Pythagoras was wrong.

Not about the theorem bearing his name, which describes the relationship between the lengths of the sides of a right triangle in two dimensions, which has been around for at least 2,500 years. That theorem was re-proved once again in five unique ways by a pair of high school students in New Orleans, whose work was published in 2024.

But not all of Pythagoras' long-standing conjectures stand up when put to the test, despite having been believed to be true for centuries.

We've gathered that story and more to present in our year-end round up of the biggest math stories of the year. In selecting these stories, we put special emphasis on stories involving practical or real-world applications of maths. We'll present the story we believe is the Biggest Math Story of 2024 at the end of this edition after reviewing the contenders we considered for that title.

Let's get started!

Moving Sofa Problem Is Solved

Is there any more real-world application than moving furniture?

At first glance, you might not think that might even involve any kind of math, and yet, there has been a geometry problem related to moving furniture that has perplexed mathematicians since 1966!

The problem is known as the "moving sofa" problem and it addresses the challenge of identifying the shape of the largest possible shape of a sofa-like piece of furniture around a sharp corner in a narrow hallway without getting stuck.

As anyone who has ever personally encountered that problem, say by trying to maneuver a sofa through a narrow hallway or through a stairwell knows, geometry is the key to success. So when South Korean mathematician Jineon Baek posted a more-than-100 page long preprint paper identifying the geometry of the sofa and confirming it is the largest possible shape that can be moved through a hallway with a narrow 90-degree corner, it captured the attention of a lot of people.

In this case, work by Joseph Gerver in 1992 had identified an optimal shape of the sofa, but the challenge lay in the proof of how large it could be. The following summary from Rutgers' University's press release gives the details:

In his work, Baek chose the Gerver sofa as a demonstration shape. The Gerver sofa is a mathematical construct developed by Joseph Gerver, a professor at Rutgers University, in 1992. It is basically a cuboid with a U-shaped front, a flat back with rounded edges and flat, front-facing arms.

After first, clearly defining the problem, Baek applies mathematical tools to move through the proof step by step before eventually arriving at the answer: For a hall of 1 unit, a Gerver sofa's maximum area can only be 2.2195 units. As part of the proof, Baek also narrowly defined the shape of the Gerver sofa he was using. Thus, different interpretations of the sofa shape would result in different answers.

Because the shape of the sofa is clearly defined at the outset, the answer Baek found could conceivably be used in the real world by people attempting to move a couch around a corner—though it would have to conform to the interpretation of a Gerver sofa as defined in the proof.

There's no telling how long it will be before some furniture designer looking to differentiate their product from the countless other sofa designs out there brings Baek's maximum-area sofa geometry to life. Until then, rest assured that the largest sofa you can wrangle around a 90-degree corner in a narrow hallway is smaller than Baek's sofa.

Discovery of the New Largest Ever Confirmed Prime Number

Prime numbers are the proverbial building blocks of mathematics. In fact, a very large portion of number theory is organized around determining how frequently they appear among all the numbers, which is one of the great, yet-to-be-proved unknowns for mathematicians.

Part of that challenge lies in the fact that there are an infinite number of prime numbers, which was first proven by another Greek mathematician, Euclid, about 2,300 years ago. Because there are so many, number theorists have had to come up with ways to sample prime numbers as far out into infinity as they can. One of those prime number sampling methods was developed by a Christian monk named Marin Mersenne some 400 years ago. In recent years, the largest numbers confirmed to be prime numbers have been Mersenne primes and in 2024, for the first time since 2018, a new Mersenne prime was confirmed!

Here's Numberphile's Brady Haran's video announcing the achievement, which includes an interview with Luke Durant, who found it:

The new record holder for world's largest prime, M(136279841), or rather, 2¹³⁶²⁷⁹⁸⁴¹ - 1, has 41-million or so digits.

That's a very big number, and it's certainly a very big math story, but since that achievement is not the biggest math story of the year, let's press on....

Pythagoras Debunked!

Pythagoras is a giant in the world of mathematics, having contributed much to the early organized study of the field in ancient Greece. Then again, Pythagoras was also a cult leader, whose persuasive ability has allowed some of his pronouncements to carry weight for centuries.

