Source: BBC

Art and mathematics. For many, this would appear to be synonymous with chalk and cheese. One is the domain of emotional expression, passion and aesthetics. The other, a world of steely logic, precision and truth. And yet scratch the surface of these stereotypes and one discovers that the two worlds have much more in common than one might expect.

Any creative artist will tell you that the emotional resonance of a piece emerges out of the construction of the work and is rarely an ingredient fed in at the beginning of a composition. The composer Philip Glass admits that he never deliberately programs any emotional content in his work. He believes it’s generated spontaneously as a result of all the processes that he employs. “I find that the music almost always has some emotional quality in it; it seems independent of my intentions.” The structure and internal logic of a piece is what drives its composition.

Perhaps more surprising is the role that emotions and passion play in the mathematics that we humans create. Mathematics is far from being just a list of all the true statements we can discover about number. Mathematicians are storytellers. Our characters are numbers and geometries. Our narratives are the proofs we create about these characters. Not every story that it’s possible to tell is worth telling.

I have spent many years as a mathematician working alongside artists and what has struck me is how similar our practices are. I have so often found artists drawn to structures that are the same ones I am interested in from a mathematical perspective. We may have different languages to navigate these structures but we both seem excited by the same patterns and frameworks. Often, we are both responding to structures that are already embedded in the natural world. As humans we have developed multiple languages to help us navigate our environment.

A different beat

“Music is the pleasure the human mind experiences from counting without being aware that it is counting.” – Leibniz

Music is probably the artistic discipline that traditionally has resonated most closely with the world of mathematics. As the German philosopher Gottfried Wilhelm Leibniz once declared: “Music is the pleasure the human mind experiences from counting without being aware that it is counting.” But this connection goes much deeper than that. The very notes that we respond to as harmonic have a mathematical underpinning, as Pythagoras famously discovered. And mathematical structures also inform the architecture of composition.

Take the 20th-Century composer Messiaen’s Quartet for the End of Time. In this piece, Messiaen creates an extraordinary sense of tension by employing one of the most important sequences of numbers on the mathematical books: the primes. In the opening movement, Messiaen uses the indivisible numbers 17 and 29 to create a sense of never-ending time.

If you look at the piano part, you’ll find a 17-note rhythmic sequence repeated over and over – but the chord sequence that is played on top of this rhythm consists of 29 chords. So as the 17-note rhythm starts for the second time, the chords are just coming up to about two-thirds of the way through its sequence. The effect of the choices of prime numbers 17 and 29 are that the rhythmic and chords sequences won’t repeat themselves until 17 times 29 notes through the piece, by which time the movement has finished.