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RTD info logoMagazine on European Research Special Issue - March 2004   
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 HOME
 TABLE OF CONTENTS
 EDITORIAL
 Science and the world, art and the ego
 The enigma of knots
 The mysteries of a mutant art
 Research in all its aspects
 Intuition and fantasy
 Science in fiction
 The seventh art
 Crossed ideas
 The paradoxes of perception
 Experiencing science through art
 Museums of the digital age
 Europe, researchers and cultural heritage

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Title  The beauty of maths

'Many mathematicians see their discipline as an art. They work according to their specific methods, but also using aesthetic theories that can be applied to artistic creation.   Conversely, some artists are attracted and/or stimulated by mathematics and use ideas developed by scientists.' That is the opinion of Michele Emmer, a mathematician and film-maker in whose company – among others – we take a look at the relationship between art and maths, images and visualisation, and aesthetics and education.  

The Möbius strip (1790-1868) is one of the most famous and easily understood geometric paradoxes. It simply involves joining the two ends of a strip together after first twisting one end through 180 degrees. This produces a 'non-orientable' surface. In moving along it, one moves from one surface of the strip to the other without any transition.   Konrad Polthier, T-U Berlin
The Möbius strip (1790-1868) is one of the most famous and easily understood geometric paradoxes. It simply involves joining the two ends of a strip together after first twisting one end through 180 degrees. This produces a 'non-orientable' surface. In moving along it, one moves from one surface of the strip to the other without any transition.
©Konrad Polthier, T-U Berlin
'The sense of beauty in maths must go back at least as far as Pythagoras. There is no lasting place for ugly mathematics. The elegance of a demonstration or the beauty of a formula can, in itself, be a pointer to the truth,' believes Osmo Pekonen, a Finnish mathematician from the University of Jyväskylä, who is also a writer. When Pekonen speaks of maths, his enthusiasm is immediately evident – and contagious. 'In mathematical physics, there are currently some mind-boggling predictive theories, such as String Theory and its most recent form, M-Theory. They allow us to dream of the existence of superior dimensions in the universe, dimensions which go beyond our usual perception of time and space where the fundamental interactions of contemporary physics – gravity, electromagnetism and nuclear interactions – would be unified. This may seem fictitious, and it remains experimentally unverifiable, but theorists want to believe in it because they are inescapably drawn to the beauty of equations.'   Such sentiments would have undoubtedly been shared by Henri Poincarré for whom 'All mathematicians experience a genuine sense of aesthetics. It is a question of sensitivity.'

Whether it is considered as art or not, mathematics plunges us into a world of balance and harmony – its links with music have long been the subject of study for example - and also of form (don’t understand this sentence). Do we not speak of mathematical objects – and of ‘objets d'art’? 

Plasticity
When rendered visible or palpable, equations become less obscure. Felix Klein (German mathematician, 1849-1925) was one of the first to understand this and, back in the 19th century, produced a collection of plaster models of complex functions that form the collection of the University of Göttingen (SE). The campus of Bangor University (UK) is also the site of James Robinson's Symbolic Sculptures, inspired in particular by the famous Borromean Rings – an idea that came from the mathematician Ronnie Brown, founder of the university's Centre for the Popularisation of Mathematics. For Professor Brown 'the major problem of teaching is converting mathematical reality into mathematical objects'. His exhibition on knot theory, based on an exhaustive and fascinating presentation of knots, also proved a resounding success.    

But sometimes the initiative comes from artists rather than mathematicians. Max Bill, an artist who worked with plastics and member of the Bauhaus school, spent many years investigating these abstractions – and sculpting, in his way, another life (see Emmer's lens). ‘Is it necessary to say that a mathematical approach to art has nothing to do with any ingenious system of calculation based on ready-made formulas?' he wrote in 1949. 'As regards composition, however, we can affirm that all schools of art have had, more or less, mathematical foundations.' 

Visibility
Illustration of another topological shape known as the Klein bottle, named after the mathematician who first described it in 1882. This surface, with no inside outside or edge, resembles a bottle whose neck has been stretched and then twisted back into itself through to the base.   Konrad Polthier, T-U Berlin
Illustration of another topological shape known as the Klein bottle , named after the mathematician who first described it in 1882. This surface, with no inside outside or edge, resembles a bottle whose neck has been stretched and then twisted back into itself through to the base.
©  Konrad Polthier, T-U Berlin
More recently, mathematicians have found a new media in computing. The infograph is a means of visualising known problems, of understanding how to solve others and sometimes of helping new research – especially in geometry. 'The point of departure was the discovery of new algorithms which generated new forms. Mathematicians and artists co-operated as never before with a new kind of Renaissance spirit, using what is known as visual mathematics,' explains   Michele Emmer, professor at the University of Rome. 'The importance of these virtual images and the possibilities for animating them are increasing all the time in mathematics. It seems only natural that all these new visual aspects should also interest artists. After all, at certain times, during the Renaissance for example, it was not easy to distinguish between an artist and a mathematician.' Hence the name of the official journal of The International Society for the Arts, Sciences and Technology, of which Emmer (and formerly Max Bill) is a member of the editorial team: Leonardo.

