# Maths is painting without the brush; Painting is maths without the chalk

Although at a superficial level mathematics and painting may be perceived as of very different nature, they are profoundly similar at a deep conceptual and functional level. The deep and significant kinship between math and painting becomes evident, when one considers that both disciplines are concerned with a symbolic description of aspects of the surrounding world. Both painting and mathematics struggle to express, by abstraction, the general behind the specific and to establish the essential and relevant. Both disciplines try to digest and analyse notions such as open versus closed, or figurative versus non-figurative, or finite versus infinite. Both activities make use of conjectures and explorations.
It is important, not least for the teaching of mathematics, to realise that mathematics is fundamentally a discipline that is profoundly similar to the arts, music and humanities and in particular to painting. Mathematics will then not be considered a unique or alien discipline, but can be approached with playfulness and experimentation along the traditions used in the teaching of art, where rigor and exploration go hand in hand.

## INTRODUCTION

Mathematics is the science of pattern; painting is articulation through pattern.

We all know that Leonardo da Vinci was as much a painter as he was a scientist. Many people may also know that Isaac Newton considered his theological interests and investigations as being of equal value and importance as his scientific work. These few remarks just to indicate that simultaneous interest and gift for art and for mathematics are far from contradictory, but rather on the contrary can be wonderfully complementary.

Fundamentally, both mathematics and painting strive to develop a representation of the surrounding reality. And in both cases one is aiming at a representation possessing a large degree of universality. We see that this endeavour has been successful indeed in cases such as Euclid’s geometry or the wall paintings of Knossos. Despite the dramatic change in worldview during the centuries that separates us from Euclid or from Knossos’ artist, we understand and appreciate the mathematics or the art. I will emphasise the following fact. Namely, that painting and mathematics share a programme aimed at developing symbolic, concise, and often very abstract, representations of reality.

## THE CONCEPTUAL PARALLELS

Holy Icon of Theotokos Threnody (ca. 1860-1880) by Ioasaf Athonites

One objective, mathematics and paintings obviously share, consists in the striving to represent aspects of reality that cannot adequately be captured by words alone. A painting of a person may at one level represent, say, a woman but at another level may seek to reach far beyond the concrete object. Perhaps most clearly this is seen in the aspiration of the icon painters of Orthodox Christianity to create imagery that reaches into a different dimension of reality. When the religiously minded person meditate on an icon of Saint Mary, it is not the portrait of a women that develops in the mind of the beholder, it is rather a set of transcendental concepts related to the Theotokos, the God-bearer or the one who gave birth to God.

The icon is used to generate, or communicate, abstractions far beyond the immediate physical world. The icon attempts to handle concepts that cannot be reduced to the low “bandwidth” of our daily language. Obviously it is not only religious art that finds its proper level of expression beyond the concrete. Any painting, whether figurative or abstract, is, to one degree or another, an endeavour to create in the mind of the observer perceptions and feelings that are larger than what can be capture by a bare listing of the shapes on the canvas.

The situation is similar in mathematics. A conceptual universe is constructed with some anchoring points in the physical observable reality, from which explorations into the realm of the abstract are launched. An example can be the route from natural whole numbers through irrational number to the complex numbers. In this example we begin with concepts that can be represented by material objects, say number of beads on an abacus, and we arrive at concepts like imaginary numbers, for which we can’t find a simple tangible representation. However, the imaginary numbers are certainly able to reach into a realm of conceptual reality that is very real. We would for example not be able to represent the quantum mechanical world of atoms, if we didn’t make use of complex numbers – or some equivalent algebraic structure.

It is also natural to mention the fact that both painting and mathematics make use of representations that, when observed by a person with the appropriate background, create mental associations not directly referred to. Recall how the above Icon of Theotokos, when viewed by, say, an orthodox Christian, can stimulate associations to entities not visually present in the picture. Now compare this transcendental content of the icon to the effect on a mathematically trained mind of the diffusion equation $latex \frac{\partial\varphi}{\partial t}=D\nabla^2\varphi$

For the uninitiated the equation probably doesn’t make much sense, whereas for one trained in mathematics concepts such as heat, random walks, cream in coffee and many more may spring to mind. And this despite the equation in no direct way refers to any of these entities.

Let us consider the following simplistic artistic creation to illustrate the fact that both painting and mathematics will combine components together to obtain a sum that is greater and typically of an entirely different nature than the components. E.g. a smiley will use “•”, ”•”, “|” and “$latex \smile$” to represent a mental mood of a person $latex \begin{array}{c}^{\bullet}{ }_{\mid}{ }^{\bullet}\\^{\smile}\end{array}$. The smiley generates emotion in the receiver, whereas the unassembled individual components: “ •”, ”•”, “|” and “$latex \smile$” don’t.

