Tricky Math of Herd Immunity for COVID19
Herd immunity differs from place to place, and many factors influence how it’s calculated.

The ‘Useless’ Perspective That Transformed Mathematics
Representation theory was initially dismissed. Today, it’s central to much of mathematics.

Let Game Theory Tell You When It’s Time to Go Shopping
By using this methodology, combined with available information from Google maps to measure store size and popularity, one can model the best time to go shopping in a pandemic.Photograph by Diego Cervo / Shutterstock Now is not the time to go to the grocery store, to restock the pantry, to get fresh milk and eggs. […]

In a Single Measure, Invariants Capture the Essence of Math Objects
To distinguish between fundamentally different objects, mathematicians turn to invariants that encode the objects’ essential features.

In Mathematics, It Often Takes a Good Map to Find Answers
Mathematicians try to figure out when problems can be solved using current knowledge—and when they have to chart a new path instead.

OutofSync ‘Loners’ May Secretly Protect Orderly Swarms
Studies of collective behavior usually focus on how crowds of organisms coordinate their actions. But what if the individuals that don’t participate have just as much to tell us?

Why COVID19 FlareUps Will Keep Happening
Like earthquakes and forest fires, outbreaks have a “heavy tail” of large events.

Math After COVID19
Modern mathematics relies on collaboration and travel. COVID19 is making it increasingly difficult.

John Conway Solved Mathematical Problems With His Bare Hands
The legendary mathematician, who died on April 11, 2020 was curious, colorful and one of the greatest problemsolvers of his generation.

Graced With Knowledge, Mathematicians Seek to Understand
A landmark proof in computer science has also solved an important problem called the Connes embedding conjecture. Mathematicians are working to understand it.

‘Rainbows’ Are a Mathematician’s Best Friend
“Rainbow colorings” recently led to a new proof. It’s not the first time they’ve come in handy.

To Beat COVID19, Think Like a Fighter Pilot
How an aerialcombat theory can help fight the outbreak.

ColorChanging Material Unites the Math and Physics of Knots
Reprinted with permission from Quanta Magazine’s Abstractions blog. One sunny day in the summer of 2019, Mathias Kolle, a professor at the Massachusetts Institute of Technology, took a couple of eminent colleagues out sailing. They talked about their research. They had some drinks. Then Kolle noticed something was off: A rowboat tied to his boat had […]

ColorChanging Material Unites the Math and Physics of Knots
Mathematicians have studied knots for centuries, but a new material is showing why some knots are better than others.

Mapping GayFriendly Cities Through History
A data analyst uncovers a timeless message about correlation.

How Pi Connects Colliding Blocks to a Quantum Search Algorithm
A curious physicist has discovered an unexpected link between theoretical block collisions and a famed quantum search algorithm.

What Are the Odds of Alien Contact?
Bringing the Drake Equation up to date.

How Inequality Imperils Cooperation
A game theorist breaks down the effects of inequality.

Are Neural Networks About to Reinvent Physics?
The revolution of machine learning has been greatly exaggerated.

Mathematicians Calculate How Randomness Creeps In
The goal of a 15 puzzle is to put numbered tiles in order. Now mathematicians have solved the opposite problem — how to scramble one.

Mathematicians Begin to Tame Wild ‘Sunflower’ Problem
A major advance toward solving the 60yearold sunflower conjecture is shedding light on how order begins to appear as random systems grow in size.

The Flawed Reasoning Behind the Replication Crisis
It’s time to change the way uncertainty is quantified.

How Randomness Can Make Math Easier
Randomness would seem to make a mathematical statement harder to prove. In fact, it often does the opposite.

Presenting the Scrabble Luck Calculator
Are you as good at Scrabble as you think?

The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
Over a decade ago, I was sitting in a college math physics course and my professor spelt out an idea that kind of blew my mind. I think it isn’t a stretch to say that this is one of the most widely applicable mathematical discoveries, with applications ranging from optics to quantum physics, radio astronomy, […]

How I Rewired My Brain to Become Fluent in Math
The building blocks of understanding are memorization and repetition.

