Interdimensional: Maths at the Heart of History

October 7, 2025

Interdimensional: Maths at the Heart of History

By Grace Olusegun

I doubt you’ve ever written down a² + b² = c² in your maths book and felt your mind wander to how the Pythagoreans were so disturbed by Hippasus’ discovery of the irrationality of 2 , he was thrown overboard. Or drifted off in class to reflect on how the scrap paper doodling of a French mathematician led to a discovery that could have changed the course of the Cold War. Well, this is perhaps because the curriculum has done a good job of portraying maths to be a two-dimensional binary system of right and wrong answers, mutually exclusive from other subjects - such as history, for instance. With that said, back to The Cold War and the fascinating role of mathematical discovery within it…

You may have assumed the nuclear arms race was inevitable. But what if I told you it never had to be? After the US dropped nukes on Hiroshima and Nagasaki, it was guaranteed that other nations would rush to develop nuclear weapons. Their unfathomable power made them the newest determinant of influence and control. With everyone rushing to get a slice of this new, potentially catastrophic technology, so came the need to regulate its development. Meetings with ridiculously literal names like“The Conference on the Discontinuance of Nuclear Weapons Tests”, initiated in Geneva on 31st October 1958 and (take a deep breath) “The Conference of Experts to Study the Possibility of Detecting Violations of a Possible Agreement on the Suspension of Nuclear Tests,” also held in Geneva in August 1958, were conducted to legislate nuclear testing everywhere possible: earthbound, underwater, in space and underground.

A partial ban was eventually signed in 1963, meaning nuclear tests were prohibited in the earth’s atmosphere, underwater and in space. But why only a partial ban? Well, it was all to do with how nuclear detonations could be detected. Atmospheric detonations aren’t difficult to verify. Radioactive isotopes produced by the explosion can be detected 1000s of kilometres away. Underwater, hydrophones pick up the definite sound caused by underwater detonations. Extraterrestrially, sensors in satellites monitor radiation consisting of gamma rays, X-rays and neutrons, emitted from nuclear detonations. Underground, well… There was no way to accurately deduce a subterranean nuclear explosion. You see, their radiation is mainly contained, the resultant impact in no way distinguishable from that of a seismic wave. And if you can’t detect something, how can you ban it? And so what did the USSR or US do? Well, they took the nuclear arms race underground of course.

Given the newfound dangers of this loophole, it was clear that a method of detecting underground nuclear outbursts was needed ASAP. And this is where mathematicians enter the scene

…Nuclear detonations and natural seismic events produce frequency wave signals that are almost identical, thus need to be broken down to tell them apart. This is where everything changes, meet the star of the show: the Fast Fourier Transform! The FFT was derived from the Discrete Fourier Transform, discovered by Joseph Fourier in 1807. The DFT is a method of Fourier Analysis, the study of how waves can be broken down into simpler sine and cosine waves. You can imagine it as taking a bucket of paint made up of various colours, and extracting each individual color in the mixture. Along with each individual color we find, is the proportion of that color in the bucket, like 10% yellow, 25% red etc.

Now is a good time to introduce three more main characters - American physicist Richard Garwin and American mathematicians John Tukey and James Cooley. Garwin and Tukey already had the DFT to work with, but at the speed of a 1960s computer, 1 single wave would take 3 years to decompose. On either side of the Iron Curtain, you would want to know immediately when and where the other side is testing. Nuclear powers wanted to spy on each other, monitor how powerful the other’s explosions were. But at this rate, that would be impossible.

In 1963, a meeting of the President’s Science Advisory Committee took place, reported to be apparently quite boring. So boring in fact, Garwin saw Tukey doodling throughout (issues of national importance were being discussed by the way). It was to Garwins surprise that Tukey had figured out a faster way to compute DFTs. Tukey had doodled and discovered a new Fourier Transform… aFast Fourier Transform. One that took a few minutes to a few hours to complete.

That’s a 99% decrease by the way. No joke.

The solution is actually as simple as doodling on scrap paper, just like Tukey was doing. The secret lies in exploiting the way in which waves repeat every given time or degree range. Sine and cosine waves have symmetry, so they overlap at predictable points. So when we are doing calculations to find if a given sine or cosine wave is a part of our frequency signal, some of these calculations can be done once, and reused if they repeat for other sine or cosine waves. This significantly reduces the processing power and time required for our Fourier Transform, taking it from a slow, Discrete one, to a Fast, more efficient one!

With their new Fast Fourier Transform ready to go, Tukey worked with Cooley to program a computer to perform the algorithm, co-publishing the method in a 1965 paper titled‘An Algorithm for the Machine Calculation of Complex Fourier Series’

And just like that the problem was solved. Although too late to adequately ease fears of an Armageddon, the FFT triggered enormous breakthroughs in many other fields - audio/image processing; chemical analysis; data compression; seismology and acoustics, marking the beginning of a new era in signal processing (simply the receiving, analysing and manipulating of signals). American mathematician Gilbert Strang described it as the “most important numerical algorithm of our lifetime” (1994), with the FFT being included in the Top 10 Algorithms of the 20th Century by the IEEE* magazine.

It doesn’t end there though.

Tukey and Cooley were too late to orchestrate a tangible ban on the testing of nuclear weapons. A shame if I must say so myself. But what if I told you this famous algorithm was originally discovered over a century and a half earlier? Remember Joseph Fourier from earlier, who we said discovered the Discrete Fourier Transform in 1807? Well, Carl Friedrich Gauss, arguably the most brilliant mathematician of all time, beat him to it, in 1805.

So if the FFT had already been discovered, surely a comprehensive test ban would have been established much earlier? Correct, but Gauss didn’t bother to publish it. He essentially disregarded it, deeming it primarily useless. To add insult to injury, his breakthrough only appeared after his death, using non-standard notation in a 19th century version of Latin, such that it was effectively unintelligible, thus was never adopted

.A sad story, yes but what it does do is perfectly illustrate one of the most beautiful things about maths: the symbiotic, mutualistic relationship it possesses with so many other fields

The story of the Fast Fourier Transform and its role in the Cold War is one of my favourite examples of this, where the purely mathematical study of harmonic analysis, frequency waves and trigonometry seamlessly plants one right in the middle of one of the most controversial periods of global political tension in history, magnificently captivating the essence of the 20th century - turbulent, revolutionary and apocalyptic.

.Indeed, I can’t help but wonder what would’ve been possible if Gauss had realised he almost erased decades worth of human destruction out of history, almost 2 centuries before it even occurred…

Glossary

SOFAR - Sound Fixing and Ranging Channel; a special, naturally-occuring layer in the ocean at which the speed of sound is at a minimum, its efficiency and capacity to travel long distances at a maximum

IEEE - Institute of Electrical and Electronics Engineers; formed from the combining of the American Institute of Electrical Engineers and the Institute of Radio Engineers

Juno, Pallas and Ceres - members of the asteroid belt → region between orbits of Mars and Jupiters, where majority of asteroids are found, in orbit of the Sun

Sinusoid - any periodic wave whose curve resembles a sine wave

Harmonic analysis - branch of maths overlapping with the Fourier Series concerned with the decomposition of a function into its components sinusoidal components (harmonics → anything concerned with sinusoidal functions + solutions to Laplace’s equation)