Beyond the Fold

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 www.mingei.org. You can visit Tom Hull’s mathematical-origami Web site.