A New Approach to Multiplication Opens the Door to Better Quantum Computers
In practice, quantum computers can’t run many programs that classical computers can, because they’re not allowed to selectively forget information. A new algorithm for multiplication shows a way around that problem.

Nassim Taleb’s Case Against Nate Silver Is Bad Math
Nassim Nicholas Taleb has overplayed his hand this time and is left looking, well, klueless.Photograph by Salzburg Global Seminar / Flickr Since the midterm elections, a feud has been raging on Twitter between Nate Silver, founder of FiveThirtyEight, and Nassim Nicholas Taleb, hedgefundmanagerturnedmathematicalphilosopher and author of The Black Swan. It began, late last year, with […]

How Search Algorithms Are Changing the Course of Mathematics
The sumofthreecubes problem solved for “stubborn” number 33.

How to Improve Political Forecasts
With a better understanding of probability, we won’t be misled.

Smaller Is Better: Why Finite Number Systems Pack More Punch
Recent progress on the “sum product” problem recalls a celebrated mathematical result that revealed the power of miniature number systems.

The Math That Takes Newton Into the Quantum World
How a math professor learned to stop worrying and love algebraic geometry.

Mathematicians Seal Back Door to Breaking RSA Encryption
Digital security depends on the difficulty of factoring large numbers. A new proof shows why one method for breaking digital encryption won’t work.

Does Scrabble Need To Be Fixed?
An experiment in controlling how much of Scrabble is luck.

New Proof Shows Infinite Curves Come in Two Types
Alexander Smith’s work on the Goldfeld conjecture reveals fundamental characteristics of elliptic curves.

Why Mathematicians Can’t Find the Hay in a Haystack
In math, sometimes the most common things are the hardest to find.

Why Mathematicians Can’t Find the Hay in a Haystack
Reprinted with permission from Quanta Magazine‘s Abstractions blog. The first time I heard a mathematician use the phrase, I was sure he’d misspoken. We were on the phone, talking about the search for shapes with certain properties, and he said, “It’s like looking for hay in a haystack.” “Don’t you mean a needle?” I almost interjected. Then he […]

The Strange Numbers That Birthed Modern Algebra
Reprinted with permission from Quanta Magazine‘s Abstractions blog. Imagine winding the hour hand of a clock back from 3 o’clock to noon. Mathematicians have long known how to describe this rotation as a simple multiplication: A number representing the initial position of the hour hand on the plane is multiplied by another constant number. But is a similar […]

Kolmogorov Complexity and Our Search for Meaning
What math can teach us about finding order in our chaotic lives.

A Short Guide to Hard Problems
What’s easy for a computer to do, and what’s almost impossible? Those questions form the core of computational complexity. We present a map of the landscape.

Our Strange Relationship to World Cup Probabilities
What this World Cup reveals isn’t that the stats were wrong—far from it, they were insightfully calculated—but rather that we relate to stats and probabilities in strange ways.Photograph by Hey! Play! / Wayfair This year’s World Cup has been full of surprises. Tournament mainstays such as the Netherlands and Italy didn’t even qualify, and Germany, […]

Three Decades Later, Mystery Numbers Explained
Zeta values seem to connect distant geometric worlds. In a new proof, mathematicians finally explain why.

Civilization Is Built on Code
Code is at once a force, or a means, of liberation and constraint.Photograph by Donnie Ray Jones / Flickr How did we humans manage to build a global civilization on the cusp of colonizing other planets? It seems like such an unlikely outcome. After all, we were prone to cycles of war and famine for millennia, […]

Pick the Statistic You Want to Be
Understanding the odds lets you play with them.

The Infinite Primes and Museum Guard Proofs, Explained
A simple, stepbystep breakdown of two “perfect” math proofs.

Robert Langlands, Mathematical Visionary, Wins the Abel Prize
Generations of researchers have pursued his “Langlands program,” which seeks to create a grand unified theory of mathematics.

