Dipartimento di matematica, Universita di Roma "La Sapienza",

Piazzale A. Moro, 00185 Rome, Italy

email: emmer@mat.uniroma1.it

To

Fred Almgren

Fred Almgren

The "Mathematics and Art" project started in 1976. Or better, that year I started thinking of the project. The reasons why I started thinking of it are essentially two, or perhaps three. The first: in 1976, I was at the University of Trento, in the North of Italy. I was working in that area called the Calculus of Variations, in particular Minimal Surfaces and Capillarity problems. I had graduated from the University of Rome in 1970 and started my career at the University of Ferrara, where I was very lucky to start working with Mario Miranda, the favourite pupil of Ennio De Giorgi; then I met Enrico Giusti and Enrico Bombieri. It was the period in which in the investigations of Partial Differential Equations, of the Calculus of Variations and the Perimeter theory, first introduced by Renato Caccioppoli and then developed by De Giorgi and Miranda, the Italian school of the Scuola Normale Superiore of Pisa was one of the best in the world. And in the year 1976, Enrico Bombieri received the Fields medal. By chance I was in the right place at the right time. All the mathematicians world-wide who were working in these areas of research had to be updated about what was happening in Italy. In July 2000, I participated in the annual congress of the American Mathematical Society in Los Angeles, entitled "Challenges for the 2000". Many of the invited speakers were asked to provide a survey of the researches in the last 50 years. Those who talked of Partial differential Equations, Calculus of Variations and Minimal Surfaces like Karen Ulhenbeck, Haim Brezis and Jean Taylor, all recalled the group of mathematicians from Pisa, born around the famous Ennio De Giorgi, and its great scientific relevance.

Always in the 1976, Jean Taylor proved a famous result that closed a conjecture that was raised experimentally by the Belgian physicist Joseph Plateau over a hundred years before: the types of singularities of the edges that soap films generate when they meet. Plateau had experimentally observed that the angles generated by the soap films are only of two kinds. Jean Taylor, using the Theory of Integral Currents introduced by Federer, and then by Allard and Almgren, was able to prove that the result was true. A few years before, Ennio De Giorgi was able to prove in its generality the existence of the solution of Plateau's problem. To prove that for any chosen boundary it is possible to find a minimal surface that has this boundary. De Giorgi was also able to prove the isoperimetric property of the sphere in every dimension n. In three dimensions, which is the case of soap bubbles. You have a surface with assigned mean curvature that must contain a fixed volume of air. Soap bubbles have always been a great fascination for everybody, from children to scientists. It is sufficient to quote the case of Gilles de Jennes who ended his presentation on "Soft Matter" for the ceremony of the Nobel prize for physics in 1992 with a poem on soap bubbles.

french
*Amusons-nous. Sur la terre et sur l'onde *

Malheureux, qui se fait un nom!

Richesse, Honneur, faux éclat de ce monde.

Tout n'est que bulles de savon.

For an exhaustive story of soap bubbles in mathematics, in art, in chemistry, in architecture and in biology, refer to the volume "Bolle di sapone: un viaggio tra matematica, arte e fantasia" (Soap bubbles: a journey into mathematics, art and fantasy).
Malheureux, qui se fait un nom!

Richesse, Honneur, faux éclat de ce monde.

Tout n'est que bulles de savon.

Let's get back to Jean Taylor and to the year 1976. In 1976, the journal Scientific American asked Jean Taylor and Fred Almgren (they got married a few months before) to write a paper on the more recent results on the topic of Minimal Surfaces and Soap Bubbles. A professional photographer was asked to take the pictures for the paper. The same year, Jean Taylor and Fred Almgren were invited to the University of Trento as visiting professors and during the summer, they gave a summer course in Cortona, near Arezzo. I already knew both of them, maybe the one I knew better was Fred, who in his Swedish manner, was always very kind with me. A few years before, in Varenna, on Lake Como (North of Italy), another summer course was held, always on Minimal Surfaces. It was the year of the Olympic Games and the feats of the USA swimmer Mark Spitz. Mario Miranda was carried away with enthusiasm, and proposed a swimming race in the lake. The only participant who accepted was Fred Almgren. Miranda (at that time I was his assistant) asked me to join the two swimmers. I answered that for ten years I was a professional swimmer so it was not fair on my part to participate to the race. Miranda insisted on asking me to swim, saying that I had been a professional swimmer a number of years before. Of course I won the race leaving the other two swimmers many meters behind. A beautiful satisfaction for a young assistant! When Almgren and Taylor came to Trento in 1976 the issue of the Scientific American had just been published. The pictures of the article and the cover were really beautiful and interesting. I do not remember why, but looking at the pictures I had the idea of making a film on soap film and to show its shapes and geometry in the greatest possible detail, more closely and using the rallenti technique. I spoke of the project with Valeria, who lived in Rome with our two sons, and she was very pleased and attracted by the idea.

