Par Michele Emmer



In the town of Baltimore, USA, it is located the Maryland Science Center, a very modern and completely interactive museum of sciences. At the third floor there is a permanent exhibition dedicated to mathematics entitled Beyond Numbers. In the exhibition you can find images and objects related to research results in mathematics obtained in the last two or three years. The exhibition is divided in three tematic sections, of course connected among them. In Playing with Abstractions the visitor has the opportunity to play with interactive computer software, to construct modern mathematical surfaces and other topological models, to make experiments. It is possibile to see the images of two recent mathematical videos Not Knot (Gunn et al, 1991) and Inside out (Levy et al, 1994) both of them realized at the Geometry Center of the University di Minnesota in Minneapolis; the first video is on knot theory while the second one is dedicated to the problem of how a sphere can be turned inside out without tears or creases.
Mathematical images have a great importance in the exhibition. The public can not only look to the mathematical videos, to visual computer software, to mathematical models; a large space is dedicated to the works of contemporary artists who have used in different ways visual mathematical ideas. The works of Heleman Ferguson are of particular interest. Ferguson is an artist-mathematician. Ferguson is an artist who uses mathematics to create his work, or he is a mathematician who uses art to create his work. In the foreword to the volume Ferguson: Mathematics in Stone and Bronze (C. Ferguson, 1994) Richard Walker wrote: «The physical reality of his sculpture is mathematics with an intuitive insight as even the descriptive terminology conjures up enigmatic possibilities....The universal language of mathematics and visual art reach beyond written works or verbal explanations, and the art of Ferguson aptly demonstrates the power of this universality when used for the clarity and sensual beauty possible in the two disciplines. »
Described in art language, Ferguson's sculptures are abstract images but in spite of his use of exacting mathematical equations and content, his sculptures are not rigid geometric configurations nor are they mathematic illustrations. «The artist chooses to use these mathematical ideas to create objects that are warm and humanistic.»
It is not a unique example of interest of an artist for modern mathematics; this new relationship between artists and mathematicians have been very profound in the last years. See for example the catalogue of the exhibition The Eye of Horus: Art and Mathematics (Emmer, 1990).


No doubt that in the last years a revival of interest for creativity in mathematics has taken place; mainly for the possible connections with the artistic creativity. The principal motivation for this new interest is the large diffusion of computers with high graphics facilities. This very large diffusion has strongly raised intuition and creativity in that part of mathematical research connected to the possibility of visualizing not only known phenomena but to make visible the insivible (Emmer, 1984). Roger Penrose in his essay The Emperor's New Mind (Penrose, 1989) wrote: «It is a feeling not uncommon amongst artists, that in their greatest works they are revealing eternal truths which have some kind of prior etherial existence... but the case for believing in some kind of etherial, eternal existence, at least for the more profound mathematical concepts, is a good deal stronger than in those other cases. There is a compelling uniqueness and universality in such mathematical ideas which seems to be of quite a different order from that which one could expect in the arts. »
Mathematical ideas are not subjects to fashions, they do not vary in centuries; a theorem proved by Euclid is valid today and it will be valid for centuries; it will never be over. How many other human activities have this caracteristic of universility, of immortality? Mathematics as the true art? «Of course the creative process must produce a work that has design, harmony and beauty. These qualities too are present in mathematical creations» wrote Morris Kline in his essay Mathematics in Western Culture (Kline, 1953).
There is no doubt that there are some peculiarities in considering the question of creativity in mathematics and in trying to compare it with the artistic one. Mathematicians state on the one hand that the real universal art is mathematics, on the other hand that they are the only ones able to understand this truth; so only the participants to the scientific community can take part in this « banquet of gods» (Le Lionnais, 1962). It seems that the only conclusion is that trying to analyze relationships between mathematical and artistic creativity is a loss of time. In my opinion it is indeed possible to provide elements to the investigation starting from the suggestions of Feyerabend: not to look for general formulas that will give the illusion to find a final solution to the problem (Feyerabend, 1989).