The Simple Algorithm That Ants Use to Build Bridges
Even with no one in charge, army ants work collectively to build bridges out of their bodies. New research reveals the simple rules that lead to such complex group behavior.

How to Understand Extreme Numbers
The late statistics wizard Hans Rosling, who died in February of 2017 at age 68, brought at least 10 toilet paper rolls to some of his beloved presentations. He would stack them into a tower on a table, each roll representing one billion people. In a 2012 talk at the Skoll World Forum, he […]

What Makes the Hardest Equations in Physics So Difficult?
Reprinted with permission from Quanta Magazine‘s Abstractions blog. Physics contains equations that describe everything from the stretching of spacetime to the flitter of photons. Yet only one set of equations is considered so mathematically challenging that it’s been chosen as one of seven “Millennium Prize Problems” endowed by the Clay Mathematics Institute with a $1 million reward: […]

What Makes the Hardest Equations in Physics So Difficult?
The NavierStokes equations describe simple, everyday phenomena, like water flowing from a garden hose, yet they provide a milliondollar mathematical challenge.

You’re Descended from Royalty and So Is Everybody Else
Anybody you can name from ancient history is in your family tree.

How Classical Cryptography Will Survive Quantum Computers
Quantum Lab: Scientists are fabricating quantum photonic circuits—consisting of waveguides and other elements—to manipulate single photons for future quantum communications and processing.Oak Ridge National Laboratory / Flickr Justin Trudeau, the Canadian prime minister, certainly raised the profile of quantum computing a few notches last year, when he gamely—if vaguely1—described it for a press conference. But […]

Mathematicians Crack the Cursed Curve
A famously difficult mathematical problem resisted solution for over 40 years. Mathematicians have finally resolved it by following an intuition that links number theory to physics.

The Prison Guard with a Gift for Cracking Gang Codes
Former correction officer Gary Klivans doesn’t want to be photographed more clearly for fear of gang retaliation.Gary Klivans As a corrections officer at a Westchester County, N.Y., prison in the 1990s, Gary Klivans was a oneman gang unit. Members of The Latin Kings and the Bloods made up a sizable part of the prison population. […]

How to Teach Science with Sugar and Cream
High school teachers are bringing ice cream into the lab.

Visionary Mathematician Vladimir Voevodsky Dies at 51
Voevodsky’s friends remember him as constitutionally unable to compromise on the truth—a quality that led him to produce some of the most important mathematics of the 20th century.

A Simple Visual Proof of a Powerful Idea in Graph Theory
Ramsey’s theorem predicts a surprising (and useful) consistency in the organization of graphs. Here’s a simple visual proof of how it works.Image by Lucy ReadingIkkanda/Quanta Magazine; Source: Jonathan Jedwab, Simon Fraser University Reprinted with permission from Quanta Abstractions A recent advance in geometry makes heavy use of Ramsey’s theorem, an important idea in another field—graph theory. […]

The Math That Promises to Make the World Brighter
The color of LED lights is controlled by a clumsy process. A new mathematical discovery may make it easier for us to get the hues we want.

Why Mathematicians Like to Classify Things
It’s “a definitive study for all time, like writing the final book,” says one researcher who’s mapping out new classes of geometric structures.

19 Women Leading Math and Physics
Reprinted with permission from Quanta Abstractions In an interview with Quanta Magazine last fall, the eminent theoretical physicist Helen Quinn recalled her uncertainty, as a Stanford University undergraduate in the 1960s, about whether to pursue a career in physics or become a high school teacher. “There were no women in the faculty at Stanford at that time in the […]

Claude Shannon, the Las Vegas Shark
The father of information theory built a machine to game roulette, then abandoned it.

Marjorie Rice’s Secret Pentagons
A California housewife who in the 1970s discovered four new types of tessellating pentagons is dead at 94.

The Tricky Translation of Mathematical Ideas
Big advances in math can happen when mathematicians move ideas into areas where they seem like they shouldn’t belong.

