Ivars Peterson's MathTrek

November 8, 1999

# Art of the Tetrahedron

Some might consider the tetrahedron a rather humble geometric figure.

Any four points in space that are not all on the same plane mark the corners of four triangles. The triangles in turn are the faces of a tetrahedron. It's the simplest of all polyhedra--solids bounded by polygons.

If each face is an equilateral triangle, the result is a regular tetrahedron, one of the five Platonic solids.

To sculptor Arthur Silverman of New Orleans, however, tetrahedra are very special. He has been investigating variations of tetrahedral forms for more than 20 years in sculptures displayed in public spaces in New Orleans and other cities from Florida to California.

"The tetrahedron is very exciting visually," Silverman insists. "It's very difficult to anticipate what you are going to see."

Until the age of 50, Silverman had been a highly successful surgeon, practicing medicine with considerable enthusiasm and skill. Then he encountered an ailing colleague near death, who advised Silverman that if there was anything he really might want to do, then he ought to do it right away, before the chance slips away.

That encounter changed Silverman's life. He returned to interests that had captured his attention when he was a teenager, had visited museums to gaze at statues, and had tried carving wood. Studying medicine at Tulane University, he met a sculpture teacher, who invited him to classes and taught him how to look.

In the midst of these early explorations, Silverman discovered the wonders of the tetrahedron, a form to which he returned with a passion many years later.

So, what can you do with (or to) a tetrahedron?

You can elongate a tetrahedron, stretching several edges to create a slim, stainless-steel tower, 60 feet high, then pair it with an identical tower to a produce an elegant structure that seems to soar in formation into the sky. Such a sculpture stands in the middle of a plaza fountain in New Orleans.

You can join tetrahedra together to form an aluminum cascade. The water wall at the Lee County Sports Complex in Ft. Myers, Fla., spotlights such geometric playfulness and activity. Or you can stack them symmetrically to produce a solemn memorial to Martin Luther King Jr. in Baton Rouge, La.

You can slice tetrahedra. A vast, interior wall at the Equitable Center in New Orleans is covered with aluminum tiles based on such a cross section.

You can divide tetrahedra, then rejoin them in various ways. You can look at what's left when tetrahedra are cut out of a column or from inside a cube. You can elongate a tetrahedron and turn it inside out. You can stand it on edge or balance it on a vertex. The possibilities seem unlimited.

Silverman has produced more than 300 sculptures based on the tetrahedron. "When I get an idea, I play with it as long as I can," he admits. The sculptor fabricates nearly all his pieces in his studio by welding together metal plates.

This is art that conveys no political, social, or historical message, Silverman remarks. "The sculpture is strictly a visual experience."

One of the most intriguing of Silverman's creations is an ensemble of sculptures he calls "Attitudes," which are spread across a grassy area at the Elysian Fields Sculpture Park in New Orleans. The pieces are identical, but they have startlingly different orientations. When you walk around and look at them, "it's hard to believe they are all the same structure," Silverman remarks. "Every time you move, you see something different."

To Nat Friedman, a mathematician and sculptor at the State University of New York in Albany, Silverman's creation represents a hypersculpture--a way of seeing a three-dimensional form from many different viewpoints at once.

To see a two-dimensional painting in its full glory, you have to step away from it in the third dimension, Friedman says. To see a three-dimensional sculpture in its totality, you need a way to slip into the fourth dimension. Friedman calls this process hyperseeing. A hypersculpture, which consists of a set of several related sculptures, is one way to approximate that experience.

References:

Friedman, N. 1997. Reunification and hyperseeing. Ylem Newsletter (November/December).

Gardner, M. 1961. The five Platonic solids. In The 2nd Scientific American Book of Mathematical Puzzles and Diversions. New York: Simon and Schuster.

______. 1965. Tetrahedrons in nature and architecture, and puzzles involving this simplest polyhedron. Scientific American 212(February):112.

Silverman, A. 1997. Tetrahedral variations. Ylem Newsletter (November/December).

Arthur Silverman's artworks are featured at http://www.lemieuxgalleries.com/artist_silverman.html,
http://www.bilhenrygallery.com/silverman/,
and http://www.anewleafgallery.com/artists/Silv.lasso.

Various mathematical properties of the tetrahedron are described at http://mathworld.wolfram.com/Tetrahedron.html.