I must say that for me thinking of making a film was quite natural. My father is a famous Italian film-maker. Mastroianni made his first film with him, "Domenica d'agosto" in 1949. When I was a child and a boy I was always involved in film making, as collaborator, as organiser, even as an actor, in several of my father's movies. Both Almgren and Jean Taylor were very interested in my project. In any case my idea was not to make a ßmall" scientific film, a sort of scientific commercial just to show some small experiments with soap bubbles and soap films. I have never been able to stand these short films on mathematics (which have fortunately disappeared with the diffusion of computers), made to illustrate theorems or results of plane geometry or similar topics. These films are very boring and not very useful, not even for teaching mathematics at all the levels. I was attracted by the phenomena of soap films because they were visually interesting and I thought that the technique of filming them would have increased the general interest and fascination about them. I was not at all interested in just filming a lesson by Almgren and Taylor, with them explaining their results, inserting a few images of soap bubbles and soap films here and there. Almgren and Taylor shared my opinion. The project was not making any progress, because the motivation for making a film like this was not clear to me. Which was the purpose, if any; just the fascination of soap films? For which audience. And what did the length of the film have to be ?

Now the second reason. I was working at the University of Trento while my family, Valeria and sons, lived in Rome. Every Friday I left Trento to go to Rome (seven hours by train) and then on Monday, I travelled back to Trento. I have been always a lover of art, of any kind, of any culture and period. Of course there are some artists that I prefer. When I was in Trento, I read in a newspaper of an exhibition, in Parma, dedicated to one of the most important artist of this century: Max Bill. I already knew some of the sculptures of the Swiss artist but I had not visited a large exhibition like the one in Parma before. As the town of Parma was more or less on my way from Trento to Rome I decided to stop on my way back to Rome to see the exhibition. The topological sculptures of Bill were a real discovery for me. Years before, I had seen a large exhibition of the works of Henry Moore in Florence and of many other artists, but the ones of Bill almost immediately gave me the impression of *Visual Mathematics. The Endless Ribbon, that Moebius Band, enormous and made of stone, granite, was a real revelation. Its shape, its physical nature, tridimensionally real, making it live in space. A mathematical form, alive. This was the idea that was missing: mathematics, mathematicians in all the historical periods and in all the civilisations have created shapes, forms, relationships. Some of these shapes and relationships are really visual, they can be made visible. The idea for the great success of the use of computer graphics in some sectors of mathematics. In these same years the mathematician Thomas Banchoff was making his first short films in animation of mathematical surfaces but at that time I was not aware of his work.
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Coming back from Parma to Rome, I spoke again with Valeria. The project was becoming clearer: to make films, two perhaps, in which to compare the same theme from a mathematical and artistic point of view, asking for the opinion of mathematicians and artists. Not just filming a long discussion between artists and scientists on the theme that is so vague of the connections between art and science, but a real confrontation on the visual ideas of the artists and the mathematicians. To make visible the invisible like the artist David Brisson says in the film Dimensions made in 1984 with Thomas Banchoff. So the general plan of the project was almost clear: to make two films on the visual relationships of the forms created by artists and mathematicians. The themes of the two films were: soap bubbles, topology in particular the Moebius band. To have more visual ideas and objects to film we finally decided to include the connections between mathematics and architecture, all the other sciences, in particular biology and physics, not excluding literature and even poetry. And, why not, cinema. Just from the beginning of the project there was the idea of focusing on the cultural aspect of mathematics, the influence and the connections of mathematics and culture, of course starting from the point of view that mathematics has always played a relevant role in culture, being an important part of it. All these using the most important visual tool: filming. As these were the general lines of the project, it was quite natural to consider as part of it the organisation of exhibitions (many were made in the next years), congresses and seminars, the publishing of books (with many illustrations!), even theses for students in mathematics, in history of art, in architectures. Today, 25 years later, it is easy to say that the project went far beyond the expectations. Starting from 1997 at the University of Ca' Foscari in Venice, we organised an annual congress on Mathematics and Culture. From an idea that started in Torino after a discussion with Valeria, Odifreddi, a mathematician in Torino, and myself. A first, not so precise idea of such a congress was already included in the project of the seventies. In 1976, the whole project seemed a very absurd one for many reasons:
