December 31, 2020



Santa Fe Botanical Garden, Santa Fe, New Mexico, 2020.

Photo by I. Peterson

December 30, 2020

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December 26, 2020

Knot Divided in Snow

Mathematical forms are no longer unusual sights at the annual International Snow Sculpture Championships in Breckenridge, Colorado. For seven consecutive years, mathematician Stan Wagon assembled teams to carve huge blocks of snow into graceful geometric shapes.

In 2005, the team took on a particularly challenging, intricately convoluted form—one that incorporated the intriguing surface known as a Möbius strip.

Dubbed Knot Divided, this snow sculpture has an intricate form that incorporates a Möbius strip. Richard Seeley.

A Möbius strip (or band) is a remarkable surface having only one side and one edge. You can make a model of one by joining the two ends of a long strip of paper after giving one end a 180-degree (or half) twist.

Standard Möbius strip.

Cutting a Möbius strip down the middle along its length produces not two separate bands but a single, twisted loop that is twice as long as the original loop and has a 360-degree twist.

Intriguingly, you can also get a one-sided, one-edged band by joining the two ends of a long strip of paper after giving one end three half-twists instead of just one. If you were to lay a string along the strip's edge until the string's ends met and pulled the string tight, you would end up with a trefoil knot in the string. If you did this with a standard Möbius band, you wouldn't get a knot.

The trefoil (or overhand) knot is the simplest possible knot.

Splitting a triply twisted Möbius strip down the middle yields a single, twisted loop tied into a trefoil knot.

In general, all bands with an odd number of half-twists (and their mirror images) are, roughly speaking, one-sided surfaces. Bands with no twist or an even number of twists have two sides. Topologists generally apply the term "Möbius band" not only to the standard form (one half-twist) but also to the symmetric version (three half-twists) and anything else "homeomorphic" to the standard form.

For the 2005 snow-sculpting competition, computer scientist Carlo H. Séquin created an elegant design based on a split, triply twisted Möbius strip. He then generated three-dimensional, scale models of the structure to guide the snow carving.

Model for Knot Divided.

The 2005 snow-carving team consisted of Wagon, Séquin, John M. Sullivan, Dan Schwalbe, and Richard Seeley.

The challenge of carving this particular form became apparent early on. A practice block crashed, but the carvers did learn some valuable lessons from the experience. Then, it was time to start on the 10-foot-by-10-foot-by-12-foot snow block made available to the sculptors for the competition.

The snow carvers in the competition had 4½ days to complete their creations. The Minnesota team spent much of the first 2 days removing more than half of the 20 tons of snow in their block to obtain a rough approximation of a triply twisted band. In the remaining time, the team split the structure's three lobes lengthwise and sculpted the resulting bands into crescent-shaped cross sections.

Knot Divided in daylight, still standing. Richard Seeley

Amazingly, the fragile structure was still standing for the judging and subsequent public viewing. It didn't win a prize, but it was still a tribute to the power and beauty of mathematics. Interestingly, the winning entry, from a Tennessee team, also featured a mathematical shape—an intricate rendition of the spiral shell of a nautilus.

Event accounts and photos by Stan Wagon, Carlo Séquin, and John Sullivan.

Originally posted February 7, 2005

December 24, 2020

Turning a Snowball Inside Out

Turning a sphere inside out without allowing any sharp creases along the way is a tricky mathematical maneuver. Carving an intricate snow sculpture depicting a crucial step in this twisty transformation presents its own difficulties.

This was the challenge facing a team led by mathematician Stan Wagon in January 2004 at the 14th International Snow Sculpture Championships in Breckenridge, Colorado. In the end, after 5 days of arduous labor, the team managed to shape a 12-foot-high block of snow into a daring, prize-winning creation.

Turning a Snowball Inside Out: An award-winning snow sculpture. Carlo Séquin.

Turning a sphere inside out is harder than it sounds. In principle, a determined beachgoer can do it to a beach ball by deflating the ball, pulling the empty bag through its opening, and pumping up the ball again. The task for mathematicians is more difficult: The perfect sphere that they work with has no orifice, and the rules are different.

Imagine a ball made of a ghostly membrane that can stretch, bend, and pass through itself. The idea is to turn such a sphere inside out without puncturing, ripping, or creasing it.

