In 1999, 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 and Macalester College student Tamas Nemeth, labored intensively for 65 hours to carve a spectacular version of a minimal geometric structure known as the Costa surface.
Invisible Handshake is a Costa minimal surface carved from packed snow (above). Below: The snow-sculpting team of (from left to right) Stan Wagon, Tamas Nemeth, Helaman Ferguson, and Dan Schwalbe stands in front of the block of snow that would, after 4½ days, become Invisible Handshake. Courtesy of Stan Wagon.
The equations for this minimal surface were discovered in 1983 by 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, you 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.
A Mathematica-generated image of the central portion of the Costa surface.
Several years before, Ferguson had created a number of Styrofoam, marble, granite, and bronze versions of the Costa surface. Carving one in snow, however, presented a host of new challenges.
A Costa surface carved by Helaman Ferguson out of Styrofoam (above) hints at the appearance of one made from snow. Below: Ferguson created this bronze version of the Costa surface as a memorial to Alfred Gray.
When Wagon first approached Ferguson with the idea of submitting a proposal to the by-invitation-only competition, Ferguson was initially skeptical and reluctant to get involved.
"I do granite; I don't do snow," Ferguson insisted. "My idea of mathematical sculpture combines timeless conceptual content with timeless-as-possible materials, such as billion-year-old granite."
"Snow?" he added. "Here today, evaporated tomorrow."
Ferguson's interest increased, however, as he and Wagon began to discuss which of Ferguson's many sculptures would look best in snow. The discussion forced Ferguson to think about what snow and stone have in common.
Stone can carry weight. It has compressive strength. On the other hand, it can't be stretched very much; it has significantly less tensile strength than compressive strength. Hence, it's possible to make an arch out of stone but not an unsupported ceiling.
Snow has similar characteristics. An igloo is a system of arches, and a minimal surface can be thought of in terms of arches.
Technically every point of a minimal surface has what mathematicians call a negative Gaussian curvature—where the surface curves up in one direction and down in a perpendicular direction, like the seat of a saddle (below).
In effect, any point of a curved minimal surface is the keystone of a cluster of arches.
So, if the object to be sculpted were a Costa surface, could compacted snow be carved as thinly as marble can, to create a seemingly delicate sculpture that still manages to support itself?
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, where Ferguson lived. He ended up retrieving several cubic feet of high-consistency snow—actually, 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 several tunnels. As it melted in the late afternoon sun, he watched its walls get thinner and thinner. The snow sculpture maintained its basic structure, however.
"It seemed that a Costa form could be carved in snow, and without any special equipment," Ferguson said.
Wagon and Ferguson formulated a proposal to create a snow sculpture based on the Costa surface with the title Invisible Handshake. The title refers to the fact that the intertwining tunnels of the Costa surface roughly correspond to the orientations of two hands about to grasp each other. The Costa surface itself represents the empty space between the hands at the instant before the consummation of a handshake.
This interpretation gives the mathematical surface a distinctly human touch. Two people can each extend a hand through an opening in the surface as if to shake hands, and only the thin material of the sculpture keeps their hands from grasping each other in a true handshake. Each can touch the figure's walls but not the other hand.
The Ferguson-Wagon proposal to create Invisible Handshake was accepted. The team that Wagon assembled to accomplish the feat 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 an ice arena to get additional Zamboni snow at various times between summer and the competition date in January.
The technical details included determining the types of tools to use, what sort of clothing to wear for the anticipated long days of snow carving under potentially blizzard conditions, and how best to coordinate the work of team members who were mathematicians, not sculptors.
At work on Invisible Handshake. Claire Ferguson.
The anticipated conditions would be tough. At nearly10,000 feet above sea level the mountain air is thin, and biting blizzard winds are a constant threat.
The packed, manufactured snow is hard (five times more dense than natural snow and about half as dense as ice), and the contestants have a strictly limited amount of time available to remove the excess material. They can use only hand tools and are not allowed to incorporate any sort of support structure.
Ferguson also could not rely on a computer to help him shape the snow sculpture, as he did with his stone and bronze creations. "Fortunately, I had learned all I needed to complete this piece from my prior experience carving the various other Costa surfaces," Ferguson remarked.
Ferguson's assistants, though inexperienced in sculpting, were all mathematicians and understood the language of algorithms and differential geometry.
"Our common mathematical language helped tremendously in my communicating my prior experience in carving the other Costas," Ferguson said.
Remarkably, it all came together in 4½ days of intense labor. On the first day, the team removed enough snow to produce a rough sphere. The second day's labor was devoted to drilling the figure's two main tunnels.
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.
Kindergarten children lined up (above) to crawl through the snow-sculpted Costa surface (below). Claire Ferguson (above); Stan Wagon (below).
The team continued to shave the outside walls until they were only 4 inches thick. Altogether, about fourteen tons of snow were removed from the original block to reveal the ultimate shape.
Invisible Handshake at night, nearing completion. Claire Ferguson.
"It was a real thrill to learn the rudiments of sculpting as we progressed," Wagon observed. "Our piece was ugly at first, slowly became less ugly, and then emerged all at once in all its curvaceous glory."
The choice of tools—a four-foot logging saw, a large wood chisel, masonry tools, and small hatchets and shovels—proved critical.
An auger designed for drilling holes through three feet of ice for fishing turned out to be particularly effective for removing large quantities of material during the early stages. Its razor-sharp blades readily chewed through the snow, performing much the same function that diamond drill bits do in carving stone.
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 of them had even imploded. The only significant change in Invisible Handshake was that its walls had become thinner still.
In its final guise, in the gray light of a wintry day, the giant, sleekly curved figure reclining on its icy couch appeared ready for slumber.
Invisible Handshake. Stan Wagon.
From some angles, the snow sculpture resembled a strangely contorted bell, a piece of surreal plumbing or an ancient urn worn smooth by time. Its smooth contours and curious system of tunnels echoed the work of sculptor Henry Moore.
"It was a real blast," Wagon said in describing the snow-sculpting effort. "We will be back next year." See Stan Wagon's report "Snow Sculpting with Mathematics." (Wolfram announcement.)
Ferguson was used to creating sculptures from granite, marble, and other materials in the expectation that they would last for millennia. His creation in snow, in contrast, eventually melted away, remaining only in memory.
Nonetheless, Ferguson prized his newfound knowledge of the surprising structural strength of negative Gaussian curvature as a sculptural fabric, even when the material is as flimsy as snow. The power of a minimal surface—both structurally and aesthetically—resides in this curvature.
See also "A Minimal Winter's Tale," "White Narcissus," "A Snowy Twist." "Whirled White Way," "Turning a Snowball Inside Out," and "Knot Divided in Snow."
Originally posted March 8, 1999
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