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- to make a film was (and is) very expensive; one thing was very clear to me. I did not intend to make an amateur's film. I wanted to make a real professional movie, of high quality, and all the technicians involved had to be well qualified.
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- I had started my professional career at the university and one of the most difficult things to do in an Italian university is to be involved in a field connecting two or more different areas. It can be the very quick end of your work at the university. This is still true today. But I was lucky because I was working on the Calculus of Variations and Minimal Surfaces, a field of great importance in the seventies.
- trying to obtain the collaboration of Italian mathematicians (for the reasons illustrated in the previous point) was very hard. It was considered not very professional for a mathematician to be involved in such a project. During the last ten years I have been invited to many Italian universities to show and discuss my movies. But when I first showed one of my movies in Rome, in 1981, to an pubblic audience, mathematicians of my department told me that it was not good for the
reputation of our department. This is the main reason why almost all my movies have been made abroad, in Europe, in the USA, in Canada, Japan, even in India. And the same is true for the publication of books and proceedings of congresses organised abroad or with the help of non-Italian mathematicians . This is the reason why it has been possible to organise the congress Mathematics and Culture in Venice, only in the last five years, not before. And in a few years the congress has become an important traditional meeting for mathematicians and students.
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- As it is clear from the previous remarks, to obtain funds and support for the project from the Italian institutions, was a desperate feat (and still is in a sense).
Notwithstanding all this, we started the project. The themes of the first two films were clear enough; we were looking for funds, this was the hardest thing to do; then it was necessary to choose the mathematicians and artists who would be involved in the project. Of course, the first thing was to obtain their collaboration. In those years RAI, the Italian public television, had a department specifically dedicated to educational projects called DSE, Dipartimento Scuola Educazione. They accepted the challenge to make a program on mathematics for the first time. Nobody before had proposed something on these topics, except the lessons filmed for strictly educational reasons. But they requested one condition: that we make a series of films. This is a magic word for television: series. If you want to do something for the Italian Television (but I have the impression it is the same everywhere) you have to propose a series. It is not important that you have no ideas about how to do a series, but you have ideas only for one or two episodes. So they asked me to make 8 films. I said no for the reason that I had no ideas to make the other six. Finally we reached an agreement to make 4 films, with an option to make 4 more in the following two years. Each film had to be 27 minutes long. The other topics we were thinking about were: Platonic Solids, Symmetry and tessellations.
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In these same years I had already discovered the works of the Dutch graphic artist Maurits Cornelis Escher. From the first time I saw his engravings my purpose was to make a film only on him. My idea was to use the technique of animation in order to make his works really tridimensional. Something that Escher himself suggested; he personally was involved in a short film with several animations of his works before his death in 1972. I discovered his ideas and the film only a few years later. The financial support of RAI was not sufficient to make my first four movies; it was out of question to make a film on Escher using animations, as this technique is very expensive. Financial support from RAI was enough just to make a film, entirely in a studio, with a person talking all the time; their idea of an educational TV series. My idea was to film all over the world where the artists and mathematicians involved in the film were working. My father, film-maker, was the producer of all of them; so I was able to find more supports. In any case the project of a film on Escher was postponed. I only filmed three works of Escher and inserted them in the film Moebius Band.
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- It will take me more than ten years to complete the film on Escher in 1990. What we needed for the project was a title; it was quite natural to choose the general title "Mathematics and Art" (even if for RAI the title was changed because the Italian television, like the Italian Universities, cannot consider the idea of treating two topics at the same time; the problem is to have a precise target! ) This added more months to the start, but at the end of 1979 I was able to show a first and preliminary version of Soap Bubbles at a Scientific Film festival at the CNRS in Paris. Also the second movie on the Moebius band was almost ready. In the film on soap bubbles I asked the collaboration of the Italian artist Arnaldo Pomodoro, who has always been fascinated by the theme of Spheres, while in the Moebius band's film, apart from Max Bill, I filmed the works of Corrado Cagli, of the French designer Moebius.