You could try simply pushing the poles of a sphere toward each other, as if to make them pass through each other and change places. At some point, however, the distorted surface would develop a sharp kink, and that's not allowed, according to the mathematician's rules. Avoiding such a kink makes the task of exchanging a sphere's inner and outer surface—called an eversion—a challenging mathematical puzzle.

No one knew if a sphere eversion was even possible until 1959, when Stephen Smale proved a theorem that indirectly leads to the proposition that it could be done. However, Smale's step-by-step path for accomplishing a sphere eversion was so complicated that no one could visualize his procedure.

Gradually, visual answers to the sphere-eversion problem began to emerge, and mathematicians continued to look for increasingly simple ways of describing and displaying how the change occurs  (see "A History of Sphere Eversions").

In the 1970s, the task fell to French topologist Bernard Morin. Morin put together what can be thought of as a set of architectural plans for a sequence of three-dimensional constructions showing the essential steps in a sphere eversion. It was the halfway point in Morin's famous representation that ended up as a snow sculpture.

This was Wagon's sixth entry in the prestigious Breckenridge competition. He was joined this time by Dan Schwalbe, computer science student Alex Kozlowski, and mathematician John M. Sullivan. Computer scientist Carlo H. Séquin designed the sculpture and served as team representative.

In Séquin's design (below), a lattice structure of struts and bands represented a sphere's inside surface and solid material represented its outside surface. The resulting form elegantly captured both the tension and the swirled flow evident at the midpoint of a complex transformation.

Out of the 14 entries in this year's competition, Turning a Snowball Inside Out earned an honorable mention for "most ambitious" sculpture. That was a fine showing for an intricate, challenging piece of mathematics!

Event accounts and photos by Carlo Séquin (see also "Torus Eversion"), John Sullivan, and Stan Wagon.

Originally posted February 9, 2004

December 22, 2020

Whirled White Way

Since the late 1960s, Brent Collins has carved gracefully curvaceous sculptures out of wood. Born of his imagination, rendered in wire and wax, then painstakingly realized in wood in his Missouri workshop, each creation demanded many weeks of labor.

Music of the Spheres by Brent Collins. Photo © Phillip Geller.

Collins was not a mathematician, yet his intuition and aesthetic sense routinely led him to explore patterns and shapes that have an underlying mathematical logic. His creations had a strong sense of natural economy, like that of a soap film spanning a warped wire frame.

Indeed, many of his sculptures displayed the characteristic curves of a minimal surface, as observed in saddle-shaped soap films hanging from wire frameworks. They also surprise the viewer by dramatically changing in appearance from different viewpoints.

In 2003, Collins had a chance to realize one such daring, ethereal form on a massive scale, carved from a 20-ton block of hard, packed snow. It happened at the 13th International Snow Sculpture Championships in Breckenridge, Colorado, where Collins was a member of the USA-Minnesota team, led by mathematician Stan Wagon.

Whirled White Web: An award-winning, ill-fated snow sculpture. Courtesy of Carlo Séquin.

This was Wagon's fifth entry in the prestigious competition. He was joined this time by Collins, veteran participant Dan Schwalbe, computer science professor Carlo Séquin, and sculptor Steve Reinmuth.

The team chose a mathematically inspired design that might remind you of an elegant, fanciful propeller—an intricate whirl of twisted and stretched saddle shapes.

This particular design had arisen out of a longstanding collaboration between Collins and Séquin. In the early 1990s, Collins had been working on a series of sculptures in which each piece consisted of a punctured ring of intertwined saddle surfaces.

Hyperbolic Hexagon by Brent Collins. Its shape can be understood as a six-level Scherk minimal surface bent into a ring.

These sculptures can be described as different ways of warping pieces of an infinite mathematical shape called Scherk's second minimal surface, named for the mathematician Heinrich Ferdinand Scherk (1798–1885).

Central portion of Scherk's second minimal surface. Courtesy of Carlo Séquin.

Intrigued by Collins' artwork, Séquin had worked with his students to develop an innovative computer program (Scherk-Collins Sculpture Generator) for creating prototypes of such sculptures, allowing a sculptor to try out new designs without having to spend weeks making physical prototypes. Séquin could also send those designs to a three-dimensional prototyping machine, which would produce small models of the sculptures.

The chosen snow-sculpture design was a three-fold symmetrical torus of twisted and stretched saddle shapes—an intricate network of ribs and internal spaces suspended from a web of mutually interwoven double loops. Courtesy of Carlo Séquin.