Let us point out that the generation of an emotional state by the smiley in this example really involves an infinite number of interacting components. These components consist of the entire cognitive machinery of the observer together with the “cultural” heritage, which enables the observer to register and to interpret the emotional content of the four geometrical shapes organised in a smiley configuration.

The Scream by Edward Munch, Oil, tempera and pastel on cardboard. National Gallery, Oslo

As another example of parallel approaches in painting and mathematics I want to mention the effort to extract the essential and leave out irrelevant details. Think of Edward Munch’s The Scream or, as it was originally named, The Scream of Nature. The bare minimum is included to allow the overall composition to render a sense of ineffable angst. The heart penetrating effect of the painting is very much an effect of particular details not being allowed to distract. Had the faces or the human bodies been elaborated in more detail, our mind would immediately start to generate associations to particular people known to us. Similarly the broad strokes of the background don’t allow us to see this as a particular geographical location. We are therefore forced to experience the painting as a prototype of general relevance to our own emotional life. If the screaming figure in the front had reminded me of my aunt, or the two dark figures in the background could be thought of as the two bad guys always making trouble Saturday nights in my little home town; I might not have realised that the figure in front is myself on my way across one of the multitude of real, or virtual bridges, I have to cross on my way through life.

The situation is very similar in mathematics. Let us think of Newton’s equation of motion $latex m\frac{d^2x}{dt^2}=F$ The equation describes the trajectory, i.e. the position $latex x(t)$ for any time $latex t$, of any object of mass $latex m$ subject to the force $latex F$. The equation makes clear that only the mass and the force matter. We do not need to worry about the colour or the shape of the object. Both Munch’s painting and Newton’s law are able to study the issue in question, world angst or particle motion, respectively, neglecting an infinity of aspects. And in fact it is a crucial part of the understanding to realise that most details are irrelevant.

In more concrete ways the painter and the mathematician also often use corresponding techniques. I want to mention superposition. That is adding components together to obtain a whole that is different, and sometimes in various ways also bigger, than the collection of the parts. Superposition in mathematics can for instance consist of adding oscillating waves together to get a new temporal behaviour. The following equation adds together three sine waves each with frequencies, $latex \theta_1$, $latex \theta_2$ and $latex \theta_3$ , a little different from the others, to produce a new time signal $latex f(t)= sin(\theta_1t)+\sin(\theta_2t)+\sin(\theta_3t)$.

The result of the three added waves together with the three original waves is shown in the figure to the right. Interference between the three components leads to a behaviour of the sum that is markedly different in nature from the behaviour of the individual components. This effect is of course what string musicians make use of when they tune their instrument by listening for the sound “beat” between nearly tuned stings.

The emergence of a new quality due to the combined effect of components can also be seen in the painting below of landmarks from London.

The painting was painted from memory after moving around in London and tries to capture the effect of the myriad of contrasts one encounters in the great metropolis. The photos are included in retrospect and collected from the Internet. Although some of these structures are at quite separate locations in London, the net impression, left on the mind of a person moving through the city, is one crowded with buildings and people. In the memory the individual landmark has a tendency to loose its integrity and is recollected as part of a blend. The non-realistic juxtaposition in the painting attempts to generate a sensation of the overload of the mind experienced after a trip through London.

Our final example of conceptual parallels between mathematics and painting is concerned with the use of hierarchies. In mathematics one encounters many different kinds of hierarchical structures. Think of how the natural numbers, $latex \mathbb{N}$, sit inside the integers, $latex \mathbb{Z}$, which are located inside the rational numbers, $latex \mathbb{Q}$, which are part of the real numbers, $latex \mathbb{R}$, which in turn are inside the complex numbers, $latex \mathbb{C}$;. Or more simply stated by use of mathematical notation $latex \mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\subset\mathbb{C}$.

Modern mathematics is built around the theory of abstract sets, which is a formalised way of dealing with things that are to be found within each other. Set theory is used persistently across all fields of mathematics and has allowed a much greater precision than was possible before the development of modern set theory. Russell’s paradox is a famous example of complications that can arise when hierarchies and self-reference are combined. Russell suggested to “consider a set containing exactly the sets that are not members of themselves”. These considerations lead Russell to a paradox similar to the one introduced by the mythical Cretan Epimenides, when he claimed: “That all Cretans are liars”. So surely, hierarchies are powerful mathematical structures.

In fact hierarchical structures are so powerful and so natural that paintings often make use of them to evoke a global and delocalised impression. Think e.g. of Picasso’s paintings such as “Portrait of Ambroise Vollard” or “The Clarinet Player”. The painting to the right is a semi-figurative example of a hierarchical composition. The same theme repeats itself in a nested fashion throughout the painting whereby it becomes difficult to identify regions more important than others. Perhaps this helps to establish an impression of the theme’s all encompassing transcendental character.