I contacted Max Bill writing him a letter. He was very kind; he invited me with my troupe to his house in Zurich and he gave me permission to film everything I was interested in, including his fabulous collection of contemporary art. With one exception: it was strictly forbidden to film a little window in which there was his collections of forms, topological forms, made in paper. Very small objects, his Data Base for future works. He was afraid that someone could see his projects and copy them. We then became friends, we made two exhibitions together, and another film on Ars Combinatoria . We both were in the editorial board of the journal Leonardo, at that time published by Pergamon press, then by MIT Press. For my book The Visual Mind: Art and Mathematics, MIT Press, 1993 (4th edition), Bill rewrote the title and made same changes to his famous paper originally written in 1949 A mathematical approach to art. Two of Bill's works are reproduced on the front and back cover of the book. A new volume The Visual Mind 2 will be published, always by MIT Press in 2002. The volume will be dedicated to Valeria and Max Bill.
Of course it is very hard to describe a film using words, it is almost impossible, even not correct. If it is almost impossible to describe a film using words, it is good, because it means that the film has been made really using a visual technique, mixing, images, sounds, music in an essential and possibly unique way. If a film can be narrated it means that something is not working well from the visual point of view. One thing was really clear to me: in making the films, all words, all explanations had to be reduced to the minimum, or even be absent if possible. Whenever possible, images must speak for themselves. If, for its nature, art does not need explanations, mathematics too has to be presented almost without words. A film is not the best tool to explain and to learn. A film can, in a short amount of time, give ideas, suggestions, stimuli, emotions. A film can generate interest, even enthusiasm. Looking at an interesting, pleasant film can stimulate the audience to learn more, both in the artistic and the mathematics fields. In this sense I consider my films educational, but only with this meaning. This was also the reason why at the beginning the films were refused by RAI.
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This, on the contrary, is the secret of their success, as for example for the
movie Soap Bubbles, even 20 years after the film was made. In fact the most beautiful sequences I have ever made, the soap films dancing to Weber's RosenKavalier waltz was included in the VideoMathfestival selection for the World Mathematical Congress in Berlin in 1998, and in the European Congress in Barcelona in 2000.
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In the making of the films of the series, in the last 25 years (actually the films are 22), I have had the help of many artists. I now want to give some examples of the kind of collaboration that I had; of course the best thing to understand this, is to look at the films. It is not enough to read what the artists are saying. For the occasion of the congress in Maubege I have assembled selected scenes taken from different films, making a selection of the artists, making in a sense a new film, new as each time you take some visual material and you manipulate it, you area creating something new following your sensibility at the time of the new edition. A film that can be called: Results of the last 20 years.
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Moebius Band
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There is no doubt that the clearest approach to the possibility of a mathematical approach to the arts has been formulated by the famous Swiss artist Max Bill. In 1949 he wrote: "By a mathematical approach to art, it is hardly necessary to say I do not mean any fanciful ideas for turning out art by some ingenious system of ready-reckoning with the aid of mathematical formulas. So far as composition is concerned, every former school of art can be said to have had a more or less mathematical basis. Even in modern art, artists have used methods based on calculation, inasmuch as these elements, alongside those of a more personal and emotional nature, give balance and harmony to any work of art.
These methods had become more and more superficial, for the artist's repertory of methods had remained unchanged, except for the theory of perspective, since the days of ancient Egypt. The innovation occurred at the beginning of the twentieth century: it was probably Kandinsky who gave the immediate impulse towards an entirely fresh conception of art. As early as 1912... Kandinsky in his book Ueber das Geistige in der Kunst indicated the possibility of a new direction which would lead to the substitution of a mathematical approach for improvisations of the artist's imagination... It is objected that art has nothing to do with mathematics; that mathematics, beside being by its very nature as dry as dust and as unemotional, is a branch of speculative thought and as such in direct antithesis to those emotive values inherent in aesthetics... yet art plainly calls for both feeling and reasoning."
We must not forget that Max Bill was first and foremost a sculptor who believed that geometry, which expresses the relations between positions in the plane and in the space, is the primary method of cognition, and can therefore enable us to apprehend our physical surroundings, so, too, some of
its basic elements will furnish us with laws to appraise the interactions of
separate objects, or group of objects, one to another. And again, since it is mathematics that lends significance to these relationships, it is only a natural step from having perceived them to desiring to portray them. Visualised presentations of that kind have been known since antiquity, and they undoubtedly provoke an aesthetic reaction in the beholder.
