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Math Trek

White Narcissus

4:14pm, February 8, 2001
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The elegant, swooping forms carved out of wood by sculptor Robert Longhurst often resemble gracefully curved soap films that span twisted loops of wire dipped into soapy water. Alhough these abstract sculptures bear an uncanny resemblance to mathematical forms known as minimal surfaces, they emerge from Longhurst's imagination rather than from mathematics.

"Curvilinear works, whether they fall into the categories of art, architecture, or design, have always held a fascination for me beyond that of straight lines," Longhurst says.

Trained as an architect, Longhurst has been carving wood and stone since 1976. His studio is on a farm in the Adirondack Mountains, near Chestertown, N.Y.

Longhurst typically begins with a conceptual sketch, which he then translates into a wax, wire, or aluminum-foil model. Once he is satisfied with the result, he selects an appropriate type of wood, which has been carefully kiln-dried and seasoned to give it the stability necessary for sculpting delicate passages and ornate curves. Longhurst's main tool is a die grinder whose carbide bit spins at 22,000 revolutions per minute, enabling him to cut away wood with great precision on interior as well as exterior surfaces.

Several years ago, computer-generated images of a true minimal surface--called an Enneper surface of degree two--inspired Longhurst to create a sculpture based on that form. Last year, Longhurst led a team in carving a huge block of packed snow into a spectacular version of the Enneper minimal surface. The snow sculpture won second place in the elite Breckenridge snow sculpture championship in Colorado (see A Minimal Winter's Tale, February 5, 2000).

In this year's championship, held in January, Longhurst was again a member of the team assembled by mathematician Stan Wagon of Macalester College in St. Paul, Minn. Longhurst and Wagon were joined by mathematician Dan Schwalbe and John Bruning of Tropel Corp. in Fairmont, N.Y. Matthias Weber of the Mathematical Sciences Research Institute in Berkeley, Calif., served as team photographer and manager. Wolfram Research, maker of Mathematica software, sponsored the team.

Wagon and his coworkers chose to carve a complex, wraparound shape designed by Longhurst. Previous experience in the Breckenridge competition had vividly demonstrated the remarkable strength and stability of intricate, thinly carved snow structures that have the saddle-like contours of a minimal surface (see Minimal Snow, March 6, 1999). Mathematically, a minimal surface is one whose area becomes greater if it is distorted. At every point, such a surface either is flat or has a saddle shape.

Longhurst's design was not based on a known minimal surface, however. Last October, Wagon showed Weber photographs of Longhurst's creation and asked if he could come up with equations to describe the surface.

"The sculpture looked quite complicated," says Weber, an expert on minimal surfaces. "There was no known minimal surface like it." From a single photograph, it was even difficult to discern how many boundary curves enclosed the shape. Fortunately, additional views showed that the surface incorporated a pair of straight lines--a feature that could be useful in looking for equations that characterize the surface.

"By making some assumptions, I derived equations for a minimal surface that the Longhurst surface must satisfy if we were sure that it was a minimal surface," Weber says.

To his surprise, Weber found that there was essentially just one equation that would work. "Usually, by looking at some minimal surface shape, there are many possible equations with many parameters, and one has to choose the equation and the parameter carefully so that the minimal surface looked at is matched by the one produced by the equation," Weber remarks.

A second surprise occurred when Weber generated a computer image of the surface and discovered that it looked very similar to Longhurst's sculpture. "It could have looked quite different," Weber says. Once he had the equation, Weber could generate all sorts of images of the minimal-surface cousin of Longhurst's carving.

At the Breckenridge snow sculpture competition in January, Wagon and his team spent four and a half days using only hand tools to carve Longhurst's design out of a 20-ton, 12-foot-high block of packed snow. They named the intricately curved snow sculpture White Narcissus.

Wagon and his team faced stiff competition from the 13 other teams at this year's championship and failed to win a prize. Nonetheless, "we had great fun constructing it," Wagon says. "The weather was super, the team worked well together, and we still love the piece!"

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