|Ivars Peterson's MathTrek|
February 7, 2000
For the second year in a row, a team assembled by mathematician Stan Wagon of Macalester College in St. Paul, Minn., participated in the international event. In its debut effort last year, the team carved a huge block of packed snow into a spectacular version of a minimal geometric structure known as the Costa surface (see Minimal Snow, March 8, 1999). This year, a new team created an award-winning snow sculpture out of a mathematical form called Enneper's surface.
Every summer, Wagon and several colleagues conduct a program called "Rocky Mountain Mathematica," teaching courses on how to use Mathematica software. Last July, John Bruning of Tropel Corp. in Fairport, N.Y., who was one of the students in the course, showed Wagon a postcard of an elegant, swooping sculpture crafted from wood by Robert Longhurst of Chestertown, N.Y. It was just the sort of design that Wagon had been looking for, and he immediately telephoned Longhurst, an experienced wood and stone carver, to see if he would be willing to sculpt in snow.
Wagon, Longhurst, and Dan Schwalbe, a Macalester colleague who had been on the previous year's team, considered a number of possible designs but quickly settled on the mathematical form on which Longhurst's sculpture had been based. Adding Macalester student Andy Cantrell to the team and Bruning as nonsculpting team manager and photographer, they submitted a proposal, which went in just before the Sept. 1 deadline. Wolfram Research again sponsored the team.
A minimal surface is one whose area becomes greater whenever it is distorted. At every point, such a surface either is flat or has a saddle shape. In effect, its curvature is like that of a potato chip, which typically starts out as a flat, thin slice of moist potato. As it dries during frying, the chip shrinks. Minimizing its area, it curls into a saddle shape.
The particular minimal surface of interest to Wagon and his team had been discovered in 1864 by Alfred Enneper (1830-1885), a mathematics professor at the University of Göttingen in Germany. The equation defining the surface looks very simple, but the highly symmetric, complicated surface that results is hard to visualize because it curls around and intersects itself.
The team's design effort involved the use of computer graphics to decide where to truncate what is mathematically an infinite surface and how to orient the result to create an aesthetically pleasing sculpture. "Truncating the surface just before the self-intersections leads to a very pleasing design," Wagon says. "It is very open and invites the viewer to explore it."
Even getting into the Breckenridge competition is tough, however. Only 17 designs were chosen from 24 proposals submitted from around the world. Wagon's team made the cut again and arrived in Breckenridge on Jan. 16 to work on a small practice block in preparation for the four-and-half-day snow-sculpting event.
Faced with strict time limits and the removal of large quantities of snow from a 10-foot by 10-foot by 12-foot block, the team used a variety of hand tools, from an ice-fishing auger to various chippers, axes, and shovels. Longhurst also brought along some specialized tools that he had devised to help with shaping the snow.
Details mattered. Cantrell spent a day and half crawling up and down the snow sculpture removing little bits of dirt from exposed surfaces and patching those spots with snow to give the sculpture a crisp, clean finish.
"It all went quite smoothly," Wagon notes. "This snow is very strong, and I think some of the other teams underestimate its strength." Moreover, a minimal surface itself has considerable strength, allowing it to be carved very thinly out of packed snow or ice.
Wagon named the result "Rhapsody in White," reflecting the sculpture's graceful curves, dramatic overhangs, and harmonious repeating pattern in the swooping clarinet solo that starts off George Gershwin's musical composition "Rhapsody in Blue."
The award-winning snow sculpture of Enneper's surface.
The team captured second place in the elite international competition, losing only to a team from Russia, which had created a soaring tribute to the new millennium. The sculpture of Ennepers surface also received two other prizes. It was voted the Artists' Choice Award by the participating snow sculptors and the Peoples Choice Award by the event spectators. The only prize "Rhapsody in White" didn't win was the Kid's Choice Award, which went to the Breckenridge team for its butterfly and rose.
"It is very satisfying to use a purely mathematical object and sculpt it in a way that looks beautiful," Wagon says.
About 10,000 people came to view the results on the final weekend, and many more showed up earlier in the week to see the snow sculptors at work.
A week after the competition, Enneper's surface was still standing in its pristine glory. "Our piece has no fine detail--no positive curvature anywhere," Wagon remarks. "There is no detail to melt out or to get overwhelmed by new snow."
Copyright 2000 by Ivars Peterson
Gray, A. 1998. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, Fla.: CRC Press.
Schwalbe, D., and S. Wagon. 1999. The Costa surface, in snow and in Mathematica. Mathematica in Education and Research 8(No. 2):56.
Additional information about this year's mathematical snow-sculpting effort, along with a variety of images, can be found at http://www.wolfram.com/news/snowsculpture2000.html.
Various definitions and renderings of Enneper's surface can be found at http://mathworld.wolfram.com/EnnepersMinimalSurface.html and http://www.iam.uni-bonn.de/grape/EXAMPLES/AMANDUS/enneper.html.
You can see examples of sculptures by Robert Longhurst at http://www.cs.berkeley.edu/~sequin/SCULPTS/LONGHURST/.
Comments are welcome. Please send messages to Ivars Peterson at email@example.com.