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And here is the definition of what must be a mathematical approach to the arts: Ït must not be supposed that an art based on the principles of mathematics, such as I have just adumbrated, is in any sense the same thing as a plastic or pictorial interpretation of the latter. Indeed, it employs virtually none of the resources implicit in the term pure mathematics. The art in question can, perhaps, best be defined as the building up of significant patterns from the ever-changing relations, rhythms and proportions of abstract forms, each one of which, having its own causality, is tantamount to a law unto itself. As such, it presents some analogy to mathematics itself where every fresh advance had its Immaculate Conception in the brain of one or other of the great pioneers."
To convince his readers, after having clarified his thoughts, Bill needed to provide some examples which pertained to his point of view as an artist, examples of what he called "the mystery enveloping all mathematical problems, the inexplicability of space that can stagger us by beginning on one side and ending in a completely changed aspect on the other, which somehow manages to remain that self-made side; the remoteness or nearness of infinity which may be found doubling back from the far horizon to present itself to us as immediately at hand; limitations without boundaries; disjunctive and disparate multiplicities constituting coherent and unified entities; identical shapes rendered wholly diverse by the merest inflection; fields of attraction that fluctuate in strength; or, again, the space in all its robust solidity; parallels that intersect; straight lines untroubled by relativity, and ellipses which form straight lines at every point of their curves.
Far from creating a new formalism, what these can yield is something far transcending surface values since they not only embody form as beauty, but also form in which intuitions or ideas or conjectures have taken visible substance.
Thus, the more succinctly a train of thought was expounded, and the more comprehensive the unity of its basic idea, the closer it would approximate to the prerequisites of the mathematical way of thinking. The orbit of human vision has widened and art has annexed fresh territories that were formerly denied to it. In one of these recently conquered domains, the artist is now free to exploit the untapped resources of that vast new field of inspiration. And despite the fact that the basis of this mathematical way of thinking in art is in reason, its dynamic content is able to launch us on astral flights which soar into unknown and still uncharted regions of the imagination."
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One of the visual ideas that Max Bill used without realising it was the Moebius ribbon. Max Bill called some of his sculptures Endless Ribbons and
in fact they are shaped like Moebius ribbons even if at that time Bill has no idea of the Moebius band. The interesting thing to note is that Bill thought he had invented a completely new shape. Even more curious was that he discovered it by twisting a strip of paper, just as the mathematician Moebius had done many years previously.
Max Bill's Endless Ribbon was put on display for the first time at the Milan Triennale exhibition in 1936.
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At the end of the seventies, I wrote to Max Bill asking him if he was interested in making part of a film on the Moebius strip in art and mathematics. He was very kind and he invited me to his home in Zurich. As a choice, in all my movies there are questions asked to the artists and mathematicians involved. Actually I prepared several questions, a sort of script to be followed in all the interviews. Then of course all the answers were edited, cut to my choice. The most important thing in a movie is that everything must seem natural, the only possible way of doing it. Of course not, it is the point of view of the filmmaker.
So here are the words, better, the words I have used, that Max Bill said in his answers.
"The first strip that I made is exactly the same as this one, which is a real Moebius strip, however that one was shorter and had a central support. A few years later I discovered that when making a Moebius strip one could orientate it in different directions. With these different possibilities of positioning one can obtain all sorts of variations of the same shape.
I made the first Moebius strip without knowing what it was. I made it by accident. I wanted to make a decorative object to put above an electric fire. An object that moved, so I tried to make it with paper, like a game for children. I tried to make something that would turn in the air and would give the impression of spiralling, and trying over an over again with the paper I came to a shape like this, a shape with only one surface which has all the characteristics of the Moebius strip.
I started by making various works always based on the same idea. They are works where the external edge crosses the surface: for example, I take a surface with six angles, I join these angles and I have a complete circle: that way, all this group of figures takes on new characteristics which no longer have anything in common with the Moebius strip, they are only folded surfaces, cut according to my imagination."
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We made another film together, Ars Combinatoria. In his words Bill explains what is happening in the part of the painting that does not exists, that he has never painted.
