# Ivars Peterson's MathTrek

January 15, 2007

### Art of the Tetrahedron, Revisited

The tetrahedron is the simplest of all polyhedra. 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.

For more than 30 years, Arthur Silverman of New Orleans has created artworks arising out of explorations of this angular form. Many of his sculptures are on display in public spaces and various buildings in New Orleans and other cities from Florida to California.

Arthur Silverman stands with one of his tetrahedron-based sculptures. Both bent columns are identical but look quite different from different viewpoints.
Photos by I. Peterson.

"The tetrahedron is very exciting visually," Silverman says. "It's very difficult to anticipate what you are going to see. Every step around a piece gives you a different view."

We're accustomed to thinking about orientation in space in terms of three perpendicular axes. A tetrahedron has no right angles. So, a tetrahedral structure jars us out of spatial complacency. It has so few faces, its aspect changes abruptly as you move around to view it from different angles.

This colorful trio of tetrahedra can be found on Poydras Street in New Orleans. It's one of the few Silverman sculptures that's painted.

Until age 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 were anything he might really want to do, then he ought to do it right away, before the chance slips away.

The encounter changed Silverman's life. He returned to interests that had captured his attention when he was a teenager. He had visited museums to gaze at statues, and he had tried his hand at carving wood. Later, when studying medicine at Tulane University, he had met a sculpture teacher who had invited him to classes and taught him to see, in the artistic sense.

Located in front of City Hall in New Orleans, this sculpture was designed by removing tetrahedral forms from a rectangular block.

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

"When I first encountered tetrahedra, I was immediately fascinated by the notion of using these forms as basic building blocks for three-dimensional designs," Silverman recalls.

It's sometimes hard to tell that Silverman's sculptures are based on tetrahedra.

Over the years, Silverman has manipulated the tetrahedron's familiar shape into myriad forms that are often virtually unrecognizable as tetrahedra. He has elongated it, stretching several edges to create a slim stainless-steel tower 60 feet high, then twinned it with an identical tower to create an elegant pair of structures, which soar in formation into the sky. With water flowing down its sides, this sculpture stands as Silverman's signature piece in the middle of a plaza fountain in New Orleans.

Arthur Silverman's "Echo" features a pair of elongated tetrahedra, each seemingly balanced on one edge. The sculpture is 60 feet tall and rests on a foundation that extends 20 feet into the ground.

Silverman has joined tetrahedra to form an aluminum cascade. He has stacked them symmetrically. He has also sliced them. A vast, interior wall of the Equitable Center in New Orleans is covered with aluminum tiles based on such cross sections. A slowly changing light washes over the surface, highlighting different facets.

Aluminum tiles based on cross sections of a tetrahedron create a striking wall design.

Silverman has divided tetrahedra, then rejoined them in various ways. He has looked at what's left when tetrahedra are cut of a column or from inside a cube. He has stretched a tetrahedron and turned it inside out. He has stood the shape on edge, and he has balanced it on a vertex.

"I find that the unique geometric relations intrinsic to the tetrahedron persist in the final sculpture, notwithstanding all the manipulations I carry out," Silverman notes. At the same time, "photographs do not do these works justice," he contends. "One must actually see, feel, and walk around these works in order to experience them in their totality."

Silverman created an outdoor menorah for Temple Sinai in New Orleans.

Silverman fabricated nearly all his pieces in his studio by welding together metal plates. Altogether he produced more than 200 sculptures based on the tetrahedron, from large-scale outside works to diminutive studio models.

Changing the orientation of this sculpture, about 20 inches high, gives observers startlingly different views.

Silverman's outdoor sculptures, including his tall twin towers, survived the onslaught of Hurricane Katrina in 2005. Water filled his studio to a depth of about 1 foot, leaving distinct waterlines on several sculptures that had sat on the floor. Some smaller sculptures were knocked off their pedestals, resting in brackish water for days. Interestingly, several of these pieces developed a deep, rich patina—the weathered look of survivors.

Partially immersed in floodwaters for days, this sculpture displays the resulting waterline.

Now 82, Silverman has stopped creating new works. He has turned his studio into a gallery, where visitors can wander about and ponder the tetrahedron's amazing versatility, as unveiled by an artist's imagination.

Silverman has turned his sculpture studio into a gallery, showcasing his many artworks.

References:

Examples of Arthur Silverman's sculptures can be seen at http://www.barnettfineart.com/ and http://www.artnet.com/artist/15571/arthur-silverman.html. Additional artworks can be seen at http://www.lemieuxgalleries.com/artist_silverman.html, http://www.anewleafgallery.com/artists/Silv.lasso, and http://www.bilhenrygallery.com/silverman/.

Peterson, I. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. New York: Wiley. See http://ivarspeterson.googlepages.com/art.

______. 2000. Four corners, four faces. Muse 3(April):26.

______. 1999. Art of the tetrahedron. MAA Online (Nov. 8).