Category Archives: math

What to Get a Maths Enthusiast for Christmas

Acme Klein Bottle

From time to time, we like to offer gift advice for what to get that one person you know who's almost impossible to find a gift for. This year, we're taking on the challenge of what to get that person if they're really into maths!

Sure, you could get them books, but the problem with that is they'll either be much too advanced or much too simple for the person for whom you're gift shopping. Partly that's because math concepts, by their very nature, go to extremes. There's not much in the middle, so your better bet is to get them maths stuff.

Maths stuff can be as complicated or as simple as you can find and it will probably be very appreciated by your maths-loving gift recipient. We're going to suggest the sweet spot is represented by the seeming impossibility of a Klein Bottle.

What makes a Klein Bottle special is how it incorporates the maths of topology in its construction. It's a surface that has only one side and does not have any edges, so it neatly introduces the concept of infinity in the form of something that can also be used to store stuff, which makes it unexpectedly practical.

You can even get one on Amazon, handmade in glass by UC-Berkeley's topologist Cliff Stoll. But if you're reading this, you'll want to act quickly because he only makes a limited quantity each year!

Speaking of which, the following 10-minute Numberphile video shows how he and Lucas Clarke made them this year from borosilicate glass tubes.

That leave the challenge of how to fill a Klein Bottle, for which there is another Numberphile video, also featuring Cliff Stoll, where you can get the answer while going on an unexpected ride through topological theory and physics.

A Mathematical Intersection Between Epidemiology and Chemistry

Early during the coronavirus pandemic, researchers in many fields got a crash course in epidemiology. More specifically, they got a crash course in how to apply the math behind the SIR model, which describes how fast a contagious condition might spread through a population before becoming endemic.

The SIR model divides a population into three categories, the Susceptible, the Infectious, and the Recovered (or Removed). Once basic data about the rates of infection and recovery are determined, the model can simulate how many people will fall within each of these categories at different points of time. Here's a primer introducing that basic math, in which we featured the following 22-minute video from Numberphile's Brady Haran and Ben Sparks on how to build your own SIR model from scratch using the online GeoGebra application:

Although each of the individual equations for each component of the SIR model involves relatively simple relationships, their interactions lead to much more complex math. Math that cannot be done simply by plugging numbers into an algebraic formula. Instead, the SIR model's math requires serious computing power to apply numerical methods, running thousands or millions of calculations to progressively reach reasonably accurate, but still not exact solutions.

That's why the press release for a paper recently published in the International Journal of Chemical Kinetics caught our attention. Its authors recognized part of the SIR model's math developed by epidemiologial pioneers W. O. Kermack and A. G. McKendrick is identical to the math used to describe the progress of an autocatalytic reaction in chemistry. Here's the Kermack-McKendrick integral, which has no direct solution:

Baird, Barlow, Pantha (April 2022) Equation 10 - Kendrick-McCormack Integral

In this equation, So and Io represent initial values for the number of Susceptible and Infectious portions of the popuation, R represents the Recovered (or Removed) portion of the population as a function of time (t), while the Greek letter lambda (λ) represents the ratio of the rate of spread among susceptible population to the rate of recovery. The letter e is Euler's constant.

That was math for which chemists James Baird, Douglas Barlow and Buddhi Pantha had developed the next best thing to a direct solution. They derived an approximate algebraic formula for quickly solving the Kermack-McKendrick integral with a small margin of error. Here's their simplified formulation:

Baird, Barlow, Pantha (April 2022) Equation 52 - Approximate Solution to Kendrick-McCormack Integral

Better still, they identified where their simplified formulation will work best:

In this report, a description is given of an accurate approximation to the Kermack-McKendrick integral which in turn can be used to determine values for R(t), I(t) and S(t) in the SIR epidemic model. The result is shown to be effective for situations where 1.5 ≤ Ro ≤ 10 with no need to numerically compute an integral.

The press release better describes how their formulation meshes with the chemistry of autocatalytic reactions:

Dr. Baird presented the model in May at the Southeastern Theoretical Chemistry Association meeting in Atlanta.

"The World Health Organization could program our equation into a hand-held computer," Dr. Baird says. "Our formula is able to predict the time required for the number of infected individuals to achieve its maximum. In the chemical analog, this is known as the induction time."