In any case it is possible to discuss the new possibilities opened for the relationships betweent art and mathematics by the new technologies. It is possible to focus on the main directions along which to obtain results of interest for each fields. On the one hand the mathematicians have obtained in the visual investigation of scientific problems images that have arose the interest not only of the scientific community but of a large audience, artists in particular; on the other hand artists, feeling themselves excluded from the possibility of using in full the new visual tools, have asked for cooperation mathematicians and experts in computer graphics. In this way from this new collaboration new interesting results have been recently obtained.
However, the great possibility that has been opened with the use of computer graphics of seing mathematical objects of which it was not even possible to imagine the enormous graphic complexity, has opened wide spaces to artistic creativity. Mathematicians very soon became aware of this not secondary aspect of their researches. Mandelbrot wrote:« I believe I can safely say that fractal geometry' contribution to science and art is absolutely original.»


Mandelbrot has repeatedly stressed the importance of fractals in art (Mandelbrot, 1989) :
«We can say that fractal geometry has given rise to a new category of art, close to the idea of art for art's sake: art for science's (and mathematics') sake. The origine of fractal art lies in the recognition that very simple mathematical formulas, seemingly dry and dust, can actually be very rich, so to speak, in huge graphic capacity. The artist's taste can intervene only in the choice of formulas, their arrangement and visual rendering. By bringing the eye and the hand into mathematics, we not only rediscovered the ancient beauty, which remains intact, but also discovered a hidden, extraordinary new beauty.»
The example of fractals is not at all the only case of intrusion by mathematicians in the arts' field. To give an idea of the growing importance of the visual aspects, to point out the possible connections between some of the most recent mathematical research and the work of artists using visual techniques influenced by mathematical ideas see the volume The Visual Mind: Art and Mathematics (Emmer, 1992 & 1993) .
In the last years, after a long period of oblivion, mathematics and mathematicians have again a relative importance for artists. Talent and creativity of mathematicians, assisted by graphic tools unimaginable untill a few years ago, have opened new fields to mathematical research and gave the chance to catch the great graphic complexity in very simple problems and formulas.
Impossible to imagine, untill a few years ago, a book like Symmetry in Chaos: a Search for Pattern in Mathematics, Art and Nature. The authors, the mathematicians Michael Field e Martin Golubitsky, wrote in the introduction (Field et al, 1992):
«In our mathematics research, we study how symmetry and dynamics coexist. This study has led to the pictures of symmetric chaos that we present throughout this book. Indeed, we have two purposes in writing this book: to present these pictures and to present the ideas of symmetry and chaos - as they are used by mathematicians - that are needed to understand how these pictures are formed.... One of our goals for this book is to present the pictures of symmetric chaos because we find them beautiful, but we also want to present the ideas that are needed to produce these computer generated pictures.» The authors recall the volume of Peitgen and Richter The Beauty of Fractals (Peitgen et al,1986) and add: «It is worth noting that the images we present have a different character from those found in fractal art. While fractal pictures have the sense of avant garde abstract modernism or surrealism, our typically have the feel of classical design.»
Who could have imagined a few years ago that such declarations could have been found in the introduction of a volume written by two mathematicians?
It is possible to distinguish between two types of images: those resulting from the solution of interesting mathematical problems; in this case mathematicians are able to give a precise explanation both of the scientific problem and of the method with which the pictures have been produced. Other pictures are obtained by the visualization of known phenomena or of algorthims, numerical methods, mappings for which the scientific problem is not so clear, in which it is not possible to show the solution. If in the last cases pictures are in general less important by the mathematical point of view, by the figurative point of view all kind of images can be of interest for artists, regardless their scientific interest. Mathematicians tend to privilege those pictures resulting from a well defined scientific problem while for artists a combinatorial or casual method of variation of the various graphic elements of a scientific picture can be more interesting.
