Minimalist mathematics

June 18, 1995|By Frank D. Roylance | Frank D. Roylance,Sun Staff Writer

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.

Baltimore Sun Articles
Please note the green-lined linked article text has been applied commercially without any involvement from our newsroom editors, reporters or any other editorial staff.