Forging links between mathematics and art.

To many people, art and mathematics appear to have very little in common. The seemingly rigid rules and algorithms of mathematics apparently lie far removed from the spontaneity and passion associated with art. However, a small but growing number of artists find inspiration in mathematical form, and a few mathematicians delve into art to appreciate and understand better the patterns and relationships they discover in the course of their mathematical investigations.

To prove the remarkable fruitfulness of such links, more than 100 mathematicians, artists and educators gathered last week at the Art and Mathematics Conference (AM '92), held in Albany, N.Y. Organized by mathematician and sculptor Nat Friedman of the State University of New York at Albany, the meeting represented his attempt to find people with whom he could share his deep interest in visualizing mathematics, whether in geometry, sculpture, computer art or architecture.

Attempts to visualize such mind-bending mathematical transformations as turning a sphere inside out without introducing a sharp crease at any point during the operation demonstrate how mathematics and computer graphics can lead to valuable insights that are potentially useful to both scientists and artists.

In 1959, when Stephen Smale, a mathematician at the University of California, Berkeley, first proved this particular operation possible, no one could readily visualize how it happens. By gradually simplifying the steps involved in turning a sphere inside out, mathematicians eventually found ways of picturing the entire process (SN: 5/13/89, p.299).

Francois Apery of the University of Upper Alsace in Mulhouse, France, has now captured the essence of the process, known as sphere eversion, in a surprisingly simple model. Imagine a globe marked with an equator and lines of longtitude, or meridians, that connect the poles. At the start of the sphere eversion, as one pole moves toward the other, the meridians twist sideways more and more.

When the poles meet, the meridians twist so much that they flip like a windblown umbrella over the coincedent poles to double up into a smaller spherical shape having an open end marked by a ring showing the new position of the original sphere's equator (see illustration). The twisting continues untill the equator closes up into a point and the meridians overlap and cross each other. At this stage, the sphere's outside becomes its inside, completing the eversion.

Apery speculates that the first half of this sphere eversion may serve as a mathematical model of the way an embryo, starting out as a ball of cells, can pull in part of its outer wall to form a cavity among its dividing, differentiating cells. Biologists call the process gastrulation.