In Germany, Konrad Polthier, of Berlin's Technische Universität, is an enthusiast of mathematical visualisation. He quickly realised that his research on shapes and surfaces, in three or more dimensions, were too complex to be represented using 'ordinary' tools. That is why he created his own software. Known as Java View, it can run on any PC with an Internet browser – and no copyright is charged for non-commercial use. Java permits on-screen visualisation of the widest range of calculations. A simple ‘mouse click’ is all it takes to 'manipulate' the virtual object – twisting or stretching it, flipping it over to show other surfaces, etc. 'This software is designed to enrich scientific publications through visualisation. I would be happy to see many researchers use it,' comments its creator. We know that multimedia technologies are set to become increasingly important in the world of publishing. In some cases, it may seem like a gadget but, when discussing multidimensional mathematical problems or describing complex processes in science and technology, visualisation will become an essential tool.'

These new shapes are sometimes very surprising and the images fascinating. 'Some of these structures are so harmonious that it is almost impossible not to view them as works of art,' adds Konrad Polthier – who, by the way, denies any pretentions to being a creative artist himself. He sees the beauty of these forms as absolute, comparable to the natural beauty of plants and minerals, rather than a subjective expression or message.  

Communication
Computerised visualisation of the triple point Boy's Surface concept, discovered in 1922. This strange mathematical object with a single non-orientable surface (like the Möbius strip, of which it is an extension) is a model of what mathematicians call the 'projective plane'. It may be compared, intuitively, with a plane representation of all the straight lines converging from all the points on a sphere towards an 'eye' located at its centre. It can be understood as an analytical resolution of the famous geometric problem of the inverted sphere (its internal surface having been interchanged with its external surface).   Konrad Polthier, T-U Berlin
Computerised visualisation of the triple point Boy's Surface concept, discovered in 1922. This strange mathematical object with a single non-orientable surface (like the Möbius strip, of which it is an extension) is a model of what mathematicians call the 'projective plane'. It may be compared, intuitively, with a plane representation of all the straight lines converging from all the points on a sphere towards an 'eye' located at its centre. It can be understood as an analytical resolution of the famous geometric problem of the inverted sphere (its internal surface having been interchanged with its external surface).
©  Konrad Polthier, T-U Berlin
Visualisation can also be useful in enabling a better understanding of mathematics by bridging the gap between abstract concepts and their practical applications, most notably through revealing the underlying fundamental notions. Very often it is the educational applications of mathematical images that are cited by scientists rather than the aesthetic ambitions. Manuel Arala Chaves of the Faculty of Sciences in Porto (PT) distinguishes two aspects. 'First, beautiful pictures attract the attention, arouse interest and can subsequently lead to investigations into concepts and mathematical results. But, in addition to this, sometimes the pictures themselves may already suggest mathematical ideas or make it possible to illustrate them geometrically.'

Manuel Arala Chaves has initiated a number of exhibitions, in particular Matematica Viva, in Lisbon in 2000. 'An important point is the target public. This exhibition was aimed at everybody and succeeded in its goals. Schools and individual visitors visited who wanted to investigate their discoveries further.' The modules were designed so that they could be appreciated and understood at various levels, depending on the interests and mathematical background of the individual.

'When I was young I often visited the Deutsches Museum in Germany and the Palais de la Découverte in Paris. They made a big impression on me, but I always felt that they did not pay enough attention to mathematics and that this oversight was not due to the specific nature of mathematics as such,' he explains. 

Fun and games
Manuel Arala Chaves also worked on and adapted the exhibition Symmetry and the play of mirrors, designed by the team from Milan University's Department of Mathematics, for exhibition in Portugal. The same team has since launched the particularly original Matemilano exhibition, which investigates four principal themes (topology, massimi e minimi, vision and symmetry) with reference to the city itself. Architecture, sculpture and painting serve as a gateway to geometry, perspective, knot theory, etc. The Roman mosaics, Renaissance painting, the layout of the city, the Gothic rose windows of the Duomo…   all the periods in the city's history play a part. 'In these exhibitions we gave a lot of space to images and we are firmly convinced that this beauty plays an important role in communicating mathematics, especially to young people and people of different cultural origin,' believes Maria Dedo, one of the key players in this initiative. In addition to beauty, the exhibition also incorporates the notion of play in connection with mathematics. With its problems and its enigmas, maths can take us on some exciting new journeys. All it takes – as Matemilano shows – is some paper, a length of cord or a few matches to have some fun and games with maths.

