White Narcissus

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.

An original design by Robert Longhurst, carved in wood.

Although Longhurst's abstract sculptures bear an uncanny resemblance to mathematical forms known as minimal surfaces, they typically emerge from his imagination rather than from mathematics.

To create a sculpture, Longhurst typically began with a conceptual sketch, which he then translated into a wax, wire, or aluminum-foil model. Once he was satisfied with the result, he selected an appropriate type of wood, which had been carefully kiln-dried and seasoned to give it the stability necessary for sculpting delicate passages and ornate curves.

Longhurst's main tool was a die grinder whose carbide bit spun at 22,000 revolutions per minute, enabling him to cut away wood with great precision on interior as well as exterior surfaces.

Some time 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 (below).

Arabesque XXIX by Robert Longhurst.

In 2000, Longhurst joined 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 championships in Colorado (see "A Minimal Winter's Tale").

For the 2001 championships, Longhurst was again a member of the team assembled by mathematician Stan Wagon. Longhurst and Wagon were joined by veterans Dan Schwalbe and John Bruning. Matthias Weber served as team photographer and manager.

Wagon and his coworkers chose to carve a complex, wraparound shape designed by Longhurst (below).

Robert Longhurst's sketch of the proposed snow-sculpture design.

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").

Longhurst's design was not based on a known minimal surface, however. In October 2000, Wagon showed Weber images of Longhurst's creation and asked if he could come up with equations to describe the surface.

"The sculpture looked quite complicated," said 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 said.

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 remarked. See "Snow Sculpting with Mathematica."

Computer-generated image of the minimal-surface cousin of Longhurst's sculpture, created by Matthias Weber.

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 said. 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 4½ 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.

White Narcissus. Photo by Stan Wagon.

Wagon and his team faced stiff competition from the 13 other teams at the 2001 championships and failed to win a prize. Nonetheless, "we had great fun constructing it," Wagon reported. "The weather was super, the team worked well together, and we still love the piece!" (Wolfram announcement.)

See also "Minimal Snow," "A Minimal Winter's Tale," "A Snowy Twist." "Whirled White Way," "Turning a Snowball Inside Out," and "Knot Divided in Snow."

Originally posted February 12, 2001