In the Breckenridge competition, each four-member snow-sculpting team had 4½ days to shape a 12-foot-high block of snow into its final form. The first task facing Wagon and his team was to carve their block into a giant doughnut (torus), reflecting the sculpture's underlying fundamental geometry. Because all of the edges of the final sculpture lay on the surface of the torus, the sculptors had to make sure their torus was as accurate as possible.

Séquin had calculated that the finished sculpture would have about 280 feet of edges. That was "by far the most of our 5 years with similar shapes," Wagon noted. "I think also that we removed more snow for this piece than for any others."

Wagon ended up having to work overnight to help complete the sculpture in time for judging at 10 a.m. on Saturday morning, Feb. 1. In the end, "the piece had frozen nicely, and we could stand on its apex and lie in the central tunnel confident in its strength," Wagon said.

Dubbed Whirled White Web, the finished sculpture weighed about 4 tons, and it sat on two lobes about 10 inches wide and 6 feet long. See Séquin's illustrated report.

Whirled White Web (side view). Courtesy of Carlo Séquin.

After the judging, the team went out for lunch but soon received bad news. The sculpture had collapsed, probably because of very warm weather at the end of the sculpting period.

"Our failure to adapt by thickening the base was our biggest snow-sculpting error ever," Wagon admitted. "The sun, perhaps focused and reflected by the sculpture, just melted the support, and once our shape wobbles a bit, it is gone."

Nonetheless, the USA-Minnesota team captured second place in the competition for the daring beauty and grace of Whirled White Web. It was a fortunate outcome. If the event had ended an hour later, the judges would have seen just an interesting pile of icy rubble.

Originally posted February 10, 2003

December 20, 2020

A Snowy Twist

Carving a massive block of packed snow into an elegant sculpture presents all sorts of challenges. It's even tougher when the goal is an intricate mathematical shape with a gravity-defying heart.

A Twist in Time. Photo by Stan Wagon

In January 2002, a snow-sculpting team led by mathematician Stan Wagon and artist Bathsheba Grossman endured a demanding schedule and frigid temperatures to create such a form at the 12th annual Breckenridge International Snow Sculpture Championships in Colorado.

This was the fourth time that Wagon had entered a team in the contest. The 2002 squad included competition veterans Dan Schwalbe and John Bruning. Ski instructor Rob Nachtwey served as team photographer, videographer, and webmaster.

Snow-sculpting team members: John Bruning, Dan Schwalbe, Stan Wagon, Bathsheba Grossman, Rob Nachtwey.

The event represented Grossman's first venture into snow as a sculpting medium. In the late 1980s, when Grossman was an undergraduate studying mathematics, she had found herself wanting to step from thinking about geometric abstractions to working with physical objects.

Sculpture showed her the way, and she ended up studying art with sculptor Robert Engman (see "Triune Twists") at the University of Pennsylvania in Philadelphia in the early 1990s.

Eltanin by Bathsheba Grossman. On display at the Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia. Photo by I. Peterson.

After leaving university, Grossman began designing her sculptures on a computer, using computer-aided design software. She employed rapid-prototyping technology to convert a design into a physical model, built up layer by layer into the full three-dimensional figure.

Usually the model was made from wax, resin-impregnated starch, or plastic. To create a metal casting of the sculpture, she typically turned to the ancient lost-wax method, which destroys the model and mold to leave a metal sculpture.

"So the process moves, as it were, backward in time: from virtual idea to hand-finished metal," Grossman noted (see "Flame Alpha," "Soliton," "Intersecting Tetrahedra with a Twist," and "Chiralized Tetrahedron").

A bronze sculpture created by Grossman served as the model for the Wagon team's Breckenridge snow-sculpting effort. Called Spancel II, the original was only 5 inches tall.

Spancel II by Bathsheba Grossman.

This sculpture had an intriguing, rarely encountered symmetry, featuring 180-degree rotations around three axes and no reflections. "I can't think of a familiar object that has it," Grossman remarked.

Initially, Grossman wasn't confident that her sculpture—with its open structure and strongly curved edges and surfaces—could be scaled up and successfully fashioned out of snow. However, her teammates assured her that "snow is very strong."