One of those pronouncements has to do with the idea of a universal musical harmony. Here, Pythagoras proposed that all humans would find a given harmony pleasing if they are based upon ratios of whole numbers. This concept has underscored musical theory in the Western world ever since.

Until 2024. Here's an excerpt from the press release announcing the findings of a team of Cambridge University mathematicians and musicians that demonstrate Pythagoras got this concept wrong:

According to the Ancient Greek philosopher Pythagoras, 'consonance'—a pleasant-sounding combination of notes—is produced by special relationships between simple numbers such as 3 and 4. More recently, scholars have tried to find psychological explanations, but these 'integer ratios' are still credited with making a chord sound beautiful, and deviation from them is thought to make music 'dissonant,' unpleasant sounding.

But researchers from the University of Cambridge, Princeton and the Max Planck Institute for Empirical Aesthetics, have now discovered two key ways in which Pythagoras was wrong.

Their study, published in Nature Communications, shows that in normal listening contexts, we do not actually prefer chords to be perfectly in these mathematical ratios.

"We prefer slight amounts of deviation. We like a little imperfection because this gives life to the sounds, and that is attractive to us," said co-author, Dr. Peter Harrison, from Cambridge's Faculty of Music and Director of its Center for Music and Science.

This finding has practical impact because it will likely reshape the direction in which music is produced. For example, the use of synthesizers and other computer-controlled instruments can easily produce sounds that satisfy Pythagoras' concept of an ideal harmony, which this study confirms doesn't really satisfy human listeners. If you've had the experience listening to music produced using these methods that can perfectly hit an "ideal" harmony, it often comes across as the musical equivalent of the uncanny valley in computer or AI-generated images. Which is to say it is anything but truly appealing.

Monkeys Will Never Write Shakespeare

There's a statistical conjecture that goes like this: If you take an infinite number of monkeys and put them in a room with an infinite number of typewriters then let them start pounding out letters at random, with enough time, they will someday produce the complete works of William Shakespeare.

It's one of those propositions that sounds incredible and yet is believed by many statisticians to be true. But since nobody has an infinite number of monkeys, typewriters, and time, nobody can definitively prove it.

This is clearly an example of an impractical problem. However, there is a practical way to test the conjecture with math, which is why a " study by the University of Technology Sydney mathematicians Stephen Woodcock and Jay Falletta caught our attention. Here's a portion of the press release announcing the publication of their paper:

"The Infinite Monkey Theorem only considers the infinite limit, with either an infinite number of monkeys or an infinite time period of monkey labor," said Associate Professor Woodcock.

"We decided to look at the probability of a given string of letters being typed by a finite number of monkeys within a finite time period consistent with estimates for the lifespan of our universe," he said.

The serious but light-hearted study, "A numerical evaluation of the Finite Monkeys Theorem," has just been published in the peer-reviewed journal Franklin Open.

For number-crunching purposes, the researchers assumed that a keyboard contains 30 keys including all the letters of the English language plus common punctuation marks.

As well as a single monkey, they also did the calculations using the current global population of around 200,000 chimpanzees, and they assumed a rather productive typing speed of one key every second until the end of the universe in about 10100 years.

The results reveal that it is possible (around a 5% chance) for a single chimp to type the word "bananas" in its own lifetime. However, even with all chimps enlisted, the Bard's entire works (with around 884,647 words) will almost certainly never be typed before the universe ends.

By defining the parameters of the problem in finite terms, Woodcock and Falletta successfully demonstrate how improbable the proposition behind the Infinite Monkey Theorem is within a more realistic world context. It's still an impractical thought experiment, as nobody is going to round up 200,000 chimpanzees to put them to work as typists, but one that verifies how even more impractical and unlikely the original proposition is.

The Geometric Langlands Conjecture Is Proven

The biggest math story of the year is also the biggest pure math story of the year. It offers the promise of finally delivering a unified field of mathematics, in which very different fields of math can be linked together, allowing the tools developed to solve problems in one field can be used to solve problems in others.

Before we go any further, it might help to take an 11 minute video tour of the map of mathematics, which provides a general overview to an endeavor that only begins with counting numbers before branching off into very different disciplines.