Probably the creativity and capability of mathematicians have put to artists the problem of an active confrontation with the ideas of mathematicians. «And since mathematics possesses these fundamental elements and puts them into meaningful relationships, it follows that such facts can be represented or transformed into images....which have an unquestionably aesthetic effect.» wrote in 1949 the famous artist Max Bill, who died in 1994 (Bill, 1949). Do his words apply to the images created by mathematicians in recent years? This is the opinion of Herbert W. Franke when he says that in future centuries, the opinions held by art critics will be very different from those of our contemporary experts: Franke, 1986) «The painters and sculptors esteemed today will nearly have been forgotten, and istead the appearance of electronic media will be hailed as the most significant turn in the history of art.»
But even more interesting the new electronic images created by mathematicians have generated new works of art which use more traditional tools. A very interesting example is a recent work of Heleman Ferguson. A few words on the mathematical background. The theory of minimal surfaces emerged two centuries ago with the mathematicians Euler and Lagrange who began thinking about mathematical language to describe stretched surfaces, surface tensions and soap films. The first to describe visually very intriguing minimal surfaces made of soap films was the Belgium physicist Joseph Plateau (Emmer, 1991) over a hundred years ago. Not all minimal surfaces can be obtained with soap films; for this to be possible, it is essential that certain topological conditions be respected. For instance, soap films tend to fill holes, so in general a minimal surface with holes cannot be obtained from soap water. Untill 1982, mathematicians knew only three minimal surfaces of the class called minimal completed immersed surfaces, meaning that the surfaces extend to infinity and never self interesect. These three surfaces are the plane, the catenary and the helicoid, and a portion of all three can be obtained with soap films. None of the three has a handle; more precisely, their topological type is zero. For almost two hundred years mathematicians have wondered whether there exist minimal complete immersed surfaces with at least one handle; that is, whose topological type is greater that zero.
Two american mathematicians, David Hoffman and William Meeks III, using the equations found by the Brazilian mathematician Costa, were able to demonstrate the existence of a class of minimal surfaces whose topological type is fairly high: minimal surfaces with holes. The two mathematicians have used graphic techniques to prove the result; it was one of the first interesting example of a theorem proved using graphic techniques. The shapes of the images successively obtained by Hoffman and Meeks aroused interest extending beyond the purely mathematical. Hoffmann himself remarked to an interviewer that «the cooperation between art and science produced something significant in both areas.» Several artists were fascinated by the forms of the new discovered minimal surfaces and they used them as models to create sculptures using different materials.
Among others Helaman Ferguson had the idea to use his technique of computer assisted sculpting to the Costa surfaces (Ferguson, 1995). Using his computer assisted technology Ferguson, thank to the help of the mathematician Alfred Gray, created several aesthetic objects. In particular for the Maryland Science Center exhibition Beyond Numbers a ten foot diameter Costa minimal surface. Ferguson has described in a recent paper the technique to obtain a bronze, alluminium, fiberglass and Carrara marble Costa surface (Ferguson, 1995).
The example of the work of Ferguson is really interesting because the connections among the creativity of the mathematician, of the artist and the technology are all linked together and it is really hard to try to separate them. We are probably facing a possible revolution in the . relationships between mathematics and art, in which the creativity of artists and mathematicians will have the possibility of a very profound cooperation; perhaps a new Renaissance?
The images that mathematicians have created (from fractals to Costa surfaces) exist only as equations and algorithms that are seen on the video screen of a graphic computer. Even if in the case of fractals strong connections with the real world have emerged, in the case of the Costa surfaces it is really difficult to talk of possible connections with Nature. In the words of Ferguson it is the marble sculpture that represents the return to Nature of the virtual minimal surfaces.
Are still valid the words of Max Bill, in 1949? (Bill, 1949): «It must not be supposed that an art based on the principles of mathematics 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
everchanging relations, rhythms and proportions of abstract forms.» Mathematics, art and computer graphics are probably meant to influence themselves more and more in the next years. Surely we will face new results in mathematics, a real progress in the mathematical sciences. For the arts the future is more difficult to imagine, but we will surely have new interesting results from the possible cooperations between the creativity of mathematicians and artists.


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