Photos by Anthony Scoggins, Mingei International Museum

Satoshi Kamiya’s wasp is so technically perfect it almost looks as if it could fly. Two sets of precisely formed wings sprout from a body whose sculptural detail might well suffice for an anatomy class. Delicately jointed legs end in tiny two-toed feet; its ridged abdomen arches in a curve, apparently poised for takeoff. That such a complex form could be folded from a single sheet of paper seems to defy belief. Kamiya’s wasp and several equally astonishing beetles are part of the current exhibition of “Origami Masterworks” at the Mingei International Museum in San Diego, a show that demonstrates how far the ancient art of paper folding has come from its traditional focus on flowers and cranes and cute little boxes.

Although much of the Mingei show is devoted to an elaborate zoo of creatures, including an enormous elephant and a riotous clutch of centipedes, many of the pieces are purely abstract structures inspired not by nature but by the formalisms of geometry. Mathematician Tom Hull’s model of five interpenetrating tetrahedrons calls to mind an alchemical symbol. Another, by French folder Vincent Floderer, provocatively titled Boom, consists of a single transparent sheet pleated into an insanely complex landscape of mountainous peaks arranged in a quasi-checkerboard pattern. Issey Miyake meets M.C. Escher. In a separate room at the back of the show are several enigmatic works by computer scientist Jeannine Mosely involving curved surfaces and circular folds, surreal variations on the classic platonic solids. Sharing the same display case is a geometric form known as a “gyroscopic egg,” a delicate hollow ellipsoid constructed out of origami latticework.

All these models, including Kamiya’s insects, have been made possible by the marriage of origami and mathematics, a partnership that has given rise to the emerging field of origami sekkei, or “technical folding.” This new breed of “computational” origamists are also developing techniques that are taking the art in hitherto-unthinkable directions. In the process they are turning paper folding into a science as much as an art. Their explorations are giving rise to whole new dimensions of origami aesthetics. At the same time, their radical methodologies have application to a slew of scientific and engineering problems — from deploying telescopes in space to folding air bags in cars and folding proteins in living cells.

Until recently, origami was an ad hoc art. Designers developed new models based purely on their intuitive understanding. But over the past decade, says master origamist Robert Lang, “mathematicians and scientists have begun mapping the ‘laws of nature’ that underlie origami, converting words, concepts and images into mathematical expressions.” They are bringing with them insights from such fields as information theory, algebraic geometry and number theory. As Lang points out, many basic issues in origami “are actually linked to deep mathematical questions.”

Lang, who designed the elephant in the Mingei show, is one of the pioneers of computational origami. Recently retired from a career in laser physics, he now spends most of his time as a professional paper folder. He is the author of a dozen books on the subject, including his recently published magnum opus Origami Design Secrets: Mathematical Methods for an Ancient Art (A.K. Peters, 2003), a vast compendium of ancient and new techniques that explains the principles behind the recent origami revolution.

Lang is also famous in the origami world as the author of a computer program called TreeMaker, which automates the design process and enables the construction of extremely complicated creatures. Let’s say you want to fold a scorpion. Traditional origami techniques are not well suited to many-limbed forms, with long, thin appendages posing particularly difficult challenges. With TreeMaker, the user simply inputs a stick-figure diagram of the desired model, and the program calculates the requisite crease pattern. TreeMaker has come up with designs that Lang says are radically different from anything used in traditional origami and may seem counterintuitive. Sometimes, he says, its designs only work when you scour all the creases, then fold the whole model up in unison. In these cases, “there are only two stable states, the flat piece of paper and the finished model, nothing in between.” That is quite different from traditional origami, where models have necessarily been constructed one fold at a time.

Lang developed TreeMaker because he wanted a tool to help him design elaborate animal models, particularly insects. The rules it applies when calculating crease patterns are based on techniques developed by Lang himself and a number of legendary Japanese folders, including Kamiya and Jun Maekawa. But TreeMaker can
also solve more abstract problems. Lang
has been hired as a consultant by a
team of researchers at the Lawrence Livermore National Laboratory
who are designing space-based

Astronomers searching for extrasolar planets need very large telescopes to catch sufficient light from these extremely faint sources. The Livermore scientists are hoping eventually to build space-based telescopes with lenses up to 100 meters in diameter. Something this large will never fit into the cargo bay of the space shuttle, so the idea is to make lenses that can fold up and be opened out once they’re deployed in orbit. Fortunately, flat film lenses, known as Fresnel lenses, are highly effective — they are used in overhead projectors — but the challenge is how to fold one up so that creases don’t interfere with the optics and spoil the resulting images. Lang used TreeMaker to come up with a viable crease pattern — it resembles a gigantic spider’s web — and the Livermore team is currently constructing a 5-meter prototype of his design.

Origami techniques are applicable anytime you have an engineering problem in which “something needs to change shape dramatically,” Lang says — particularly if it “needs to start small and then get very big.” Tents, deployable shelters and antennas are all potential applications. And the German company EASi Engineering has employed origami techniques to make three-dimensional models of folding patterns for air bags.


To make all this work, computational origamists have had to learn a great deal more about the formalities of folding. Tom Hull, a mathematician at Merrimack College in Andover, Massachusetts, and the organizer of an international conference on origami science, notes that when developing new models, one critical problem is “How do you assign the paper space?” Lang’s scorpion, for example, has more than a dozen parts, each of which — the head, body, claws and so on — has to be assigned space on the initial sheet of paper. It turns out that mathematically this is equivalent to the problem of efficiently packing a bunch of circles into a square. That might sound like a trivial question, but Hull points out that mathematicians have no idea how to solve the circle-packing problem for more than about two dozen circles.

Japanese folders like Kamiya and Maekawa also use circle packing and other mathematical techniques when designing their models, only without the aid of computers. The crease patterns for Kamiya’s wasp or Maekawa’s famous demon are themselves works of art, Hull says admiringly. For me, the crease patterns are almost better than the finished models, for in these intricate geometric designs we witness the raw creative power of mathematics in action. The great American logician Charles Sanders Pierce believed that mathematics was the realm of pure potential — anything that could be imagined, Pierce thought, could be expressed in mathematical form. Computational origami seems a slyly playful confirmation of this Piercean view. With the right folding algorithm, a simple flat sheet can be transmuted into a wasp, a demon or a unicorn — even a window onto the universe.

“Origami Masterworks” continues through February 8
at the Mingei International Museum, located in Balboa Park, San Diego, (619) 239-0003 or You can visit
Tom Hull’s mathematical-origami Web site.

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