Diana ben-Aaron / PORTFOLIO / Lingua Franca / article killed when the magazine changed editors


FOLDING THE UNIVERSE


Past Human Evolution, through Meteorites, not quite as far as Rocks and Minerals, there's a discreetly marked door in the Museum of Natural History in New York City. Behind it lies a circular room containing a brontosaurus, a reindeer, a cockroach, a Statue of Liberty, a rose, and hundreds of other things sculpted from paper by practitioners of origami, the Japanese art of paperfolding. On the shelves of the room are over a thousand books detailing patterns for everything from water birds to stellated icosahedra (twenty-pointed stars), as well as folding instructions for paper airplanes, napkins, dollar bills and Paris metro tickets. The room is the Origami Center of America, founded in 1980 by Lillian Oppenheimer, a philanthropist and New School instructor who is credited by members of the society with popularizing origami in the United States, and Michael Shall, a high school teacher who became her disciple. It owes its location to the late Alice Gray, an entomologist on the museum's staff and an origami aficionado who specialized in folding paper insects, which she wore as jewelry.

For most of its history, paper folding has been the province of children and eccentrics, who folded the same patterns again and again, but lately the Center has witnessed what amounts to an international origami revolution. In recent years the folding techniques have become far more sophisticated: cranes and sailboats have been superseded by zebras with accurately folded stripes and twenty-one-piece tyrannosaurus rex skeletons. (Only the most rigorous modern incarnation of origami requires models to be folded from a single square sheet of paper: both classical Japanese and Moorish origami use many shapes of paper and allow cutting and multiple sheets.) More importantly, at least for those who want paper folding regarded as a legitimate occupation for grownups, origami has become quite modish among mathematicians, who use formulas to analyze origami techniques and, conversely, use the folding process to analyze mathematical problems.

Jeannine Mosely, for instance, a former engineering lecturer at MIT and UC-Berkeley who was an early innovator of polyhedral folds, has found a way to divide the edge of the origami square into arbitrary nths: thirds, sevenths, seventeenths. A Japanese folder named H. Abe has discovered a particularly elegant way of trisecting angles, and others have discovered ways to construct the cube root of 2. In the last twenty years abstract geometric models have proliferated in origami, beginning with the five regular Platonic solids and proceeding through the various stellations and hybrid polyhedra to ever more elaborate faceted shapes. The laws of crystallography appear to persist in the world of paper folding, with odd-numbered symmetries like pentagons and heptagons being difficult to construct, but not impossible.

One productive line of inquiry has been so-called recursive folding, which was discussed by the Japanese and American mathematicians who attended the Origami Center's June convention. Chris Palmer, a former design researcher who has traveled to Spain to study Moorish tiling patterns, demonstrated a spiral fold that twists the center of a square into a flat rosette and then pinches up successive smaller flat rosettes on top of that. The rosettes could get smaller without limit if the paper were infinitely large and thin and the folder infinitely dextrous. Michel Mendes of the University of Bordeaux has performed the deceptively simple experiment of folding a strip of paper in half longways, and then in half again and again and again. The sequence of mountain and valley creases produced after each new fold (M, MVV, MVVVMMV and so on) turns out to resemble known sequences in chaos theory like the Rudin-Shapiro sequence and the fractal "dragon" sequence.

"Yoshizawa is somewhat paranoid - he rarely displays his models, fearing they will be copied, and teaches only a few trusted students." 

Origami has begun to turn up in other branches of science as well. David Huffman, formerly head of the department of computer science at UC Santa Cruz and the inventor of the powerful computer science technique known as Huffman coding, has given lectures on paper folding in connection with his interest in robot vision. Robert Lang, a semiconductor physicist at NASA's Jet Propulsion Lab in Pasadena, California and an origami designer since childhood, is known for his Black Forest Cuckoo Clock with deer head, weights and leaves. Lang likes to start from the finished origami shape and ask what crease pattern is needed to fold it. He charts the location of the important points (for a fox, say, nose, ear tips, shoulder and hip joints, toes, tail tip) and then figures out how far they would have to travel in space during the folding process.

Perhaps the most single-minded origami scholar in the United States is Thomas Hull, a Ph.D. candidate in mathematics at the University of Rhode Island, who has been making origami models since he was eight but only began doing serious research into the subject a year ago. At the moment he is investigating the relationship between the origami model and the lines that appear creased in the paper when it is unfolded and laid flat. He thinks of these crease patterns as graphs (in mathematics, a graph is a set of points connected by lines) and considers ways to color the graphs to indicate which directions the folds would have to be creased (up out of the plane, as mountains, or down from it, as valleys) in order to form the model. Then he turns the problem around: If you just draw a crease pattern on paper, can you predict whether it will fold into a valid model (i.e., no cuts, the paper doesn't intersect itself), and if so, what kind? Hull has high hopes for the advancement of mathematics through paperfolding. "In the history of math, a problem is often solved by an approach not so much difficult as novel: a new way to crack a nut," he explains. "And origami seems to have a structure. When I look at the pattern for a really complicated fold there seems to be a lot going on."