The final version, which Grossman dubbed A Twist in Time, was 12 feet high. It remained standing even a week after the competition. "Our sculpture here was by far the hardest and riskiest of any," Wagon commented. See Wagon's event diary.

"The event was especially exciting for me because…I've done very little work at large scales," Grossman said. "Prototyping technology tends to keep me inside an 8-inch box, so it was very liberating and educational to see the design so huge."

The 2002 championship trophy went to a whimsical sculpture of a bass-playing musician, created by a team from Canada. A Twist in Time earned an honorable mention for "expressive impact" and tied for second in voting by spectators.

Grossman came away with a new appreciation of snow as an artistic medium. "As carving media go, I like it," she said. "It cuts reasonably easily with a sharp shovel, but it gives enough resistance to cut pure curves, unlike soft media such as soap or clay."

While making A Twist in Time, the snow sculptors noticed that Grossman's piece can be viewed as the first two stages of an infinitely nested form—a three-dimensional fractal. Digging deeper into this structure to unveil additional levels of intricacy may be a fruitful avenue of artistic exploration.

Originally posted February 18, 2002

December 18, 2020

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.)

Originally posted February 12, 2001

December 16, 2020

A Minimal Winter's Tale

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.)

Originally posted February 7, 2000

December 14, 2020

Minimal Snow

For a truly cool experience, there's nothing like transforming a 12-foot, 20-ton block of manufactured snow into a giant sculpture. That's the premise underlying the Breckenridge snow sculpture championships in Colorado, held annually in January.

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.

Originally posted March 8, 1999

December 12, 2020

Maps of Many Colors

A deceptively simple mathematical problem lurks within the brightly colored maps showing the nations of Europe or the patchwork of states in the United States. It's the sort of problem that might trouble frugal mapmakers who insist on painting their maps with as few colors as possible.

The question is whether four colors are always enough to fill in every conceivable map that can be drawn on a flat piece of paper so that no countries sharing a common boundary are the same color.

Several conditions turn this mapmakers' conundrum into a well-defined mathematical problem. 

A single shared point doesn't count as a shared border (above left). Otherwise, a map whose countries are arranged like the wedges of a pie would need as many colors as there are countries. Also, countries must be connected regions, unlike the map shown (above right); they can't have colonies scattered all over the map.

In addition, the unbounded area surrounding a cluster of countries and filling out the rest of the infinite sheet counts as a separate country that must also be colored.

The four-color problem fascinated and stumped professional and amateur mathematicians alike ever since it was first suggested in 1852 by Francis Guthrie, who had noticed he needed only four colors to fill in the counties on a map of England.

In a letter to his younger brother, Frederick, Guthrie asked whether it was true of any map. Frederick in turn described the problem to his mathematics instructor, Augustus De Morgan. The problem intrigued De Morgan, and he quickly appreciated that it wasn't quite as simple to solve as it appeared at first glance.

Toward the end of the nineteenth century, Lewis Carroll turned the map-coloring problem into a kind of game. A passage in one of his manuscripts gives the rules:

"A is to draw a fictitious map divided into counties.
B is to colour it (or rather mark the counties with names of colours) using as few colours as possible.
Two adjacent counties must have different colours.
A's object is to force B to use as many colours as possible. How many can he force B to use?"

Three colors are certainly not enough for every possible map. It's easy to come up with a simple map that requires four colors. In fact, Lewis Carroll (in his guise as mathematician Charles L. Dodgson) worked on the problem and figured out that four colors would be needed for the type of map shown below.

This map requires four different colors.

It's not at all obvious, however, whether a fifth color is needed for more complicated maps. Because the conjecture that four colors suffice hadn't yet been proved, Carroll didn't know with certainty whether the answer to his question (rules, above) was four or five.

To get a sense of how many colors are needed to color any map, you can experiment by drawing and coloring various configurations. It readily becomes apparent that the shape of the adjacent regions doesn't matter, but their arrangement does.

Here is one way to complete the map shown using four colors.

Indeed. when mathematicians work on the problem, they often draw graphs—networks of lines and points—to represent different configurations. That involves putting a a point (often called a vertex or node) at each country's capital and drawing a line (described as an edge) to connect the capitals of any pair of countries that has a common border.

Any map on the plane can be converted into a planar graph. In this example, involving a stylized map of the western United States, lines join points (nodes) within any two states that share a boundary.

Four colors turn out to be necessary in any situation in which a region has common borders with an odd number of neighboring regions.

