**PROJETS / STATUT ESTHETIQUE DE L'ART TECHNOLOGIQUE / BIBLIOTHEQUE DU
COLLOQUE**

Par Michele Emmer

CREATIVITY: A BRIDGE BETWEEN ART AND MATHEMATICS

1. INTRODUCTION

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).

2. CREATIVITY IN MATHEMATICS.

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.»

3. VISUAL MATHEMATICS AND ART

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.

REFERENCES

- 1. M. Bill, Die Mathematische Denkweise in der Kunst unserer Zeit, Werk, n. 3, 1949; reprinted in english in (Emmer, 1993).
- 2. M. Emmer, Dimensions, video, 27 min. Film 7 International, Rome, 1984. Quotation from the interview with the artist David Brisson.
- 3. M. Emmer, ed., L'occhio di Horus: arte e matematica, Istituto Enciclopedia Italiana, Roma, 1990.
- 4. M. Emmer, Bolle di sapone: un viaggio tra arte, scienza e fantasia, La Nuova Italia, Firenze, 1991.
- 5. M. Emmer, ed., Visual Mathematics, Leonardo, special issue, 25 n. 3/4 (1992).
- 6. M. Emmer, ed., The Visual Mind: Art and Mathematics, The MIT Press, Cambridge, USA, 1993.
- 7. C. Ferguson Helaman Ferguson: Mathematics in Stone and Bronze, Meridian Creative Group, Erie, Penn, 1994.
- 8. H. Ferguson, S. Ferguson, & A. Gray, Costa Minimal Surfaces: D4 Symmetry. SCulpture by Virtual Image Projection, Part I & II, in G. Darvas, D. Nagy & M. Pardavi-Horvath, eds., Symmetry: Natural and Artificial, Symmetry: Culture and Science, vol. 6, n. 2 (1995) pp. 202-209.
- 9. P. Feyerabend, Creatività: fondamento delle scienze e delle arti o vacua diceria, in P. Feyerabend & C. Thomas, eds., Arte e scienza, Armando editore, Rome, 1989.
- 10. M. Field & M. Golubitsky, Symmetry in Chaos: a Search for Pattern in Mathematics, Art and Nature, Oxford University Press, Oxford,1992.
- 11. H. Franke, Refractions of Science into Art, in Peitgen, 1986.
- 12. C. Gunn, W. Thurston, D. Epstein, D. Maxwell, S. Levy, J. Sullivan et al., Not Knot , The Geometry Center, University of Minnesota, Minneapolis, video, 20 min., 1991.
- 13. M. Kline, Mathematics in Western Culture, Oxford University Press, Oxford, 1953.
- 14. S. Levy, D. Maxwell & T. Munzner, Inside out, The Geometry Center, University of Minnesota, Minneapolis, video, 22 min., 1994.
- 15. F. Le Lionnais, ed., Les grands courants de la pensée mathématique, A. Blanchard, Paris, 1962.
- 16. B.B. Mandelbrot, Fractals and an Art for the Sake of Science, ACM Siggraph 89, Leonardo, Supplement Issue (1989), pp. 21-24.
- 18. H.-O. Peitgen, P.H. Richter, The Beauty of Fractals, Springer verlag, Berlin, 1986.
- 19. R. Penrose, The Emperor's New Mind: concerning Computers, Minds, and the Laws of Physics , Oxford University Press, Oxford, 1989.

[Accueil Colloque] - [Présentation] - [Programme] - [Participants] - [Interventions]

Pour tout renseignement concernant le colloque, contactez Jacques Mandelbrojt - <jmandelbrojt@cyberworkers.com>

Document mis à jour : Wednesday, February 16, 2011 09:46 PM

Copyright ISAST/LEONARDO © 1997-98

pour toute re-publication de cette page, merci de contacter Annick Bureaud <bureaud@altern.org>

pour les problèmes concernant le site, Carmen Tanase<carmen_tanase@hotmail.com>

Hébergement : www.xcom.fr