Hull doesn't know if he's going to do his dissertation on paper folding, but he is working on what will be the first book in English on origami and mathematics. Not only are there no books in English on math and origami, there are no translations of Japanese works on the history of origami, according to librarians at the Origami Center. And there are no works of origami criticism, even though as early as 1968, one Gershon Legman called for a critical posture, complaining in the Origami Center's newspaper that "most of the published works are stuffed with a gruelling excess of perfectly worthless foldings ... which should really never have been retained or published."

One author has made a stab at various forms of origami theory: the science writer and architect Peter Engel, who began seriously studying origami as a pupil of design guru Arthur Loeb at Harvard. His book Folding the Universe: Origami From Angelfish to Zen (Vintage), contains essays on everything from the possibility of classifying models according to the basic crease patterns used to produce them to the origins of origami (some say it began as a way to fold letters so the contents would remain secret--once the shape was opened, it couldn't be easily recreated; it was also used to adorn gifts in the Heian court where paper was a luxury and the wrapping could be more important than the gift). The main thrust of Engel's research is toward solving problems in origami design: the second half of his book consists of plans for elaborate models like a coiled 40-inch cobra constructed from a 10-inch sheet of square paper.

In one essay, Engels describes his visit to Akira Yoshizawa, a reclusive 83 year-old Japanese who was the first truly creative modern origami designer. Yoshizawa's prowess is legendary among origami devotees. He creates twenty new models a day; he records the procedures for less than a tenth of what he creates--there simply isn't time to diagram it all. In his youth he invented over a thousand animal models and destroyed them as unworthy of him. He took 30 years to perfect his cicada. He has made a woolly mammoth that experts say has to be seen to be believed, as well as a model of a wolf suckling Romulus and Remus. Yoshizawa is somewhat paranoid--he rarely displays his models, fearing they will be copied, and teaches only a few trusted students. Several decades ago, he accused the Argentinian folder Ismael Adolfo Cerceda, a descendant of the school of paperfolding that arose separately in Moorish Spain, of stealing his models and advertising them as his own. An outraged Cerceda replied: "Mr. Yoshizawa published his pig in 1945, but I did not steal it from him . . . and if I had, I would not have sent it to everybody, claiming ownership. I am not insane, yet."

Whether models are invented or discovered is a long-running philosophical question in origami, but most folders accept that similar models may be developed independently, and most happily publish their work. Still, due in large part to language barriers, the dissemination of findings amongst origami mathematicians has been slow and troublesome. Besides the annual meetings of the Origami Center, there are several internationally known clubs (such as the British Origami Society, Mouvement Francais des Plieurs de Papier, Centro Diffusione Origami in Florence, Asociaci—n Espa–ola de Papiroflexia, and Origami Tanteidan, or Origami Detectives, an informal group of young-turk designers in Japan). Some folders participate in the Internet list "Origami-L," whose subscribers are beginning to circulate folding diagrams over the World Wide Web.

December 1989 saw the First International Meeting of Origami Science and Technology, held in Ferrara, Italy, at which event the discussion veered into the unlikely territory of origami and mental health. Papers included "Origami- therapy applied to a drug addict" and "Origami and Insanity," in which Danish folder Thoki Yenn describes how his obsession drove him to a mental hospital. No one was able to offer a satisfying diagnosis for those people who become fanatical folders. Risa Miller, the director of the Origami Center and a doctoral candidate in anthropology at New York University, thought the concentration on pleating and smoothing actions should help busy people to relax. However, in a discussion of origami as mental and physical therapy, Tony Cheng, a child psychologist at Gouverneur Hospital who is also an officer of the Origami Center, warned that "models need to be selected carefully with anxious or obsessive patients"; that is, therapists should choose models where neatness is not crucial.

A second international origami conference will be held at Seian University of Art and Design in Japan this December. An international origami research community is beginning to coalesce, and the dialogue around paper folding is moving onto the Internet and into journals, away from the hobbyist clubs that sustained it at first. Meanwhile, though, back in New York, the hidden Origami Center in the Natural History Museum continues to receive unsolicited models, letters, and mathematical proofs every day in the mail. Volunteers preserve most of this correspondence in the center's storage space in the bowels of the museum. They haven't time to evaluate it, but don't want to discard it, remembering that even journeys to the frontiers of three-space can begin with a simple paper crane, a simple paper boat.--Diana ben-Aaron