A U.S. map can be colored so that only two states require a fourth color. In this case, the two states are California and Ohio (shown shaded).

A map showing the states of the United States features several such instances. For example, Nevada has borders with five states, and Kentucky with seven. Not counting the water but including islands, Rhode Island has boundaries with three states—Massachusetts, Connecticut, and New York. In each case, a fourth color is always necessary to fill in the map.

Because Bolivia and Paraguay are surrounded by an odd number  of countries, it takes four colors to complete a map of South America.

It is not at all obvious, however, whether a fifth color is needed for more complicated maps. It's not sufficient to show that you cannot have five countries that each shares a border with the other four. Even if a map contains no such set of countries, it might still require five colors. Mathematicians could prove, however, that no more than five colors are always sufficient to color any map.

The four-color problem defied solution for more than a century, despite years of map sketching, graph drawing, and logical argument.

In one such effort, Alfred Bray Kempe, a British lawyer and amateur mathematician, announced in 1879 that he had found a step-by-step map-coloring procedure guaranteeing that no more than four colors would be needed for any map.

Kempe's argument was convincing, but 11 years later, someone found a loophole. There were a few, special cases that his method did not cover.

In 1976, Kenneth Appel and Wolfgang Haken announced they had finally proved the four-color theorem. It was the kind of news that mathematicians would greet with champagne toasts. However, there was a shock awaiting anyone curious about how the four-color theorem had been conquered.

Hundreds of pages long, the proof was truly intimidating. Even more dismaying to many mathematicians was that, for the first time, a computer had been used as a sophisticated accountant to enumerate and verify a large number of facts needed for the proof. There was no simple, direct line of argument leading to a readily verifiable conclusion.

The computer allowed Haken and Appel to analyze a large collection possible cases, a collection shown by mathematical analysis to be sufficient to prove the theorem.

The task took about 1,200 hours on one of the fastest computers available at the time; it would have taken practically forever by hand. It also meant the proof could not be verified without the aid of a computer. Furthermore, Appel and Haken had tried various strategies in computer experiments to perfect several ideas that were essential for the proof.

For a long time, some mathematicians remained unsatisfied with this state of affairs. "Even if the theorem is true, as it almost certainly is, it still awaits a simple proof that does not require hours of computer time," Martin Gardner remarked in The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Games, Puzzles, and Word Plays.

Appel himself noted, "It has troubled our profession that a problem that can be understood by a school child has yet to be solved in a way that better illuminates the reason that only four colors are needed for planar maps."

In 1996, Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas presented a new proof of the four-color theorem. Robertson and his colleagues had started out to verify the Haken-Appel proof but soon gave up. They decided it would be more profitable to work out their own proof, following a somewhat streamlined version of the approach taken by Haken and Appel.

Instead of checking the 1,936 graphs (later cut to 1,476 configurations) required by the original proof, the team reduced the number to 633 special cases and proceeded from there. Still, the new proof contained computer steps that can't be verified by humans.

"However, from a practical point of view, the chance of a computer error that appears consistently in exactly the same way on all runs of our programs on all the computers under all the operating systems that our programs run on is infinitesimally small compared to the chance of human error during the same amount of case-checking," the mathematicians insisted.

"Apart from this hypothetical possibility of a computer consistently giving an incorrect answer, the rest of our proof can be verified in the same way as traditional mathematical proofs," they added.

Mathematicians have continued working on various other aspects of the map-coloring problem. They found, for example, that not every map drawn on a surface requires the full complement of colors. A map consisting of just one country requires, of course, only one color. The four-color theorem merely sets an upper bound.

A map with intersecting straight lines as borders requires only two colors.

One question was whether there exists a quick, efficient way to tell whether a given map requires the full complement of colors. For maps on a flat surface, the answer appears to be "no." For maps on a surface shaped like a doughnut (torus), a double torus, and similar shapes, however, the answer is "yes."

Mathematicians also proved some time ago that any map drawn on a torus requires at most seven colors, on a double torus, eight colors, and so on.

As shown in this torus knitted by Carolyn Yackel, seven colors suffice.

It's nice to see there's still life in the old problem of coloring a map. The search for a simple, incisive proof of the four color theorem goes on, suggesting new puzzles and leading to novel mathematical techniques that turn out to be useful in applied mathematics and computer science.