Architecture or Music? Painting or Sculpture?

Which of these arts would have contributed first to nurture the field of Mathematics, or, in reverse, which one would have drawn technique and inspiration from this intellectual art, well before its rivals?

Without any doubt, many a pleasant, knowledgeable and enthusiastic discussion will attempt to solve this issue. However one cannot ignore that in the course of history, the development of classical art has always been related to the mathematical field.

The Renaissance era abounds with examples of art created along with the re-discovery of the polyhedrons and the creation of the theory of the linear perspective. More than a few paintings and engravings of the time illustrate scientific projections and testify to the symbiosis between painting and mathematics. The famous engraving of Dürer titled "Melancholy" (1513-1514) is rich in content and mathematical references. Luca Pacioli, the central character of the splendid painting of Jacopo de Barbari (museum of Naples), is a XV century mathematician and author of a famous work (Divine Porportione). The drawings of the polyedra which appear in that book were made by the hand of Leonardo da Vinci."

Differential Geometry, Topology and Algebra were introduced to the modern world in the XIX century as new mathematical objects were being created: Scherk’s surface, Klein’s bottle, Cayley’s cubic, hyperbolic plan. Those are just some of the names of objects familiar to mathematicians today that were discovered then.

Almost a century had to pass before artists started to incorporate again some of these intellectual preoccupations into their work: Salvador Dali represented an hypercube in his painting, Mauritz Escher used the richness of tessellation of the hyperbolic plan. They are among some of the brilliant pioneers of this new form of expression.

Many new mathematical objects made their appearance in the course of the second half of the XX century following innovative development in the fields of knots and minimal surfaces. Interested readers may want to consult the site www isama.org where they will discover, maybe with some degree of surprise, a great number of artists, painters, sculptors and architects, who found inspiration for their works in this new mathematical universe.

The artists exhibited reveal the astonishing diversity of these mathematical objects in original and unexpected forms of precise lines and perfect equilibrium and through the brilliance of their accomplishment and reputation, bring them to life in bright stone, sparkling metal, or in cheerful illustration of sharp and shimmering colors

Mathematical art today is contributing to renewal of visual art and there is no doubt that artists will continue to find inspiration in mathematical works. The initial purpose of this exhibition, the first in its kind perhaps in recent years, is to support this effort. By stressing the extent of this collaboration and making public the direct, attractive character of its contents, we hope to help reconcile the audience with the world of mathematic, whose image is often distorted by unfounded preconceived ideas.

This exhibit also looks forward to carry an element of curiosity meant to encourage young people and mathematicians alike to explore further the specifics of their work’s universe and passion, and to engage into new investigations.

One of the common features of many works shown here is the absence of immediate reference to familiar objects. Being works of mathematician-artists makes it hardly surprising. Many will be captivated by the evenness of their beauty. Perhaps other will prefer works that convey a more emotional experience where the existing environment is present and at the same time compellingly expend the creative aspect of the imagination.

Most participants in this exhibition are mathematicians, except for six of them, Philippe Charbonneau, draughtsman, Jean-François Colonna, data processing specialist, Patrice Jeener, engraver, Jean Constant, Irene Rousseau and Dick Termes painters.

The underlying mathematical inspiration of the works are sometimes very apparent. George Hart exhibits two original polyhedrons: traditional geometry and the theory of the group theory are clearly the background of these works.

Mathematicians identify the tiling of plane surface as tessellation. Mike Field, investigates this group theory and the rather recent theory of the singularities within dynamic systems. He dazzles us by the unique and captivating tessellation of his richly textured drawings.

Bill Casselman and its colleague David Austin illustrate another type of tessellation known as Penrose’s (aperiodic tiling) in a watery effect that surprises and delights the eye.

One will also find in this exhibit the December 2004 cover of “the Notes of the American Mathematical Society”. A very timely choice for the review to help brighten children’s eyes at the occasion of the Christmas season!

David Wright’s work shows the filling of the hyperbolic plan mentioned above with small discs of sharp color.

Other works have tapped another chapter of mathematics: modern geometry, the theory of surfaces and their extensions in spaces of several dimensions and their topological properties.

One will find a stunning sculpture by John Robinson, the core of an object that mathematicians call a node, more specifically the clover node (or Moebius strip). Mathematicians can turn over the sphere without folding it or tear it.

François Apéry and John Sullivan display sparkling metal sculptures of a completely original design and images that remind us in the privileged moments of the inversion of weaving techniques.

The concept of extremality, which is not without close relation with the concepts of stability, optimality and symmetry, is very present in some artists’ works as the engravings of Patrice Jeener where static contemplation (view of Chalancon, hypercube) are opposite to dynamic perspective (olive-trees and minimal surfaces), by nature much sharper and livelier.

A form of Geometry known as algebraic, bases its expression on the structural properties of the mathematical objects. It is represented here by the luminous and very elegant work of Philippe Charbonneau.

Thomas Banchoff and David Cervone have dedicated the last fifteen years of their activities to the fields of topology, geometry, visualization of objects and phenomena of multiple dimension and unusual forms. Their striking visualizations not only stimulates artistic creativity but also help mathematicians to better understand those spaces.

Jean-François Colonna has developed over time a great number of visualizations for physicists and mathematicians. His complex and painterly creations have become true works of art that have been exhibited here before.

To find the values of the unknown who cancels a polynomial, Bahman Kalantari has extended a method already employed by Newton in a simple case: the algorithm is sustained by the creation of fields that can be colored. The visual results are often elegant. This method has led him to develop a new and powerful method of artistic creation, still unknown in France, a playful and instructive technique where mathematics help art, which gratefully, comes to shoulder the mathematical process.

Nathaniel Friedman is a mathematician and a sculptor. Sculptors are doting on the sheen of the matter as much as the radiance of the form and its fractal dimension sometimes revealed by nature itself. In spite of an apparent similarity one can find in this work the delicate and alluring expression of an infinite variety of patterns.

The work of the painter Dick Termes is unique: he does not paint on canvas but on spheres to represent the totality of the space that surrounds these spheres. In doing so, he raises the curiosity of the mathematician as he instinctively reproduces processes used by topologists, which leads from the technical point of view, to call for several points of perspective. The artist is very prolific and inventive.

Jean Constant builds his creations starting from perfectly visible mathematical templates. These templates were provided to him by mathematician Richard Palais and are three-dimensional visualizations of traditional objects or surfaces obtained by Richard Palais itself in his work solving equations related to partial derivatives of the physico-mathematical theory of the solitons. Jean Constant enriches these templates superbly and creates an impressive series of genuine colorful paintings full of vitality.

Leaving this exhibit, the visitor will not fail to wonder about the reasons of the artistic activity which one meets here and the diversity of its manifestation. The is no doubt for me that the main motivations shared by all participants is the underlying need, the will to seize space and control it, which implies to comprehend it in its representation.

C.P. BRUTER

Paris XII University, Paris. France.