Is Visual Mathematics Artistic?

Siddiq Islam

In 1963, Stanisław Ulam sat down with a grid and counted the squares in an outwardly spiralling order. Every time he reached a prime number, he would colour the square in, forming what is now known as the Ulam spiral. The spiral’s intrigue arises from the strong diagonal, vertical and horizontal lines of primes – numbers whose patterns usually evade us. Regardless of its mathematical properties, I think the spiral is quite pretty and if you were to draw me one and frame it, I might be tempted to put it up on my wall.

Here is Ulam’s spiral with the numbers 1 to 225. I believe I’ve translated Ulam’s original, creative idea into a work of art, albeit a simple one. Arguably, the texture of the hatching adds additional artistic value, but the core idea is still there.

So here is the question: is Ulam’s spiral, alongside other visual representations of mathematics, art? As you might imagine, the answer is not clear-cut and will rely on our answers to other philosophical queries…

There are many ways of thinking about mathematical objects, but one thing most of us will agree on is that they are not physical entities. Whilst my desk might be rectangular, the rectangle as a shape is not a tangible thing. Mathematical objects are abstract objects. It’s like the colour green. Something can be green but greenness on its own does not exist… or does it?

Is this a triangle? A drawing of a triangle? A random image that we can describe as having properties similar to a triangle?

Platonic realism (named after Plato) is the belief that properties like greenness and rectangularity, referred to as universals, really exist in some abstract world of their own. When we see a green apple or a rectangular desk, we are seeing a representation of that thing, not the real, ideal deal. (There are other versions of mathematical realism such as Aristotelian realism (named after Aristotle), wherein these abstract properties of objects are found in real things, but don’t actually exist on their own like Plato believed.)

So, if rectangles, quadratic equations and negative numbers are universals, then Ulam’s spiral probably refers to the idea of colouring the prime numbers in a square grid, rather than the original paper on which he drew it. It’s the same way that a poem is made of abstract words and when we write it down, it is a copy of the original thing, but we would still consider it the same poem. Since Ulam and the poet have dealt with abstract ideas in a similar way, we could call them both artists, one dealing with words and the other with squares in 2D.

The way people define art can vary, but most definitions specify that art has to be man-made, so what remains to decide is whether mathematicians create their work or discover it. Plato and other realists would tell us that mathematics is innate, meaning its ideas are already out there in the universe and we discover them, rather than inventing them ourselves. T-rexes had two feet even when we weren’t there to count them, so the number two must have existed long before we could think about it. Ulam’s spiral, or, to give another example, Pascal’s triangle is an observation of life.

Every value in Pascal’s triangle is calculated by adding the two numbers above it, starting with 1’s on the edges. Within this triangle, countless patterns can be found including tetrahedral numbers, the Fibonacci sequence and Sierpinski’s triangle. It is a goldmine for mathematical observations and yet it is very hard to attribute all of these to Pascal when they seem to be retrospective realisations. Ulam only noticed his spiral was special after he drew it.

The first five rows of Pascal’s Triangle. Was this numerical poem constructed or were its verses innate patterns for us to discover?

If maths is innate, then it cannot be art because it already existed, and we discovered it. It’s like a beautiful sunset. It may be pretty, but it is not artwork (or we might call it God’s artwork). Regarding poems, they are also observations and descriptions of our world around us. Given that the English language already existed, it could be argued that Shakespeare discovered his plays rather than creating them.

This idea seems to distinguish physical art from other artforms. The Mona Lisa is actually in the Louvre. It is not an idea, it is a unique, physical artefact and if I were to recreate it myself, it would be forgery. It also distinguishes the musical composer from the musical performer. The composer authors an arrangement of notes on the stave and the performer then turns it into tangible, audible art. Perhaps all art requires a composer and a performer. Da Vinci embodied both by conceiving and executing his paintings.

If Plato is correct, then maths is not man-made and therefore not art, but neither is poetry nor written music, which most would agree are artforms. So, we might instead turn to nominalism. This implies that maths is a human construct that we have invented to describe our world around us, much like the English language. This allows mathematics to fall into an artistic category along with poetry and written music.

Perhaps you are still not convinced. Doesn’t art need to be more personal and unique and not so reproducible? Squares and circles seem far too commonplace to be considered artwork…

Well, what about the work of M. C. Escher, which is often attributed with mathematical concepts? The pieces Waterfall and Ascending and Descending are inspired by the Penrose triangle and the Penrose steps respectively. Devised by Lionel and Roger Penrose, these impossible shapes are just ideas, but like the pianist brings Mozart’s work to life, Escher injected the Penrose objects with vigour by drawing them as inhabited buildings, interacted with by people and water. To reject Penrose’s shape as art is equivalent to calling Shakespeare talentless because Hamlet was nothing but an abstract idea before the actors really brought it to life.

There are two Penrose triangles in Escher’s painting Waterfall.

Whether Penrose’s triangle or Ulam’s spiral evoke any emotion is a different question. Mathematical objects are often dismissed as pieces of art because they do not concern human feelings, but this is not a necessary aspect of art. Think about purer artforms, such as Barbara Hepworth’s beautiful string sculptures, and whether they are at all emotional, or whether they are purely aesthetic. Perhaps the level of complexity of art influences your decision of whether it is good or bad, but sometimes there is beauty in simplicity that dramatic, emotional artwork oversteps.

So, here’s how I see it. Nominalists believe Plato got it wrong with his realism, because all of our mathematics relies on rules that we have made up. Even if it is a fact that all triangles add up to 180 degrees, we invented triangles in our heads. This means mathematics requires human imagination to manipulate numbers, shapes and patterns, just as writers manipulate words. Then, when these ideas are beautiful, we like to draw them out and create mathematical proofs and diagrams, which really are pieces of art in their own right. Do you agree with me?

Part 1: Is Visual Art Mathematical?

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