INDRA'S PEARLS

Interview with Professor Caroline Series, Department of Mathematics

We usually associate geometry with squares, triangles and circles, having often spent many hours with a protractor and compass in school. This is known as Euclidean geometry. However, in the late 1800s, a German mathematician, Felix Klein, revolutionised mathematical thinking by arguing that diverse types of geometry are possible without the conventional rigid properties. Yet it is only since the invention of computer graphics that Klein's concepts of never-ending reflections and infinite repetitions can be visualised. Professor Caroline Series, Department of Mathematics, has been exploring Klein's concept of multiple co-existing symmetries in her co-authored book Indra's Pearls.

The Greek mathematician, Euclid (circa 325 BC), is often referred to as the “Father of Geometry”. For over 2000 years his axioms influenced geometric mathematics and people’s ideas of space. Up until the end of the 18th century, geometry consisted of certainty and perfect answers. However, in the early 1800s a young Hungarian mathematician, János Bolyai, discovered non-Euclidean geometry which challenged Euclid’s theorems about the relationships and behaviour of shapes and constructs. This was taken further by various mathematicians, notably the Italian Eugenio Beltrami. But it was a German mathematician, Felix Klein, who made the most important conceptual breakthrough. He pulled together the ideas of Bolyai, Beltrami and others and presented a whole new unified concept of geometry, which incorporated not only both Euclidean and non-Euclidean space, but many other constructs as well. His work revolutionised mathematical thinking. Combined with the development of fast computer graphics a century later, some of his ideas paved the way for new departures in experimental mathematics.

The new geometry disobeyed some of these fundamental rules; it challenged the idea of absoluteness of space...

Professor Caroline Series, Professor of Mathematics at the University of Warwick and co-author of the book Indra’s Pearls (exploring Klein’s concept of multiple co-existing symmetries), explains, “Klein was expanding the idea of what geometry was; you might have studied it and thought it was to do with triangles, squares, circles. That’s Euclidean. But Klein’s interest was much wider; he brought together a range of mathematics and synthesised the idea that you could have general, diverse types of geometry without the previous rigid properties. He proposed that to do geometry, all you need is two things. First you need a`space’, and a collection of objects and constructs within that space. Then you need what is called a ‘symmetry group’; a way of moving these things around and saying ‘this one is the same as that one.’ But what these spaces and things are could be extremely broad; they could include non-Euclidean geometry, or projective geometry (used by artists to represent perspective) and many other types of spaces besides. Klein introduced his ideas in a famous lecture in 1872 which became known as the Erlanger Program.

“The discovery of non-Euclidean geometry revolutionised our ideas of space. Philosophers had always said that it had to be the way Euclid stated - parallel lines never meet and triangles have angles that add up to 180°. The new geometry disobeyed some of these fundamental rules; it challenged the idea of absoluteness of space and created huge controversial mathematical and philosophical debate.

“Beltrami’s work in the 1860s showed that non-Euclidean geometry was consistent only if Euclidean was (meaning if you believed in one you had to accept the other). But people remained extremely suspicious and it was Klein’s influence and ideas that eventually led to its acceptance and gave people the confidence to take the ideas further. So when the idea that space could be curved emerged, and Einstein put forward his Theory of Relativity in the early twentieth century, people were willing to accept it.”

Klein’s great rival Henri Poincaré showed how 3D non-Euclidean geometry could be partially visualised using 2D pictures. Klein and his students made many hand calculations and intricate hand-drawn illustrations of these 2D objects, but they knew that there was a limit to their efforts. “The objects he wanted to draw were made by doing infinite repetitions of moves which were in themselves simple, but the images got smaller and smaller the further you went – at some point if you have to do everything by hand you’re going to say enough is enough. His research was sitting there waiting for something to come along and carry it further. A century later the development of computer graphics and high speed computers provided this platform, and ideas which before were confined to the mathematician’s imagination could be fully explored.”

The first computer generated mathematical images of a similar type were produced in the 1970s, displaying the so-called Julia sets obtained by repeating the operations of squaring and adding complex numbers which had been introduced by Gaston Julia in the 1920s. These pictures led to the discovery of the famous Mandelbrot set. In the early 1980s, David Mumford, a Professor at Harvard University, had the imagination and insight to sit down and begin to explore Klein’s symmetrical world using computer graphics. The pictures produced were extraordinary and allowed study of the intimate connections between never-ending reflections, eventually leading to an analogue to the Mandelbrot set and new discoveries in 3D non-Euclidean geometry. Profs Mumford, Series and David Wright (who did most of the programming) published their work in their book Indra’s Pearls which, unlike most mathematics books, they managed to make also an understandable and enjoyable read.

We can now create elaborate and beautiful computer studies previously beyond our realm.

Computer programmes were enabling mathematicians to delve deeper; “Previously, people were content in saying, for example,`there exists some enormous number such that everything less than that number has a certain property’; it was outside the realm of hand calculation to actually find what the number was. But now it was possible to ask ‘How big is that number?’ It enabled mathematicians to think about problems in a more precise and different way.

“I don’t think I’d say these areas would never have been explored further without the development of computers but without any doubt computer generated images have provided a huge impetus.

“Before this point, published work in mathematical journals would always need to prove something and be perfectly rigorous. It’s not like science where you have experimental results; there had to be a perfect result or nothing. But around 1990, David Epstein, Emeritus Professor and Professorial Research Fellow at the University of Warwick, realised there was a need to have a place where you could publish this kind of experimental work. The Experimental Mathematics Journal was born. It’s still pure maths but now you could publish something very interesting which took a lot of skill to find but which you were not able to prove; it’s worth recording but it’s in a sort of in-between state where someone else could perhaps come along and develop it further.

“Younger generations of mathematicians are becoming much more interested in experimental studies. It has led to a whole new angle on mathematics, spilling over into computer science. We can now create elaborate and beautiful computer studies previously beyond our realm."