The formula is capable of predicting the number of hospitalizations, death rates, community exposure rates and related variables. It also calculates the populations of susceptible, infectious and recovered individuals, and predicts a clean separation between the period of onset of the disease and the period of subsidence....

"The rate of infection initially accelerates until it reaches a point where the infection rate is balanced by the recovery rate of infected individuals, at which point the number of infected people peaks and then starts to decay," he says.

That mechanism reminded him of the mechanism that governs an autocatalytic reaction.

"I subsequently learned that the mathematical description of the spread of infectious diseases was first described by Kermack and McKendrick," Dr. Baird says.

"When I read their paper, I realized that their mechanism was exactly the same as that of an autocatalytic reaction, where a catalyst molecule combines with a reactant molecule to produce two catalyst molecules," he says. "The rate of production of catalyst molecules accelerates until it is balanced by the rate of decay of the catalyst to form the product."

And that's how the algebraic formula that can quickly approximate the solution to the Kermack-McKendrick integral for the epidemiological SIR model with minimal computing power came to be published in a chemistry journal.


James K. Baird et al, Analytic solution to the rate law for a fundamental autocatalytic reaction mechanism operating in the "efficient" regime, International Journal of Chemical Kinetics (2022). DOI: 10.1002/kin.21598.

James K. Baird, Douglas A. Barlow, Buddhi Pantha. A Solution for the Principle Integral of the Kermack-McKenrick Epidemiological Model. [Preprint (PDF)]. DOI: 10.31224/2264. April 2022. [This second paper is an ungated preprint that focuses on the epidemiological application of the authors' approximation of the Kermack-McKendrick integral.]

The Dumbest Way to Solve a Maze

Suppose, for a moment, that you are the super-villain in a James Bond movie. You want to make 007 suffer from prolonged dramatic tension before ultimately killing him. How would you do it?

In the following 15-minute Numberphile video, Matt Henderson explains how the dumbest way to solve a maze might be your ticket. Particularly if that maze involves flooding poison gas into one part, which then has to make its way to where Bond is tied up in another part of the maze to succeed. [To be fair, Henderson uses the example of poison gas killing a canary rather than James Bond, but the hypothetical outcome would be the same and we might as well raise the stakes for what's really a discussion about coding a simulation to solve a maze!...]

The path the "winning" poison gas molecule takes is determined by a random walk, or Brownian motion, in which it can change direction at any time.

Believe it or not, there are real world applications that utilize what Henderson calls the dumbest way to solve a maze. Labyrinth seals use maze-like structures called tortuous paths designed to either minimize the leaking of something you don't want to get out or to minimize potential contamination from something you don't want to get in.

But our favorite application is the Labyrinth Security Door Chain. Imagine being a Bond villain who puts those on every door in their secret lair!

Can Statistics Catch a Cheater at Chess?

Aside from the rise of chess-playing computers capable of beating human grandmasters in the late 1990s, the world of chess has largely stayed out of the news since the days of the Cold War in the 1970s. Until September 2022, that is, thanks to cheating allegations levied against Hans Niemann by the chess world's reigning grandmaster Magnus Carlsen.

Niemann lacks a clear track record when it comes to cheating at chess. He has acknowledged cheating using electronic devices while playing in online tournaments, including some as recently as in 2020, which has resulted in banning him from competing in these events. But the 19-year old Niemann denies having done so in so-called 'over the board' matches, where such cheating would be much more difficult for him to pull off.

That hasn't stopped some pretty wild speculation for how such cheating might take place. But that doesn't address the question of whether Niemann is cheating in these real-life matches.

That's where the maths of statistics comes into play. The following 11.5 minute video from Chess & Tech's Albert Silver summarizes what he learned from interviewing Kenneth Regan, a chess master who has successfully used statistics to identify players whose cheating was subsequently confirmed.

The TLDW (Too Long, Didn't Watch) summary is that Regan finds Niemann's level of play is consistent with his ranking, his outlier success in winning has largely come from his competitors making more mistakes than he has while playing against him.

Which is to say that if Niemann is benefitting from cheating in his analyzed over-the-board games, that cheating is very different in nature than what is being alleged most prominently in the media. The most likely scenario that would align with the statistical results is that Niemann's competitors are throwing their games when they play him. If that's true, given the number of players that would have to be involved, it will not be long before such a conspiracy would be found out.