Chaos Makes the Multiverse Unnecessary
Science predicts only the predictable, ignoring most of our chaotic universe.

Cash for Math: The Erdős Prizes Live On
Paul Erdős placed small bounties on hundreds of unsolved math problems. Over the past 20 years, only a handful have been claimed.

The Impossible Mathematics of the Real World
Nearmiss math provides exact representations of almostright answers.

The Impossible Mathematics of the Real World
Nearmiss math provides exact representations of almostright answers.

The Mathematics of Juggling
Juggling has advanced enormously in recent decades, thanks in part to the mathematical study of possible patterns.

A Simple Visual Proof of a Powerful Idea
Ramsey’s theorem predicts a surprising (and useful) consistency in the organization of graphs. Here’s a simple visual proof of how it works.

Physicists Attack Math’s $1,000,000 Question
Physicists are attempting to map the distribution of the prime numbers to the energy levels of a particular quantum system.

Yves Meyer, Wavelet Expert, Wins Abel Prize
The French mathematician was cited “for his pivotal role in the development of the mathematical theory of wavelets.”

19 Women Leading Math and Physics
Top women in mathematics and physics discuss how they got to where they are—and why there aren’t more of them.

The AlmostProof of Fermat’s Last Theorem
19thcentury mathematicians thought the “roots of unity” were the key to solving Fermat’s Last Theorem. Then they discovered a fatal flaw.

How to Weigh the World
Atlas knew the answer. Straining under the task of holding up the Earth, this Titan god likely got a good idea of how much the Earth weighed. But none of us are so conveniently situated. How could a mere mortal, a tiny person residing on Earth’s surface, carry out their own estimate of Earth’s weight? […]

This Man Is About to Blow Up Mathematics
Harvey Friedman is about to bring incompleteness and infinity out of quarantine.

How to Understand Extreme Numbers
The late statistics wizard Hans Rosling, who died this month at age 68, brought at least 10 toilet paper rolls to some of his beloved presentations. He would stack them into a tower on a table, each roll representing one billion people. In a 2012 talk at the Skoll World Forum, he used the rolls […]

The Hidden Twist to Making a Möbius Strip
The simple Möbius strip illustrates a deep mathematical challenge that has long tormented the field of symplectic geometry.

How to Build a Probability Microscope
The surprising mathematics of the extremely rare.

How Curvature Makes a Shape a Shape
The ancient study of an object’s curvature is guiding mathematicians toward a new understanding of simple equations.

The Deceptions of Luck
Nature makes chance, humans make luck.

Does a Cartoon Penguin Make Math Education Great Again?
Matthew Peterson is a pretty inspirational guy. As a dyslexic child he found math class difficult, so as an adult he resolved to totally change the way math is taught. After completing his studies in biology, electrical engineering, and Chinese language and literature at the University of California, Irvine, Peterson cofounded the nonprofit MIND Research […]

Complexity Theory Problem Strikes Back
The legendary graph isomorphism problem may be harder than a 2015 result seemed to suggest.

Test Your Mathematical Sculpting Skills
Can you turn a twodimensional fractal into a 3D object? Break out your scissors and tape for a chance to win a 3D printed sculpture.

The Math That’s Too Difficult for Physics
How do physicists reconstruct what really happened in a particle collision? Through calculations that are so challenging that, in some cases, they simply can’t be done. Yet.

The Devil in the Polling Data
The same problem that caused the 2007 financial crisis also tripped up the polling data ahead of this year’s presidential election.

Air Traffic Control for Random Surfaces
Mathematicians have had a hard time finding commonalities in large groups of random shapes — until recently.

All Is Not Fair in CakeCutting and Math
When divvying something up, there's more than one way to define what's fair.

How to Fake It So No One Notices
Seek balance in all things—including subterfuge.

