Venturing into the Unknown

The unknown has always been the fuel for artistic and scientific endeavour. The things we don’t know are what drive us to our desks, our labs, our studios, our canvases each morning. For me as a mathematician it is the conjectures, the guesses at what might be true, that make my subject a living, breathing subject. The challenge is try to prove that those conjectures are right. The mathematical Everest sits there taunting you. Is there a pathway to its summit?

In 1637 Fermat conjectured that his equations had no solutions. It took a 350 year intellectual journey before my colleague Andrew Wiles in Oxford finally revealed Fermat’s belief to be true. That journey has come to an end, but it is the conjectures that are still open that continue to obsess the mathematician. The desire to know whether the Riemann Hypothesis is true or whether there might be a surprise waiting out that will change our perspective on prime numbers is what keeps me as a mathematician exploring. 

The unknown I believe is also what drives the viewer to keep staring at Zavattari’s mathematical landscapes. His canvases do not easily yield their secrets. What are the mathematical inspirations that inspired these cryptic images? The Poincare Conjecture? Chebyshev Nodes? What is the meaning of the strange symbols that are scattered across the canvases?  Not knowing is why we keep looking. Each canvas represents its own conjecture to the viewer.

Zavattari has used the word “isomorphism” to describe these images. This is a beautifully apt word. An isomorphism is a mathematical map which takes two seemingly different structures and finds a way to translate one into the other. An isomorphism is often a powerful tool for a mathematician. A structure that at first sight seems impenetrable can yield its secrets once mapped by an isomorphism into a second structure. 

For the non-mathematician Zavattari’s isomorphisms provide a window into some of the strange and beautiful structures of mathematics. Euler’s formula which fuses five of the most important numbers of mathematics – pi, e, i, 0 and -1 – in a single equation. The empty set containing nothing from which the mathematical logicians can build the whole mathematical edifice. The Fibonacci numbers whose connections to Nature are a story some viewers will already have encountered. 

But isomorphisms work both ways. So for the mathematician viewing these canvases there is also something to discover by looking the other way through the window. I know these mathematical structures like well told stories. But to see them in a new light through the visual representations of Zavattari helps to look at these familiar structures with new, fresh eyes. And it is this change of perspective that is often crucial if we are going to find a new idea to complete the journeys of those unsolved conjectures like the Riemann Hypothesis. 

Zavattari’s work gives the lie to the idea that there are two cultures: art and science. By creating isomorphisms between these two seemingly different worlds we can see that they are just two different languages for understanding similar structures. And it is by understanding the dictionary between the two that will help us to make progress on both sides of the equation.

Marcus du Sautoy - 2017