And if so, what a story for the ages! We'd have to go back to 1919's Black Sox scandal or to 1994's ASU basketball point shaving scheme to find a near equivalent to that hypothetical sports crime. Given that the biggest cheating scandal to ever rock the chess world is known as "Toiletgate", it certainly would be a step up for the little noticed sport.

Maths Prove Slowly Rotating Black Holes Are Stable

Black holes. They're massive, they're mysterious, and they sound really eerie.

And up until now, we didn't know how stable they might be. Fortunately, 2022 has seen a monster achievement in mathematical physics. Quanta Magazine's Steve Nadis sets the intergalactic stage for a remarkable mathematical proof:

In 1963, the mathematician Roy Kerr found a solution to Einstein’s equations that precisely described the space-time outside what we now call a rotating black hole. (The term wouldn’t be coined for a few more years.) In the nearly six decades since his achievement, researchers have tried to show that these so-called Kerr black holes are stable. What that means, explained Jérémie Szeftel, a mathematician at Sorbonne University, “is that if I start with something that looks like a Kerr black hole and give it a little bump” — by throwing some gravitational waves at it, for instance — “what you expect, far into the future, is that everything will settle down, and it will once again look exactly like a Kerr solution.”

The opposite situation — a mathematical instability — “would have posed a deep conundrum to theoretical physicists and would have suggested the need to modify, at some fundamental level, Einstein’s theory of gravitation,” said Thibault Damour, a physicist at the Institute of Advanced Scientific Studies in France.

Would Einstein be proven wrong? As a wise man once said, "bet on Big Al and give the points". In this case, Team Einstein has chalked up a new win in the form of a mathematical proof that such slowly rotating black holes are stable. And all it took was a team of three mathematical physicists working for years to invent the needed mathematical tools before finally producing a single 912 page paper to prove it.

In a 912-page paper posted online on May 30, Szeftel, Elena Giorgi of Columbia University and Sergiu Klainerman of Princeton University have proved that slowly rotating Kerr black holes are indeed stable. The work is the product of a multiyear effort. The entire proof — consisting of the new work, an 800-page paper by Klainerman and Szeftel from 2021, plus three background papers that established various mathematical tools — totals roughly 2,100 pages in all.

We may be understating the amount of work that resulted in the proof by calling it a monster achievement. What we found particularly interesting was the path by which they reached the proof:

The three mathematicians relied on a strategy — called proof by contradiction — that had been previously employed in related work. The argument goes roughly like this: First, the researchers assume the opposite of what they’re trying to prove, namely that the solution does not exist forever — that there is, instead, a maximum time after which the Kerr solution breaks down. They then use some “mathematical trickery,” said Giorgi — an analysis of partial differential equations, which lie at the heart of general relativity — to extend the solution beyond the purported maximum time. In other words, they show that no matter what value is chosen for the maximum time, it can always be extended. Their initial assumption is thus contradicted, implying that the conjecture itself must be true.

Proof by contradiction is one of the oldest tools in maths. The mathematician Euclid used it to prove there are an infinite number of prime numbers roughly 2,300 years ago. Because it's a great introduction into how the tool of proof by contradiction works, here is a seven minute video of Trefor Bazett working through the basic steps Euclid followed to prove there are an infinite number of primes in Book IX of Elements:

One could reasonably argue that the multiyear effort to prove that slowly rotating Kerr black holes are stable really took hundreds, if not thousands, of years to reach.

We are however putting the cart before the horse at this point. These are preprint papers that are now going through peer review phase, so until it passes muster, the Giorgi-Klainerman-Szeftel proof is under challenge. That will almost certainly add some additional years to the total assuming it all comes together and the proof is confirmed.


Elena Giorgi, Sergiu Klainerman, Jeremie Szeftel. Kerr stability for small angular momentum. Arxiv. [PDF Document]. DOI: 10.48550/arXiv.2205.14808. 30 May 2022.

Sergiu Klainerman, Jeremie Szeftel. Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes. Arxiv. [PDF Document]. DOI: 10.48550/arXiv.2104.11857. 30 May 2022.

Image credit: Stable Diffusion Demo: "A rotating black hole in space". Generated 28 August 2022.

A rotating black hole in space - Generated by Stable Diffusion Demo