The organizers of the Breckenridge snow sculpture championships in Colorado must be getting used to having a mathematical element in their annual competition.
For the second year in a row, in 2000, a team assembled by mathematician Stan Wagon participated in the international event. In its debut effort in 1999, 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").
This time, a new team created an award-winning snow sculpture out of a mathematical form called Enneper's surface.
For many summers, Wagon and several colleagues conducted a program in the Breckenridge area called "Rocky Mountain Mathematica," where they taught courses on how to use Mathematica software. In the summer of 1999, one of the participants was John Bruning. Knowing Wagon's interest in sculpted mathematical forms, Bruning showed him a postcard of an elegant, swooping sculpture crafted from wood by Robert Longhurst.
Arabesque XXIX by Robert Longhurst.
It was just the sort of design that Wagon had been looking for, and he immediately contacted Longhurst, an experienced wood and stone carver, to see if he would be willing to sculpt in snow.
Trained as an architect, Longhurst had been carving since 1976. Longhurst's patiently crafted, sinuous, abstract sculptures typically emerged from his imagination rather than from mathematics, even though they often bore an uncanny resemblance to the types of shapes that soap films or minimal surfaces can display.
"Curvilinear [forms], whether they fall into the categories of art, mathematics, or design, have always held a fascination for me beyond that of straight lines," Longhurst said.
Loop II by Robert Longhurst.
Wagon, Longhurst, and Dan Schwalbe, who had been on the previous year's team, considered a number of possible designs for a snow sculpture but soon settled on the form depicted on the Longhurst postcard—a piece that Longhurst had titled in his usual understated fashion Arabesque XXIX.
In this particular case, Longhurst had taken his inspiration from mathematics. He had learned about this intriguingly curved shape from mathematician and sculptor Nat Friedman (see "Points of View" and "Nat Friedman (1938-2020)"). Friedman had seen a video depicting computer-generated variants of Enneper's surface and snapped photos of frames from the video. Familiar with Longhurst's penchant for curvaceous forms, he sent them to Longhurst.
Fascinated by the shapes, Longhurst ended up fashioning from bubinga wood a graceful, 12-inch-high model of one depiction that he found particularly appealing.
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. 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 is hard to visualize because it curls around and, unlike the Costa surface, 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 said. "It is very open and invites the viewer to explore it."
A computer-generated version of the central portion of Enneper's surface just before (above) and just after (below) its lobes intersect.
Adding Macalester student Andy Cantrell to the team and enlisting Bruning as non-sculpting team manager and photographer, Wagon, Longhurst, and Schwalbe submitted a proposal just before the Sept. 1 deadline.
Even getting into the Breckenridge competition was tough, however. Only 17 designs were chosen from 24 proposals submitted from around the world. Wagon's team made the cut and arrived in Breckenridge on Jan. 16, 2000, to work on a small practice block in preparation for the 4½-day snow-sculpting event.
Dan Schwalbe's design, generated and rendered using Mathematica software, for Rhapsody in White, based on Longhurst's Arabesque XXIX version of a variant of the Enneper minimal surface.
Having learned from the 1999 effort, the team came prepared with a variety of hand tools, from the ice-fishing auger that had been so helpful the previous year to various chippers, axes, and shovels. Longhurst also brought along some specialized tools that he had devised to help with shaping the snow.
Robert Longhurst working on the snow sculpture at night.
Details mattered. Cantrell spent a day and half toward the end 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 noted. "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.
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.
The award-winning snow sculpture of Enneper's surface, Rhapsody in White.
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 Enneper’s surface also received two other prizes. It was voted the Artists' Choice Award by the participating snow sculptors and the People’s Choice Award by 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.
The snow-sculpting team of (from left to right) John Bruning, Andy Cantrell, Dan Schwalbe, Stan Wagon, and Robert Longhurst posed with the completed, award-winning Rhapsody in White.
"It is very satisfying to use a purely mathematical object and sculpt it in a way that looks beautiful," Wagon said.
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 remarked. "There is no detail to melt out or to get overwhelmed by new snow."
In his award acceptance speech, Wagon noted that even for mathematicians, true understanding can be obtained only by interacting with a geometric form in a truly three-dimensional way. "This is what snow allows us to do," he added. "In a very short period of time and with a minimum of tools, we can sculpt a complicated shape and so know more about it."
"It's a glorious opportunity and tremendous fun," Wagon declared. (Wolfram announcement.)
See also "Minimal Snow," "White Narcissus," "A Snowy Twist." "Whirled White Way," "Turning a Snowball Inside Out," and "Knot Divided in Snow."
Originally posted February 7, 2000
No comments:
Post a Comment