Within each of the fields on the theoretical side of the map, mathematicians have developed specialized tools for solving the problems they've tackled. But the tools developed to work in one theoretical field have often only proven to be useful within that field. Mathematicians working in other fields haven't been able to deploy them to solve the problems they are working upon.

That's disappointing because some of those tools are very powerful. What if it were possible to translate problems from other fields into ones that those powerful tools can solve?

That has been the challenge and promise of the Langlands project, which seeks to connect different fields of math by uncovering a shared language between them. Much as archaeology's Rosetta Stone made it possible to translate ancient Egyptian hieroglyphs and demotic scripts by connecting them to written Greek.

In 1967, mathematician Robert Langlands conjectured it would be possible to connect several different fields in mathematics, which if successful, would make it possible to used the tools developed in each to gain insights in the others.

From here, lets turn to Quanta Magazine's Erica Klarreich's reporting on a mathematical breakthrough:

A group of nine mathematicians has proved the geometric Langlands conjecture, a key component of one of the most sweeping paradigms in modern mathematics.

The proof represents the culmination of three decades of effort, said Peter Scholze, a prominent mathematician at the Max Planck Institute for Mathematics who was not involved in the proof. “It’s wonderful to see it resolved.”

The Langlands program, originated by Robert Langlands in the 1960s, is a vast generalization of Fourier analysis, a far-reaching framework in which complex waves are expressed in terms of smoothly oscillating sine waves. The Langlands program holds sway in three separate areas of mathematics: number theory, geometry and something called function fields. These three settings are connected by a web of analogies commonly called mathematics’ Rosetta stone.

Now, a new set of papers has settled the Langlands conjecture in the geometric column of the Rosetta stone. “In none of the [other] settings has a result as comprehensive and as powerful been proved,” said David Ben-Zvi of the University of Texas, Austin.

“It is beautiful mathematics, the best of its kind,” said Alexander Beilinson, one of the main progenitors of the geometric version of the Langlands program.

The proof involves more than 800 pages spread over five papers. It was written by a team led by Dennis Gaitsgory (Scholze’s colleague at the Max Planck Institute) and Sam Raskin of Yale University.

Please do click over to Quanta to find out more. Their coverage is an excellent entry point for understanding Langlands conjecture and how the mathematicians who published their proof of it some 57 years later tackled the challenge.

It's a massive proof and if it holds, a massive achievement with massive practical applications. The proof of the Geometric Langlands Conjecture is easily the biggest math story of 2024.

Other Takes

We're not the only ones surveying the year's mathematical developments. For a take that focuses more on pure math achievements, here is Quanta Magazine's video overview of the year in math:

Here is Scientific American's article on what they consider to be the seven coolest mathematical discoveries of 2024.

Previously on Political Calculations

The Biggest Math Story of the Year is how we've traditionally marked the end of our posting year since 2014. Here are links to our previous editions and our coverage of other math stories during 2024:

• The Biggest Math Story of the Year (2014)
• The Biggest Math Story of 2015
• The Biggest Math Story of 2016
• The Biggest Math Story of 2017
• The Biggest Math Story of 2018
• The Biggest Math Story of 2019
• The Biggest Math Story of 2020
• The Biggest Math Story of 2021
• The Biggest Math Story of 2022
• The Biggest Math Story of 2023
• The Biggest Math Story of 2024
• How Do We Know the Square Root of Two Is Irrational?
• Omniperiodic Oscillators in the Game of Life
• Happy Leap Year!
• A Sudoku Secret Pattern in Plain Sight
• Decoding the Language of Maths
• A Major Step Toward a Grand Unified Theory of Mathematics
• U.S. Patent Awards Correlated with American Cheese Consumption
• The Rule of 72
• Outside the Box Thinking: Fully Controllable Xmas Tree Lights

This is Political Calculations' final post for 2024. Thank you visiting us this past year, we hope found the stories and analysis we've presented to be either thought-provoking, informative, or entertaining. We'll see you again in the New Year, which we'll kick off with our annual tradition of presenting a tool to help you find out what your paycheck will look like in 2025.