By MARGARET WERTHEIM
June 22, 2004

SANTA CRUZ, Calif. - On the mantel of a quiet suburban home here stands a curious object resembling a small set of organ pipes nestled into a neat, white case. At first glance it does not seem possible that such a complex, curving form could have been folded from a single sheet of paper, and yet it was.

The construction is one of an astonishing collection of paper objects folded by Dr. David Huffman, a former professor of computer science at the University of California, Santa Cruz, and a pioneer in computational origami, an emerging field with an improbable name but surprisingly practical applications.

Dr. Huffman died in 1999, but on a recent afternoon his daughter Elise Huffman showed a visitor a sampling of her father's enigmatic models. In contrast to traditional origami, where all folds are straight, Dr. Huffman developed structures based around curved folds, many calling to mind seedpods and seashells. It is as if paper has been imbued with life.
In another innovative approach, Dr. Huffman explored structures composed of repeating three-dimensional units - chains of cubes and rhomboids, and complex tesselations of triangular, pentagonal and star-shaped blocks. From the outside, one model appears to be just a rolled-up sheet of paper, but looking down the tube reveals a miniature spiral staircase. All this has been achieved with no cuts or glue, the one classic origami rule that Dr. Huffman seemed inclined to obey.

Derived from the Japanese ori, to fold, and gami, paper, origami has come a long way from cute little birds and decorative boxes. Mathematicians and scientists like Dr. Huffman have begun mapping the laws that underlie folding, converting words and concepts into algebraic rules. Computational origami, also known as technical folding, or origami sekkei, draws on fields that include computational geometry, number theory, coding theory and linear algebra. This weekend, paper folders from around the nation will gather at the Fashion Institute of Technology in New York for the annual convention of Origami USA. At an adjacent conference on origami and education, Dr. Robert Lang, a leading computational origamist, will give a talk on mathematics and its application to origami design, including such real-world problems as folding airbags and space-based telescopes.

Dr. Lang, a laser physicist in Alamo, Calif., who trained at the California Institute of Technology, gave up that career 18 months ago to become a full-time folder. "Some people are peculiarly susceptible to the charms of origami," he said, "and somewhere along the way the ranks of the infected were joined by mathematicians." Dr. Lang is the author of a recent book on technical folding, "Origami Design Secrets: Mathematical Methods for an Ancient Art."

Most computational origamists are driven by sheer curiosity and the aesthetic pleasure of these structures, but their work is also finding application in fields like astronomy and protein folding, and even automobile safety. These days when Dr. Lang is not inventing new models using a specialized origami software package he has developed, he acts as an origami consultant. He has helped a German manufacturer design folding patterns for airbags and advised astronomers on how to fold up a huge flat-screen lens for a telescope based in space.

Dr. Lang has been studying Dr. Huffman's models and research notes, and is amazed at what he has found. Although Dr. Huffman is a legend in the tiny world of origami sekkei, few people have seen his work. During his life he published only one paper on the subject. Dr. Huffman worked on his foldings from the early 1970's, and over the years, said Dr. Lang, "he anticipated a great deal of what other people have since rediscovered or are only now discovering. At least half of what he did is unlike anything I've seen."

One of Dr. Huffman's main interests was to calculate precisely what structures could be folded to avoid putting strain on the paper. Through his mathematics, he was trying to understand "when you have multiple folds coming into a point, what is the relationship of the angles so the paper won't stretch or tear,'' said Dr. Michael Tanner, a former computer science colleague of Dr. Huffman who is now provost and vice chancellor for academic affairs at the University of Illinois in Chicago.

What fascinated him above all else, Dr. Tanner said, "was how the mathematics could become manifest in the paper. You'd think paper can't do that, but he'd say you just don't know paper well enough."

One of Dr. Huffman's discoveries was the critical "pi condition." This says that if you have a point, or vertex, surrounded by four creases and you want the form to fold flat, then opposite angles around the vertex must sum to 180 degrees - or using the measure that mathematicians prefer, to pi radians. Others have rediscovered that condition, Dr. Lang said, and it has now generalized for more than four creases. In this case, whatever the number of creases, all alternate angles must sum to pi. How and under what conditions things can fold flat is a major concern in computational origami.

Dr. Huffman's folding was a private activity. Professionally he worked in the field of coding and information theory. As a student at M.I.T. in the 1950's, he discovered a minimal way of encoding information known as Huffman Codes, which are now used to help compress MP3 music files and JPEG images. Dr. Peter G. Neumann of the Computer Science Laboratory at the SRI International (formerly the Stanford Research Institute) said that in everything Dr. Huffman did, he was obsessed with elegance and simplicity. "He had an ability to visualize problems and to see things that nobody had seen before," Dr. Neumann said.

Like Mr. Resch, Dr. Huffman seemed innately attracted to elegant forms. Before he took up paper folding, he was interested in what are called "minimal surfaces," the shapes that soap bubbles make. He carried this theme into origami, experimenting with ways that pleated patterns of straight folds can give rise to curving three-dimensional surfaces. Dr. Erik Demaine of M.I.T.'s Laboratory for Computer Science, who is now pursuing similar research, described Dr. Huffman's work in this area as "awesome."
Finally, Dr. Huffman moved into studying models in which the folds themselves were curved. "We know almost nothing about curved creases," said Dr. Demaine, who is using computer software to simulate the behavior of paper under the influence of curving folds. Much of Dr. Huffman's research was based on curves derived from conic sections, such as the hyperbola and the ellipse.

His marriage of aesthetics and science has grown into a field that goes well beyond paper. Dr. Tanner noted that his research is relevant to real-world problems where you want to know how sheets of material will behave under stress. Pressing sheet metal for car bodies is one example. "Understanding what's going to happen to the metal,'' which will stretch, "is related to the question of how far it is from the case of paper," which will not, Dr. Tanner said.

The mathematician G. H. Hardy wrote that "there is no permanent place in the world for ugly mathematics." Dr. Huffman, who gave concrete form to beautiful mathematical relations, would no doubt have agreed. In a talk he gave at U.C. Santa Cruz in 1979 to an audience of artists and scientists, he noted that it was rare for the two groups to communicate with one another.

"I don't claim to be an artist. I'm not even sure how to define art," he said. "But I find it natural that the elegant mathematical theorems associated with paper surfaces should lead to visual elegance as well."