Volume 23, Number 2 Exploring Origami - Page 4 of 5

he Japanese have never been quite as fussy about using just one piece of paper as American and British folders. In one of the most striking branches of origami, called modular origami, many identical, interlocking units fit together to form a complex model. (Each of the cubes to the left is made from twelve identical parts; the different sizes of the cubes come from the different sizes of the original paper squares.) The units themselves are not very interesting to fold, but the thrill of modular origami is assembling the units. The best designs have ingenious systems of pockets and tabs that lock the whole contraption together without glue and give it astonishing strength for a paper figure.

 Jeremy Shafer uses a similiar concept of linking with interlocking rings which he demonstrates in this RealVideo. Shafer's rings, however, use only one sheet of paper. Watch this clip in RealVideo through the link above, or in the following formats: Quicktime 4.0: 1.5MB or 1MB

Though modular origami dates back hundreds of years, a folder named Mitsunobu Sonobe invented the locking mechanism that has made it popular. Variants of his unit, as well as completely new designs, are being invented all the time.

obert Lang, a laser physicist from Pleasanton, California, has moved origami into the electronic age. A computer program he has written, called TreeMaker, can take any stick figure and calculate a pattern of creases that will produce that figure. For example, to create the lobster below, Lang drew a stick version to determine how many appendages it needed, how they connected, and how long they should be. Then he gave the information to TreeMaker, which printed out this folding pattern (left).

 The lobster (shown below) is a complex creature that was designed by Robert Lang using TreeMaker, the software he developed.

If you fold along the lines the right way, you get a primitive lobster model with flaps of paper that correspond to each limb of the stick figure. From there, Lang says, he uses his origami experience - not the computer—to transform the flaps into claws that look as if they might pinch you if you get too close. "After a while, you learn the repertoire of how you put in elbows or feet," Lang says. "The hard part is getting the flaps in the right places to begin with."

The secret of TreeMaker lies in the circles that have been drawn on the crease pattern. Each circle corresponds to one appendage. For example, the gigantic circle eventually turns into the lobster's tail. Inside each circle, the creases fan outward like the spokes of an umbrella. When collapsed, each of these "umbrellas" becomes a narrow, straight projection or flap, which Lang subsequently shapes into the appendage of choice. The computer packs the circles together as tightly as possible (to avoid wasting paper), while respecting the overall geometry of the lobster.

This type of mathematical problem is called "constrained optimization"—finding the optimum design that satisfies certain constraints, such as the proportionality of the lobster's limbs. It's also exactly the type of problem Lang solves in his day job as a designer of laser systems. "Is it just lucky coincidence," Lang asks, "or is it that to a man with a hammer, everything looks like a nail?"