Bill "The idea of this picture is that it is a square balanced on a pivot. It is not a normal square as one can see by doing this like for windows, for doors…Here it is balanced on a pivot. When a figure is balanced on a pivot in space it becomes elongated in all four directions; that is the basic principle behind this square on a pivot. There's the other part, here, there are colours that cross it, there is an area of colours and what is beyond the colours is the same quantity as we have here….
The inside there is an octagon, and in this octagon there is this square; this square fixes the whole figure, and these areas cross the figure. And then outside there is an area with colours, and there are these triangles…These triangles are equivalent that has been cut away here on the side. So …inside there are rhythms made up of a rotation…. It is a rotation from yellow to red, as far as blue and green here on the inside; when one looks at it from the outside one sees the same series, there is yellow, red, blue and green.
And on the inside there's this composition using colours, the green in the blue, and here there's yellow in green…and one has a rotation similar to the previous one.
So this is the system, this lengthening of strips and one creates a mysterious area because we do not know what is happening here on the outside, one can imagine that there is mixture of colours but we can't be sure because this has been cut. So there is something mysterious about it, because this doesn't exist yet, but there is a tendency that's increasing, that's growing there, somewhere…….
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A few years later I studied the possibility of fixing thought precisely, in a way that wasn't too personal, that in a sense is objective. At that time, I was looking for the basis of the development of figures, and I went back to music knowing that it is based on the laws of mathematics, and that it is possible to find rules behind the organisation of a theme. This was the beginning of the idea of variations. I then tried some types of variations but in a limited system. In other works, I tried to push a theme to its limit.. For instance, I did something with squares and, once I had carried out the operation that I had established as the basis of the theme, once it had been repeated eleven times, then it was exhausted, there was no other new variation to be tried."
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In the film Ars Combinatoria another famous Italian artist, Luigi Veronesi, explains his methods for variations.: "How did I came to the subject of variations…I took the advice of Leger, the French artist, when I was studying under him in Paris. This was his advice, 'Veronesi, don't stop at a single image, but consider the image you are thinking of, that you have in your mind, as a theme on which to work variations, exactly like a musical theme. Working like this is the only way to see an image from every side, from every aspect.
It's an advice that I accepted with enthusiasm, so much so, that 50 years later I am still working with idea of variations. I can say that one of my first positive experiences was the series of 14 variations on a pictorial theme that I did in 1936 and that the Italian musician Malipiero later set to music.
The research that I have been carrying out for many years is into the
relationship between sound and colours, studied on a mathematical basis.
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Here are some chromatic variations on a rectangle, considered as a minimal
element, amongst the least evocative geometrical figures. For me, the rectangle is one of the minimal elements of my compositions, not only in the resolution of chromatic differences between sound, but also in all my painting. Of course I do not only use the rectangle; I also use curved lines. In fact there's hardly a picture of mine in which there are no circles and other curved lines, parabolas, ellipses, hyperboles and so on. Sometimes I also use triangles and squares but mostly as a counterbalance to the other figures that make up the composition."
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In the film on this theme, the Italian painter Fabrizio Clerici realised a very large painting for the film and for the exhibition "The eye of the Horus" that was held in Italy in 1990. I made a film for the Italian public television, RAI, in the series "The great exhibitions of the year" on the exhibition. For the anthology exhibition of Clerici, at the National gallery of Modern Art in Rome I made a short animated movie on Clerici's combinatorial drawings using music with variations by Wolfang Amadeus Mozart. These are the words of Clerici on the theme of the labyrinth: Ï think that for an artist who was first educated as an architect, began as an architect and wound up as a painter, for him the labyrinth is a natural step: it's the kind of architecture a painter is continually rebuilding in various ways. I return to it periodically. I think I drew my first labyrinth when I was a child, unthinkingly, without knowing it was a labyrinth; like a geometrical design. But an actual labyrinth, a construction in the form of a puzzle, a kind of trap, that happened much later, around 1945, just after the war. For a long time I was in doubt about what to place in the centre of these corridors, this amphitheatre, and then suddenly the solution came all by itself. Since the edifice was a labyrinth, the personage that must dominate the entire situation had to be the Minotaur; but why the Minotaur all alone in the centre of the arena? Because it is accusing someone and that someone can be no other but the mother which bore this monster.

