Paul Dirac had an eye for beauty. In one essay, from May 1963, the British Nobel laureate referred to beauty nine times. It makes four appearances in four consecutive sentences. In the article he painted a picture of how physicists saw nature. But the word beauty never defined a sunset, nor a flower, or nature in any traditional sense. Dirac was talking quantum theory and gravity. The beauty lay in the mathematics.
What does it mean for maths to be beautiful? It is not about the appearance of the symbols on the page. That, at best, is secondary. Maths becomes beautiful through the power and elegance of its arguments and formulae; through the bridges it builds between previously unconnected worlds. When it surprises. For those who learn the language, maths has the same capacity for beauty as art, music, a full blanket of stars on the darkest night.
“The slow movement of the Mozart clarinet concerto is a really beautiful piece of music, but I don’t print off a page of the score and put that on my wall. It’s not about that. It’s about the music and the ideas and the emotional response,” says Vicky Neale, a mathematician at Oxford University. “It’s the same with a piece of mathematics. It’s not how it looks, it‘s about the underlying thought processes.”
Brain scans of mathematicians show that gazing at formulae considered beautiful by the beholder elicits activity in the same emotional region as great art and music. The more beautiful the formula, the greater the activity in the medial orbito-frontal cortex. “So far as the brain is concerned, maths has beauty just like art. There is common neurophysiological ground,” says Sir Michael Atiyah, an honorary professor of mathematics at Edinburgh University.
Ask mathematicians about the most beautiful equation and one crops up time and again. Written in the 18th century by the Swiss mathematician, Leonhard Euler, the relation is short and simple: eiπ+1 = 0. It is neat and compact even to the naive eye. But the beauty comes from a deeper understanding: here the five most important mathematical constants are brought together. Euler’s formula marries the world of circles, imaginary numbers and exponentials.
The beauty of other formulae may be more obvious. With E=mc2, Albert Einstein built a bridge between energy and mass, two concepts that had previously seemed worlds apart. Maggie Aderin-Pocock, the space scientist, declared it the most beautiful equation and she is in good company. “Why is it so beautiful? Because it comes to life. Now energy will have mass and mass can be put into energy. These four symbols capture a complete world. It’s difficult to imagine a shorter formula with more power,” says Robbert Dijkgraaf, director of the Institute for Advanced Study in Princeton, where Einstein was one of the first faculty members.
“One of the reasons there’s almost an objective beauty in mathematics is that we use the word beautiful also to indicate the raw power in an idea. The equations or results in mathematics that are seen to be beautiful are almost like poems.The power per variable is something that is part of the experience. Just seeing a huge part of mathematics or nature being described with just a few symbols gives a great sense of elegance or beauty,” Dijkgraaf adds. “A second element is you feel its beauty is reflecting reality. It’s reflecting a sense of order that’s out there as part of the laws of nature.”
The power of an equation to connect what seem like completely unrelated realms of mathematics comes up often. Marcus du Sautoy, a maths professor at Oxford, has more than a soft spot for Riemann’s formula. Published by Bernhard Riemann in 1859 (the same year Charles Darwin stunned the world with On the Origin of Species), the formula reveals how many primes exist below a given number, where primes are whole numbers divisible only by themselves and one, such as 2, 3, 5, 7 and 11. While one side of the equation describes the primes, the other is controlled by zeros.
“This formula turns these these indivisible prime numbers, into something completely different,” says du Sautoy. “On the one side, you’ve got these indivisible prime numbers and then Riemann takes you on this journey to somewhere completely unexpected, to these things which we now call the Riemann zeros. Each of these zeros gives rise to a note – and it’s the combination of these notes together which tell us how the primes on the other side are distributed across all numbers.”
More than 2,000 years ago, the ancient Greek mathematician, Euclid, solved a numerical puzzle so beautifully that it still makes Neale smile every time it comes to mind. “When I think about beauty in mathematics, my first thoughts are not about equations. For me it’s much more about an argument, a line of thinking, or a particular proof,” she says.
Euclid proved there are infinitely many prime numbers. How did he do it? He began by imagining a universe where the number of primes was not infinite. Given a big enough blackboard, one could chalk them all up.
He then asked what happened if all these primes were multiplied together: 2 x 3 x 5 and so on, all the way to the end of the list, and the result added to the number 1. This huge new number provides the answer. Either it is a prime number itself, and so the original list was incomplete, or it is divisible by a smaller prime. But divide Euclid’s number by any prime on the list and always there is a 1 left over. The number is not divisible by any prime on the list. “It turns out you reach an absurdity, a contradiction,” says Neale. The original assumption that the number of primes is finite must be wrong.
“The proof for me is really beautiful. It takes some thinking to get your head around it, but it doesn’t involve learning lots of difficult concepts. It’s surprising that you can prove something so difficult in such an elegant way,” Neale adds.
Behind beautiful processes lie beautiful mathematics. Well, some of the time. Hannah Fry, a lecturer in the mathematics of cities at UCL spent years staring at the Navier-Stokes equations. “They’re a single mathematical sentence that is capable of describing the miraculously beautiful and diverse behaviour of almost all of Earth’s fluids,” she says. With a grasp of the formulae, we can understand blood flow in the body, make boats glide through the water, and build awesome chocolate enrobers.
In his 1963 essay, Dirac elevated beauty from an aesthetic response to something far more profound: a route to the truth. “It is more important to have beauty in one’s equations than to have them fit experiment,” he wrote, continuing: “It seems that if one is working from the point of view of getting beauty in one’s equations, and if one has really a sound insight, one is on a sure line of progress.” Shocking at first pass, Dirac captured what is now a common sentiment: when a beautiful equation seems at odds with nature, the fault may lie not with the maths, but in applying it to the wrong aspect of nature.
“Truth and beauty are closely related but not the same,” says Atiyah. “You are never sure that you have the truth. All you are doing is striving towards better and better truths and the light that guides you is beauty. Beauty is the torch you hold up and follow in the belief that it will lead you to truth in the end.”
Something approaching faith in mathematical beauty has led physicists to draw up two of the most compelling descriptions of reality: supersymmetry and string theory. In a supersymmetric universe, every known type of particle has a heavier, invisible twin. In string theory, reality has 10 dimensions, but six are curled up so tight they are hidden from us. The mathematics behind both theories are often described as beautiful, but it is not at all clear if either is true.
There is a danger here for mathematicians. Beauty is a fallible guide. “You can literally be seduced by something that is not correct. This is a risk,” says Dijkgraaf, whose institute motto, “Truth and Beauty” features one naked and one dressed woman. “Sometimes I feel that physicists, like Odysseus, must tie themselves to the mast of the ship so they are not seduced by the sirens of mathematics.”
It may be that mathematicians and scientists are the only groups that still use the word “beautiful” without hesitation. It is rarely employed by critics of literature, art or music, who perhaps fear it sounds superficial or kitschy.
“I’m very proud that in mathematics and science the concept of beauty is still there. I think it’s an incredibly important concept in our lives,” says Dijkgraaf. “The sense of beauty we experience in maths and science is a multidimensional sense of beauty. We don’t feel it’s in any conflict with being deep, or interesting, or powerful, or meaningful. For the mathematician it’s all captured by that one word.”