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Minimalist mathematics


A credit line was inadvertently omitted Sunday from computer images accompanying an article on minimal surfaces displayed in an exhibit at the Maryland Science Center. The images were created by James T. Hoffman of the University of Massachusetts.

The Sun regrets the error.

Samba dancers' headdresses, soap-film architecture and alien "flowers" that swallow your hands.

These aren't things most of us wrestled with in math class. They are, however, part of the bizarre scenery you encounter when you enter the world of Costa surfaces, among the latest discoveries in a cranny of modern mathematics called "minimal surfaces."

On computer screens, or modeled in fiberglass, a Costa surface looks like some sort of mutant flower, with flaring petals and interior tunnels that fall away unexpectedly to the outside.

Starting today at the Maryland Science Center, visitors will get a chance to experience several mind-bending models of Costas. They are part of a gallery of mathematical brain stretchers that make up a new permanent exhibit called "Beyond Numbers," sponsored by the National Science Foundation and IBM.

Two years in development, the $1.3 million exhibit includes three dozen displays -- 15 of them interactive -- targeted for visitors 10 and older. They were developed in collaboration with George Washington University's math department.

A duplicate version will tour science centers nationally beginning in February.

"This exhibit lets you feel intrigued and comfortable with mathematics, and we fully expect that, for a lot of our visitors, that's a new experience," said D. D. Hilke, exhibits director at the science center.

"The first time I held it [the Costa model], I knew I'd never held anything like that before," said Ms. Hilke, who directed fabrication of several Costas for the exhibit. "I expected to see my hand come out, and it disappeared. It's like a fun house."

It also represents a landmark discovery in a very serious realm of mathematical research that has practical applications in the real world.

"A Costa surface is an example of what mathematicians call minimal surfaces . . . simply a mathematician's idealization of soap film," said Dr. David Hoffman of the Mathematical Science Research Institute at the University of California at Berkeley.

Dr. Hoffman is a co-discoverer of Costa surfaces, with Dr. William H. Meeks III and computer graphics expert James T. Hoffman, both of the University of Massachusetts, and Dr. Celso Costa of the Universidade Federal Fluminense in Brazil. The Hoffmans are not related.

Soap film -- the same stuff you blow bubbles with -- intrigues architects and engineers because it represents "a surface of the absolute least possible area given the boundaries you've found," Dr. David Hoffman said. It always shrinks to minimize itself across the wire loop or any other gap created for it.

Maximum benefits

Minimal surfaces are important because builders who model their structures on them reap the benefits of minimal materials, minimal weights and, frequently, minimal costs.

The graceful tent roofs at the new Denver International Airport and the Columbus Center in Baltimore are examples of the increasing applications that architects are finding for minimal surfaces.

Eb Zeidler, a partner in Zeidler, Roberts Architects, the Columbus Center's designer, said the tent roof covers a broad area with fewer supports than a conventional roof. It also "sheds light and has a very pleasing performance and appearance at the same cost. You couldn't do that with a solid roof."

Understanding minimal surfaces that form naturally at a microscopic level may help scientists and engineers develop new high-tech, high-performance materials. Dr. Hoffman is currently turning his expertise to federally sponsored research into the microstructure of substances called compound polymers.

But en route to understanding and exploiting minimal surfaces, mathematicians need to be able to describe them in the language of mathematical equations. It is a job that starts out easy, but quickly gets very murky.

On a flat, circular wire, soap film will form into a flat disk that can be described by simple geometric equations. But bend the wire a bit, and the soap film changes into a complex undulating surface. It's still a minimal surface, Dr. Hoffman said, but "to describe it mathematically is . . . something that requires serious analysis."

Strict criteria

Mathematicians have strict criteria for a true minimal surface. In addition to having the least possible surface area for its mathematical bounds, it must extend to infinity without looping around and intersecting itself. It also must divide space in two.

The oldest-known surface meeting these criteria is the plane, understood since the ancient Greeks. Imagine you are an ant on a sheet of paper infinitely wide and long. "As you travel on it, you never come to an edge," Dr. Hoffman said. You could walk on the top or bottom, but could never get from one to the other.

Eighteenth-century mathematicians discovered two more -- the catenoid and the helicoid.

The catenoid is a bit like an hourglass, or a cylinder pinched at the center. It forms naturally when soap film connects two parallel wire rings. But in the ideal, its top and bottom walls continue to widen and flare until they extend endlessly out into space. An ant on a catenoid lamp shade could explore the inside, or the outside, but never get from one to the other.

The helicoid is a sort of double spiral, like a circular parking garage in which the "up" ramp and "down" ramp are stacked but separate from one another. In the ideal, however, the ramps' floors and ceilings would extend outward to an infinite horizon. An ant on a helicoid could explore the floor and ceiling of his own spiraling ramp, but never reach the twin ramp just above his ceiling and just below his floor.

By the early 1980s, mathematicians had equations they thought might describe parts of a fourth minimal surface. But the problem was so complex they could not visualize the entire surface, or prove that it did not intersect itself.

Costa's puzzle

That's the puzzle Celso Costa was wrestling with as a graduate student in 1983. He had devised equations that hinted at a new surface that had top and bottom portions that looked like catenoids. But he couldn't see how they might be joined in the middle.

His flash of insight came at the movies in Rio de Janeiro, Brazil, during a break from his work. The film was a documentary about Samba dancers' preparations for Rio's annual Carnival. In a 1987 article, Dr. Hoffman wrote that the Brazilian mathematician "saw a dancer with an outlandish headdress that was made to look like two crows -- one head up, the other head down, with their XTC wings meeting in an expanding circle in the middle."

That pointed toward a possible solution -- two catenoids joined at a planelike surface that extends into space. But the details of the center, where the planes and catenoids merged, were unclear.

Alerted to the promise in Dr. Costa's equations, Dr. Hoffman and Dr. Meeks obtained a copy of his thesis and began working to convert the equations to a graphic computer display. They teamed up with UMass computer graphics expert James Hoffman to piece together the surface until they could see whether any part of its center intersected with itself.

'People thought I was nuts

It was a pioneering use of computer graphics in mathematical research. Instead of using the graphics to illustrate what they'd discovered on paper, they were using the computer's imaging power to explore an unknown realm and lead them to a discovery.

"It's hard to believe today, but the idea to . . . do the integration and write the [computer] graphics to actually look at it [the Costa] was not in the arsenal of typical mathematicians. People thought I was nuts, but I fooled them," Dr. David Hoffman said.

Inventing the graphics programs they needed as they went along, they pursued the clues that emerged from the graphics, and gradually brought the unknown interior of the first Costa surface to light.

As it took shape, they were able to rotate, explore and inspect it in 3-D to be sure the surface didn't intersect itself. It didn't.

"We just knew it was right," he said. In weeks, they were able to use what they'd learned from the graphics to analyze Dr. Costa's equations and prove mathematically that they'd discovered the first new class of minimal surface since the 1700s.

A few months later, Dr. Hoffman and his colleagues were able to show their Costa was just the first of an apparently infinite family of Costas. They come with more and more holes, handles and rims. They can be stacked like fever-dream cups and saucers, or reflected like sculptures set on a mirrored table.

All of them capture the eye and engage the imagination. And each divides space in two; an ant exploring the rims, handles and holes of his side of the Costa can never reach those on the reverse side, just beneath his feet.

"This is an alien object," said Ms. Hilke of the models she has seen and held in preparing the science center's exhibit. "It came out of someone's mind when they broke the bounds of this world."

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