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Mathematicians have long fought to refute the idea that their subject is for nerds, separated from the excitement of the arts by a gulf of mutual incomprehension. In the latest salvo, two of Britain’s finest maths writers show how inextricably numbers are intertwined with artistic creativity, looking both at practitioners’ personalities and their works.
In A Perfect Harmony, David Darling focuses on music, while Marcus du Sautoy’s Blueprints also includes painting and sculpture, architecture and literature. But both authors describe vividly how the intrinsic rhythms and patterns of the universe are interpreted by mathematicians and artists — who are often the same individuals.
The best-known composers of the 18th century, Johann Sebastian Bach and Wolfgang Amadeus Mozart, were obsessed with maths and delighted in embedding mathematical structures in their music. As du Sautoy describes, Mozart’s The Magic Flute includes musical manifestations of both the “golden ratio” (in which the ratio of A to B is the same as A+B to A) and Fibonacci numbers (a sequence in which each number is the sum of its two predecessors). While mathematically initiated listeners may detect some of these patterns, others may respond subconsciously “to the moment that divides the overture into the golden ratio as the perfect turning point, even though we are not consciously aware of why it feels right”.
At the same time, mathematicians and physicists are often keen musicians. Albert Einstein, an accomplished violinist, loved the interplay between maths and the arts, as both authors show. He described pure mathematics as “the poetry of logical ideas” and said: “Life without playing music is inconceivable to me. I live my daydreams in music.” In literature, Einstein was fascinated by Fyodor Dostoevsky’s treatment of non-Euclidean geometry, which deals with more complex forms than those analysed by the pioneering Greek mathematician Euclid around 300BC. “Dostoevsky gives me more than any scientist,” he observed.
Du Sautoy takes a thematic approach arranged around “blueprints” — fundamental mathematical principles ranging from the golden ratio to Fibonacci numbers, Platonic solids to hyperbolic geometry — and focuses mainly on the past three centuries. Darling, on the other hand, writes chronologically and is particularly interesting about pre-Renaissance maths and music, which will be less familiar to many readers.
Humans originally made music with their voices, Darling argues, through singing. The first manufactured instruments were hollow bones with holes made on the side in a sequence that generates a musical scale when you blow into one end. The oldest discovered so far is the “Neanderthal flute” from Divje Babe cave in Slovenia, dating back more than 50,000 years — a claim challenged by some who say the holes were made by a biting carnivore but accepted by Darling.
The intervals and scales of modern western music were already being used in ancient Mesopotamia, a region that is home to modern-day Iraq, Kuwait, Turkey and Syria. Musicians playing wind, string and percussion instruments 3,500 years ago would have sounded, in terms of their typical sequence of pitches, much like European folk music today, Darling says.
Artefacts excavated in recent decades, including instruments and clay tablets with instructions for tuning and playing them, show the essential role of music in Mesopotamian life, where its uses ranged from soothing sounds to heal the sick to loud performances at feasts and processions.
Ancient Greeks — above all Pythagoras and his followers — were the first to apply rigorously the intimate connections between music and maths. Enthralled by numbers and the ratios between them, Pythagoreans realised that simple ratios of vibrating strings corresponded to harmonious intervals. One Greek innovation was to subdivide the octave into smaller divisions than the tones and semitones used in modern western scales. These “microtonal” modes were adopted by Arabs in the Middle East, where they still thrive, but largely disappeared in the west.
“The music we tend to think of as being uniquely Middle Eastern in flavour, with its sinuous turns and swirls and unexpected notes, is a direct product of the West — of ancient Greek modality,” Darling writes. “On the other hand, the wellspring of modern music in the West, based on diatonic modes, wasn’t Greece at all but Mesopotamia.”
Looking forward in time and space, Darling speculates about extraterrestrial civilisations — and how music might be a good medium for communicating with them. “Because music is fundamentally mathematical and because maths is universal, it seems likely that if other intelligent species have evolved elsewhere in the galaxy and beyond, they too will have come up with music of some form. The variety is likely to be immense, just as it is on Earth,” he writes.
“It’s often assumed that the first message we receive from the stars will be scientific or mathematical in content. But what better way to extend a greeting than by sending a really good piece of music, one that hasn’t just a logical basis but is full of the passion of its creators,” Darling declares.
Back on Earth, some of du Sautoy’s strongest writing is about the way maths is reflected in the visual arts. My favourite story shows how mathematical detective work exposed as fakes a trove of new paintings supposedly by the American abstract expressionist Jackson Pollock, discovered in 2005.
Pollock achieved his distinctive “drip painting” style by pouring, sloshing or dribbling paint on to a canvas on his studio floor. Some critics claimed that anyone could toss off such works, which can be worth many millions of dollars — and many fakers have tried. But in 1999 Richard Taylor, a physics professor at the University of Oregon, discovered something remarkable about Pollock’s paintings. They were fractals, displaying a mathematical structure discovered in the 20th century. A fractal pattern has no sense of scale, repeating itself at all possible magnifications. If you zoom in close on a real Pollock, the structure looks very similar to seeing the whole work from a distance.
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When Taylor experimented with volunteers trying to create paintings in the same way as Pollock, he found that all the imitations could be distinguished from genuine works because they failed to reproduce the master’s unique fractal patterns. Analysis of the cache of 32 claimed Pollocks, revealed 20 years ago by the son of family friends, showed that none was genuine.
People find Pollock’s work so appealing, du Sautoy believes, because he produced fractals with a structure close to those in plants, landscapes and elsewhere in the natural world. Although Pollock was painting before the word “fractal” had emerged, he knew what he was doing. “My concerns are with the rhythms of Nature,” he said. “I am Nature.”
That declaration sums up the message of these two fascinating books. The creative arts thrive on illuminating the intrinsic links between mathematics and the natural world — and indeed the whole universe.
Clive Cookson is the FT’s senior science writer
Blueprints: How Mathematics Shapes Creativity by Marcus du Sautoy 4th Estate £22/Basic Books $32, 400 pages
A Perfect Harmony: Music, Mathematics and Science by David Darling Oneworld £10.99, 288 pages
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