Why Blind People Are Better at Math
Bernard Morin developed glaucoma at an early age and was blind by the time he was six years old. Despite his inability to see, Morin went on to become a master topologist—a mathematician who studies the intrinsic properties of geometric forms in space—and earned renown for his visualization of an insideout sphere. For sighted people, […]

An “Infinitely Rich” Mathematician Turns 100
At the Hotel Parco dei Principi in Rome, in September of 1973, the Hungarian mathematician Paul Erdős approached his friend Richard Guy with a request. He said, “Guy, veel you have a coffee?” It cost a dollar, a small fortune to a professor of mathematics at the hinterland University of Calgary who was not much […]

Why We Need Quantitative Sports History
The early 19th century golfer Harry Vardon was the Tiger Woods of his day, and not just because he had marital difficulties. He even had a biography written about him, which recounted, among other things, how he handled losing his first child and living with tuberculosis. But Vardon’s life would be more useful to sports […]

Revisiting “Moneyball” with Paul DePodesta
Shattering preconceptions about players isn’t all about the numbers.

Men Are Better At Maps Until Women Take This Course
A bit of education can erase a definitive cognitive gap between men and women.

The Word “Million” Didn’t Exist Until We Needed It
I would cut off my right hand if you find it.” That was the guarantee retired Columbia history lecturer Jens UlffMøller made that there was no word for “million” in Old English, a medieval predecessor of the language you’re currently reading. Some AngloSaxon writers understood the idea of a million, and they had a term for it: a “thousand […]

The Artist of the Unbreakable Code
Composer Edward Elgar still has cryptographers playing his tune.

The Most Symmetrical Objects in the World
If you’ve ever tried to give yourself a haircut, you know just how hard it is to make something precisely symmetrical. We value symmetry so highly in part because it’s really hard to achieve. Here are five of the most symmetrical objects humans have ever crafted, and why they were so hard to make. 1. […]

Found: The World’s Favorite Number
Ecelop via Shutterstock Go ahead, admit it. Like a lot of people, you have a favorite number. Maybe you’re not as extreme as Sheldon Cooper, the archnerd character on television’s Big Bang Theory, who loves the number 73: “73 is the 21st prime number, its mirror 37 is the 12th and its mirror 21 is […]

A Formula That Shows How to Cheat & Triumph at Tournaments
Gryffindor and Slytherin are about to play their annual badminton match. The best players from each house are supposed to face off on court one, the second best on court two, and so on. Slytherin’s coach knows that Gryffindor will put their players on the right courts, in order of their skill, because Gryffindors are […]

The Cost of Cryptography
Computer security is trying to avoid wasting your time.

The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
Nine years ago, I was sitting in a college math physics course and my professor spelt out an idea that kind of blew my mind. I think it isn’t a stretch to say that this is one of the most widely applicable mathematical discoveries, with applications ranging from optics to quantum physics, radio astronomy, MP3 […]

The Artist of the Unbreakable Code
Composer Edward Elgar still has cryptographers playing his tune.

Purest of the Purists: The Puzzling Case of Grigori Perelman
Grigori Perelman became famous, despite his adamant opposition, for proving a conjecture from Henri Poincaré, pictured here. In November 2002, a Russian mathematician named Grigori Perelman posted the first of three short preprints to the arXiv (an online repository for drafts of academic papers in math and science), offering a proof for the famous Poincare […]

Revisiting “Moneyball” with Paul DePodesta
Shattering preconceptions about players isn’t all about the numbers.

Why a Traffic Flow Suddenly Turns Into a Traffic Jam
Those aggravating slowdowns aren’t one driver’s fault. They’re everybody’s fault.

How to Insure Against a Rainy Day
Never mind tornadoes—companies now seek shelter from daily weather.

Joys of Noise
The reliability of some technologies depends on just the right amount of randomness.

The Coin Toss and the Love Triangle
There are two flavors of uncertainty in our lives. Math helps with both.

The Deepest Uncertainty
When a hypothesis is neither true nor false.

The Deepest Uncertainty
When a hypothesis is neither true nor false.

Math as Myth
What looks like the golden ratio is sometimes just fool’s gold.