 

Transformation of a catenoidal surface into a helicoidal surface. Already conceptualised by the mathematician Euler in 1740 and then by Joseph Plateau in the 19th century, this geometric form is by definition a so-called minimal surface, in the form of a 'diabolo' consisting of two superposed circles. In physical terms, it is equivalent to the stable form acquired by a film of soap stretched between the two circles so that the tension energy applied to the film is minimised. The complex mathematical analysis of the minimal surfaces was not formalised until 100 years later. In this way, a given surface is transformed from a catenoidal surface into a helicoidal surface.    Konrad Polthier, T-U Berlin Transformation of a catenoidal surface into a helicoidal surface. Already conceptualised by the mathematician Euler in 1740 and then by Joseph Plateau in the 19th century, this geometric form is by definition a so-called minimal surface, in the form of a 'diabolo' consisting of two superposed circles. In physical terms, it is equivalent to the stable form acquired by a film of soap stretched between the two circles so that the tension energy applied to the film is minimised. The complex mathematical analysis of the minimal surfaces was not formalised until 100 years later. In this way, a given surface is transformed from a catenoidal surface into a helicoidal surface.
©  Konrad Polthier, T-U Berlin

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  Emmer's lens

'In mathematics and in science we can perhaps speak of progress, but in art it is totally absurd to do so. Technology is in the service of art as it is in the service of mathematics, but creativity and inventiveness are vital to both,' believes Michele ...
 
  Roman Opalka or time measured

Roman Opalka, Polish, born in 1931

His life counts in a different way - or rather he counts his life in a different way - since that day in 1965 when, on a 195 x 136cm canvas, with a No 0 brush, dipped in white paint, on a black background, he ...
 

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      Emmer's lens

    'In mathematics and in science we can perhaps speak of progress, but in art it is totally absurd to do so. Technology is in the service of art as it is in the service of mathematics, but creativity and inventiveness are vital to both,' believes Michele Emmer.

    This Italian mathematician has had many opportunities to experience relationships between art and science through his work as a film-maker and his friendships with artists. For the past 20 years he has been engaged in the project entitled 'Art and Mathematics' which is seeking to gather together the results of his research linking these two fields, in the form of films, books, exhibitions and conferences.(1)

    The son of Luciano Emmer, the maker of fiction as well as documentary films – most notably Picasso (1954) – Michele found himself immersed in this environment from a very young age. 'When I wanted to make a film on Max Bill, I simply wrote him a letter. He promptly opened the doors to his home – and to his incredible collection of modern art. We made two films together, he stayed in the studio for three days and we became friends.'  

    'My personal experience with artists is very interesting because I discovered many similarities in the way we regard shapes and forms. Their approach to visual and plastic questions is clearly different, but there is such scope for a genuine exchange of ideas and experiences. These were not just brief encounters.'

    (1) His films include Moebius Strip, on which he collaborated with Max Bill, Soap Bubbles with the mathematicians Fred Almgren and Jean Taylor, and Geometry with the Japanese architect Koji Miyazaki. His books include: The Visual Mind Art and Mathematics (ed. Leonardo – MIT), The Fantastic World of M.C. Escher (Springer), and Mathematics and Culture (Springer, 2003).

      Roman Opalka or time measured

    Roman Opalka, Polish, born in 1931

    His life counts in a different way - or rather he counts his life in a different way - since that day in 1965 when, on a 195 x 136cm canvas, with a No 0 brush, dipped in white paint, on a black background, he painted the number 1 right in the top left. And then he carried on - 2, 3, 4...- until he reached the bottom right. On a second canvas, he continued the sequence of numbers from where he had left off.

    And the counting never ends. So as not to lose the thread, Opalka says the numbers out loud. The tape recorder records them , as well as the occasional silences (dead time). After each session, he sets up the camera and photographs his face, always in the same way - same light, same frame, same expression, same shirt.

    Some numbers have great beauty, such as 55555 which came up after 7 years. Opalka calls this work his life plan. On each canvas the black background is lightened by 1%. Opalka's works therefore become whiter, just like his face. The passage of time leads to the illegible, the invisible, destiny.

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