The theme of the Minotaur accusing the mother was a much wider theme but it didn't stop there. I have played with this theme in many other versions, with or without the Minotaur. Sometimes a labyrinth is made up of letters of the alphabet, or of stones, stones that are created with a tendency towards gently, curving movement, to meandering….I have made several of them and I'll go on making them. We say meander for anything that is a function of the labyrinth…so I have decided to create the Meander, right on the river bed itself, a labyrinth as though the river itself were the Meander…that's the word and motif that I've given to this contorted, endless construction."

This film was made with Thomas Banchoff and Linda Henderson. The two American artists Harriet and David Brisson were involved in the film. David: "The reason why I am interested in visualisation of the materials is that a lot of it is hard to understand in a verbal sense. But when they were made into visual images then it was possible for me to understand very complicated ideas. And I felt that there were a lot of other people who could profit by making or by having the experience of these ideas expressed to them visually when they were not ready to understand these ideas in purely verbal or mathematical terms. A lot of these mathematical ideas are very interesting and beautiful when converted to 3 dimensional form, such as this projection which is not quite correct, of a 4 dimensional figure, the analogue of a ball, this is a more correct one, which gives you the same form. My personal interest is in visualising ideas so that I can understand them easier because I am a visual person."

In this paper I was interested in putting together some of the ideas that were the basis for the beginning and the making of the project Mathematics and Art : for the exhibitions, for the books, for the congresses and in particular for the films series. It is not so easy to find, in the last twenty-five years, a common and precise line of evolution of the arts that are more closely linked to science, primarily because this common line probably does not exist. In any case the interviews, the filming of the artists and their works, seen some years later, seem to me an interesting and stimulating heritage of how mathematics has had a cultural influence for many artists. This aspect can be of course of general interest for the arts tout-court. In this paper and in the video I have prepared I made a selection of the artists with whom I have been working in the last years. I have not included any mathematicians even if they are the most important motivation for all the project on mathematics and art. This was done first because I have written elsewhere about my collaborations with mathematicians on the project, and also because I was interested in choosing some of the artists with whom I was so lucky to work. With the publication of the volume The Visual Mind: Art and Mathematics I have tried, with the help of artists and mathematicians, to indicate the tendencies of evolution of the connections between art and mathematics in the last years. In the year 2002, a new book will be published The Visual Mind 2: Art and Mathematics, always by MIT Press. A new occasion to look to the future, to try to understand the new ideas for the new Millennium. In this paper my idea was to look at the recent past and remember some of my fellow-companions; without all of them my ambitious project could not have even been started.

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*, Proceedings of the congress, Amsterdam, North-Holland (1986)*- []
- , special issue "Leonardo", Pergamon Press, Oxford, vol. 25 n. 3/4 (1992)