Escher and Coxeter: a mathematical conversation

On Monday 5 June 2017, Professor Sarah Hart from Birkbeck’s Department of Economics, Mathematics and Statistics gave a prestigious Gresham Lecture at the Museum of London. Andrew Silverman, Learning Development Tutor in the School of Business, Economics and Informatics reports on the lecture.

Gresham College crest in hyperbolic geometry. Credit: Sarah Hart

Gresham College crest in hyperbolic geometry. Credit: Sarah Hart

Gresham College was founded in 1597 and has been providing free lectures within the City of London for over 400 years. Walking down from the dusty roads of the Barbican into the cool and quiet of the Weston Theatre, the audience was transported into a conversation between an artist and a mathematician, Escher and Coxeter. Told with infectious excitement and humour, Professor Hart wove the story of the lives of these two figures and their friendship through the mathematics and the artwork that fed into one another.

Born on 17 June 1898 in Leeuwarden, Holland, the youngest of five brothers and moving with his family to Arnhem when he was five, Maurits Cornelis Escher would eventually go on to study at the School for Architecture and Decorative Arts in Haarlem. His father was a civil engineer and his brothers all became scientists. Escher himself almost became an architect before switching to graphic arts. He later quipped that it was only by a hair’s breadth that he escaped becoming a useful member of society.

Escher began by producing woodcuts and lithographs featuring mainly landscapes. An example of this was the 1931 lithograph Atranti, Coast of Amalfi. But in 1936 his work went in a new direction, becoming more abstract; according to Escher, he had replaced landscapes with mindscapes. The woodcut Metamorphosis I, produced in 1937, exemplifies this change and is clearly a ‘mindscape’ adapted from the 1931 piece.

What could have triggered this change? In 1922, Escher visited the Alhambra palace in Granada, Spain; his second visit was in 1936. The buildings we know today were constructed in the mid-11th century by the Moorish king Mohammed ben Al-Ahmar. One of the key points about Moorish art, and Islamic art more generally, is that it is not permitted to contain images of living things; it is instead rich in symmetry and tessellations of tiles. Escher was able to combine this richness of geometric design with images of ‘living’ things (albeit at times mythical living things), thereby leading to works such as Angel and Devils (1941), produced in ink rather than wood.

Donald Coxeter, or Harold Scott Macdonald Coxeter, was born on 9 February 1907. His name was originally going to be Harold Macdonald Scott Coxeter, until they realised that this would have been HMS Coxeter, more a ship name than a baby name, and so the name was changed. As a schoolboy, Coxeter became so engrossed in geometry, at the expense of other subjects, that one teacher told him he was only allowed to think in four dimensions on Sundays.

In 1936, the year Escher’s art took a new direction, Coxeter took up a post at the University of Toronto. When asked what the point of pure mathematics is, Coxeter responded: “No one asks artists why they do what they do. I’m like any artist; it’s just that the obsession that fills my mind is shapes and patterns.”

In 1954, the International Congress of Mathematicians was held in Amsterdam. To coincide with this, a major exhibition of Escher’s work was held in the Stedelijk Museum in Amsterdam. It was here that Coxeter and Escher first met, when Coxeter bought a couple of Escher’s prints.

Incidentally, another mathematician who visited the exhibition was Roger Penrose, who was a Reader and then Professor of Applied Mathematics at Birkbeck from 1964 to 1973, and later became Professor of Geometry at Gresham College, London. After seeing an impossible staircase in Escher’s Relativity print, he came up with the concept of a ‘Penrose triangle’.

Professor Hart explained the mathematics behind the regular tilings in three geometries: plane (Euclidean), spherical and hyperbolic geometry. She managed to put the concepts across in such a way that even someone with no prior knowledge could walk away with a good basic understanding, and the images presented were an excellent way of getting a more intuitive sense of what was really going on.

Escher learnt a great deal from Coxeter, to the extent that when Escher created a picture based on a new geometrical concept, he would refer to it as ‘Coxetering’. But in turn, Coxeter also learnt from Escher. For example, Escher’s 1959 work, Circle Limit III, led Coxeter to a new understanding of the hyperbolic disc. By looking at the spines of the fish in the image, Coxeter realised that Escher had found equidistant curves and produced them incredibly precisely. In this way their friendship was a true exchange of ideas – a mathematical conversation.

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