Ivars Peterson's MathTrek

March 8, 1999

# Minimal Snow

For a truly cool experience, there's nothing like transforming a 12-foot, 20-ton block of manufactured snow into a giant sculpture. That's the premise underlying the Breckenridge snow sculpture championships in Colorado, held annually in January.

This year, the international competition featured 15 four-member teams representing 10 countries. It also had a striking mathematical component. Noted mathematical sculptor Helaman Ferguson, joined by math professors Dan Schwalbe and Stan Wagon (see Riding on Square Wheels, July 13, 1998) and student Tamas Nemeth of Macalester College in St. Paul, Minn., labored intensively for 65 hours to carve a spectacular version of a minimal geometric structure known as the Costa surface. Wolfram Research sponsored the team, and Ferguson's wife, Claire, photographed the event.

The equations for this minimal surface were discovered in 1984 by the Brazilian mathematician Celso J. Costa. The figure's curvature resembles that of a potato chip, which typically starts out as a flat, thin slice of moist potato. As it dries out during frying, the chip shrinks. Minimizing its area, it curls into a saddle shape. Every little section of the Costa surface has this saddle configuration. Indeed, one can imagine the surface as the sum of an infinite number of saddles.

From certain angles, the Costa surface has the splendid elegance of a gracefully spinning dancer flinging out her full skirt so that it whirls parallel to the ground. Gentle waves undulate along the skirt's hem. Two holes pierce the skirt's lower surface and join to form a tunnel that sweeps upward. Another pair of holes, set a right angles to the first pair, lead from the top of the skirt downward into a second tunnel.

An image of the Costa surface generated using Mathematica.

Several years ago, Ferguson created a number of marble and bronze versions of the Costa surface. Carving one in snow, however, presented a host of new challenges.

In fact, when Wagon first approached Ferguson with the idea of submitting a proposal to the by-invitation-only competition, Ferguson was initially somewhat reluctant to get involved. "I do granite; I don't do snow," he replied.

Ferguson's interest, however, increased as he and Wagon began to discuss which of his many pieces would look best in snow. It forced Ferguson to think about what material properties snow and stone might have in common.

Stone, for example, can carry weight. It has compressive strength. But it can't be stretched very much, so it has significantly less tensile strength. You can make an arch out of stone. Snow has similar characteristics. An igloo is really a system of arches. And a minimal surface can also be thought of in terms of arches. In effect, every point seems to be the keystone of a cluster of arches.

What about sculpting snow into the Costa surface? Marble can be carved fairly thin. Could the same be done with compacted snow?

Ferguson wanted to test the feasibility of carving snow into the required shape, but there was a dearth of snow in the month of May in Maryland. He ended up retrieving several cubic feet of high-consistency snow (shaved ice) dumped by a Zamboni ice-smoothing machine outside a local ice rink.

On a warm afternoon, using a giant kitchen spoon and spatula, Ferguson carved a minimal-surface form with lots of holes. As it melted in the late afternoon sun, he watched its walls get thinner and thinner. The sculpture, however, maintained its basic structure.

"It seemed that a Costa form could be carved in snow, and without any special equipment," Ferguson says.

The Wagon-Ferguson proposal to create a Costa form, titled " Invisible Handshake," was accepted. The team was the only one in the competition with no prior snow-sculpting experience.

 To work out the technical details of carving snow into the Costa surface, Ferguson got permission from the manager of the ice arena where the Washington Capitals hockey team practices to get additional Zamboni snow at various times between May and the competition date in January. Those details included determining the types of tools to use, what sort of clothing to wear for the anticipated long days of snow carving, and how best to coordinate the actions of team members, who were mathematicians, not sculptors. Remarkably, it all came together in four and a half days of intense labor. By the end of the competition's third day, the rough Costa shape was visible, just in time for the arrival of a class of kindergarten children, who crawled and slid along the surface's intriguing tunnels. The team continued to shave the outside walls down until they were only 4 inches thick. The final day of the competition was warm, bringing with it the threat of melting. The Costa surface, however, held up nicely. A week after the event, the other sculptures had all lost detail, and one had even imploded. The only significant change in the "Invisible Handshake" was that its walls had become thinner still. "It was a real blast," Wagon says. "We will be back next year." The completed Costa surface snow sculpture. Photo courtesy of Stan Wagon.

References:

Ferguson, C. 1994. Helaman Ferguson: Mathematics in Stone and Bronze. Erie, Penn.: Meridian Creative Group.

Gray, A. 1998. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, Fla.: CRC Press.

Holden, C. 1999. Melting minimalist. Science 283(Feb. 5):787.

Peterson, I. 1998.The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. New York: W.H. Freeman.

______. 1996. The song in the stone. Science News 149(Feb. 17):110.

______. 1985. Three bites in a doughnut. Science News 127(March 16):168.

You can find photographs of Helaman Ferguson and his Wolfram Research team working on their snow sculpture at http://www.wolfram.com/news/snowsculpture.html.