The fine line between art and maths

Sir Roger Penrose is a mathematician, but he also has a talent for drawing. As an exhibition of his work opens, Justin Mullins reports on the scientist laying a claim to art

When the Link Gallery at the Royal College of Art opens its doors this weekend, listen out for the indignant howls of derision. Today marks the start of an exhibition designed to send Britain's crop of angry young artists running for cover. Gone are the respectable media of modern art - the elephant dung, the formaldehyde and the dirty linen - and in their place are old-fashioned notions of dangerous ideas, of single-minded rebelliousness and of sublime creativity. Surely this cannot be art?

And sure enough, the author of this work is not an artist, at least not in the professional sense. He is Sir Roger Penrose, one of the world's best known mathematicians and a man with more experience than most at challenging established ideas and risking the wrath of conventional opinion.

The exhibition in Kensington, London, features a selection of drawings that Penrose, professor of maths at Oxford, has used in his lectures and to illustrate the books he has written to popularise his often controversial ideas. It is part of a major arts and science festival, "Creating SPARKS", organised by the British Association for the Advancement of Science, which runs through September.

Penrose's drawings are the product of one of the most creative and controversial minds in science and of a man unusually well placed to bridge the yawning gap between the two cultures.

As a young man, he had an introduction to life as an artist. At the age of 14, just after the second world war ended, he and his parents moved to Sussex to live with his uncle, Sir Roland Penrose, a surrealist painter, a collector of art and a driving force behind the establishment of the Institute of Contemporary Art in London. "I remember he had works by Picasso, Max Ernst and Man Ray around the house and Henry Moore in the garden," he says. These works were a significant influence on Penrose, who became fascinated by surrealism and Picasso's attempts to portray one object from various viewpoints in a single painting.

While his uncle was deeply involved in the world of art, Penrose's father, a professor of human genetics at University College London, was his link to the world of science.

Penrose's older brother became a distinguished mathematician and his younger brother became the British chess champion. "My father was always interested in puzzles and mathematics as well as art. I would often accompany him when he went sketching houses or bridges or landscapes. Of the three brothers, I was considered the most artistic. But it wasn't just drawing, I used to make things - pop-up books, cartoons, polyhedra."

In the mid-1950s, the relationship between mathematics and art was brought into sharp focus when Penrose discovered the work of mathematical artist Maurits Cornelis Escher at a mathematical conference in Amsterdam.

"Escher's exhibition had quite an influence on me. It just struck me as remarkable. There was a tightness of organisation and a twist to the logic that appeals to a mathematician," he remembers.

He left the Amsterdam conference with Escher's work firmly in his mind and began work on drawings that might impress the artist. Working with his father, he came up with the Penrose tribar, a triangle consisting of three bars at impossible right angles, and an impossible square based on the same idea. They sent the ideas to Escher who incorporated them into works such as the impossible staircase, which rises endlessly, and a waterfall in which water appears to flow upwards.

The tribar appears in the Kensington show in the drawing that Penrose picks out as his favourite. It shows an abstract depiction of three worlds: the platonic world, the physical world and the mental world. "Each one emerges from the other and the relationship is deliberately drawn as a tribar to show that there is something mysterious and paradoxical about the whole thing," he says.

This link between the mental, the platonic (or mathematical) and the physical worlds is one that causes many scientists to stare at their toes and shuffle their feet, but it is a recurring theme in Penrose's career. As a mathematician, he has specialised in geometry and, in particular, the problem of tiling a flat surface without leaving gaps, a topic that comes up in many of his drawings. "Almost all the work I do in mathematics is visual in the sense that I'm looking at pictures and I'm trying to see what happens when certain bits move in certain ways," he says.

The tiling problem is simple enough with repeating patterns of shapes such as squares and triangles, but Penrose's interest was in the more difficult task of tiling a plane without relying on patterns that repeat, so-called quasi-periodic tiling. The extraordinary thing about this problem is that it cannot be solved by a computer, even in principle.

For Penrose, this fact is profound. Were a human to come up with a solution, it would show that the human mind must work in a fundamentally different way to the computers on which sit the hopes of the artificial intelligence community. Armed with only a pencil and paper, Penrose eventually came up with the answer, which involved a non-repeating pattern of two shapes. He has even marketed the result in the United States as a mind-boggling jigsaw puzzle. For him it is proof that the workings of the human brain cannot be replicated on computer.

This is another controversial idea, and one that is represented in the exhibition by the simple example of a shape in which it is easy to determine the points that are inside it and those that are not. While this is a task humans can usually do in seconds just by looking, there are versions of this problem that are impossible for computers.

Penrose is equally contentious when it comes to quantum mechanics, the strange laws of physics that describe the universe on the smallest scale. He believes it is an incomplete theory and has diagrams to help explain his ideas. Penrose has famously argued that a full explanation of quantum mechanics might help to explain the phenomenon of human consciousness, exactly how we experience the colour red, the smell of a rose, the tingling of pins and needles. It is from this idea that the exhibition takes its title: "Shadows of the Mind" - also the title of one of Penrose's books on the subject.

Paul Bonaventura, from the Ruskin School of Drawing and Fine Art in Oxford, who organised the event, explains that the aim of the exhibition is to show that drawing is not the preserve of the artist or the architect.

"It is interesting to see the comparison between the way artists use drawing and the way mathematicians use drawing," he says. "Penrose's work is unique. Very few mathematicians spend time hand-crafting their images."

But while Bonaventura is keen to point out that he is not presenting the work as fine art, there is a profundity about Penrose's topics that few artists can match. "In mathematics, some arguments are beautiful and they will seem beautiful to anybody who can appreciate them. There is something absolute about it, something that is independent of humanity," Penrose explains. The extraordinary thing is that it is possible to get a glimpse of this mathematical beauty using drawings, although Penrose worries that the diagrams do not always convey quite the same aesthetic. "There are only a few examples that do convey the same aesthetic and they did choose one for the exhibition," he says.

It involves a proof that the sum of consecutive hexagonal numbers always adds up to a cubic number. Of course, words cannot do justice to beauty of this kind. So for anyone interested in the proof and its beauty, the visual form is hanging on the wall of the Link Gallery in Kensington until September 19. Angry young artists take note.

Creating Sparks is at the Link Gallery, RCA, September 6-30. Ticket hotline: 0870 2220; www.creatingsparks.co.uk. British Association for the Advancement of Science: www.britassoc.org.uk/holding/index.htm