Robert J. Lang

Mathematica and Origami

“I love Mathematica,” Lang says. “TreeMaker is a surgeon’s scalpel—a very narrow tool that does what it does incredibly well, but nothing else. Mathematica is the world’s largest Swiss Army knife. You can do anything in it: analyze, compute, simulate, visualize. And almost any project that involves something I’ve never done before probably involves Mathematica at some point.

“With Mathematica,” Lang continues, “I can import things. I can manipulate them, I can do visualization. I can solve equations, I can do algebraic manipulations and 3D renderings. I can plot surface graphs. It’s just mind-boggling.”

For the Pteranodon, Lang wrote a quick Mathematica program that displayed its stick figure structure and a corresponding crease pattern. “By changing the points in the crease pattern,” he says, “I could see its effect on the dimensions of the stick figure. I could make the wings longer, the neck shorter. The current version of Mathematica has live sliders so you can adjust them to find the perfect dimension on the crease pattern. You just watch the crease pattern and shape of the folded thing move in response to the sliders.”

The other way to have designed the Pteranodon, Lang says, would have been to change a dimension on a piece of paper, fold it for two hours and see what the result looked like. “But with my two or three hours of investment in programming, I go ‘badoop’ and see the result. And that’s just incredible.”

Lang says it took him three hours, tops, to set the simulation up. “Then I told Mathematica to scale the crease pattern to a four-meter square and spit out a list of measurements of the points around the edges of the square so I could mark off with a ruler where the folds need to be.

“And I start folding.”

Lang can design and fold some projects in an hour. More complicated patterns take months. “When I have a commercial job, of course,” he adds, “I work on it until the schedule says I have to deliver. I can guarantee that I can produce something in a couple of days that’s going to meet the needs and expectations of the client.”

No Limits

One of Lang’s most visible projects involved creating the folding pattern for a thin, refractive lens of a telescope developed at Lawrence Livermore National Laboratory. Called the Eyeglass, the ultimate lens would measure 100 meters across—the length of a football field—and would float in geostationary orbit 25,000 miles above the earth. A camera two kilometers away captures images of space through the lens.

But how do you get it there? Lang designed an origami structure that could be used to fold the lens (using flexible metal hinges between glass plates) into a cylinder small enough to be carried by a rocket. Lang designed a single, well-controlled folding pattern for a round shape—one that could be scaled to 100 meters in diameter—that could pack the lens into a small folded bundle.

“The creative tools on a Mac are there, and they work together, and things work intuitively so they don’t get in your way.”

Perhaps Lang’s most elaborate commercial origami assignment was creating an entire landscape for a Mitsubishi commercial. He, colleague Linda Tomoko Mihara, and a small team of model-makers folded mountains, clouds, wheat, several hundred trees—lacy trees so people could peer through the branches—tree bark, cobblestones, eight-foot-tall skyscrapers, Victorian homes, simple and complex leaves, several dogs, a deer, perching birds, flying birds and a dragon.

Lang started folding origami when he was six. “What got me hooked,” he explains, “was the idea that from the same simple beginning— an uncut square—you can make such a variety of different shapes. It’s like getting something for nothing, because the raw material, the paper, was free. You could use scrap paper and yet you could make all these different cool representations. And what kept me going over the years was that you never ran out of what was possible.”

Later, studying electrical engineering and physics in college (B.S., Caltech; M.S., Stanford, Ph.D., Caltech), where he learned to use mathematics to describe diverse natural phenomena and then exploit the math to expand knowledge of the field, it seemed natural to apply that same approach to origami.

“And in origami, we’re nowhere near the limits of what’s possible. In just the last 50 years, we’ve seen the number of published origami design grow from about 100 to more than 36,000. One of the revolutions that helped origami was the transition from hand-drawn pen-and-ink diagrams to computer-drawn diagrams. That started in the late 1980s. Everyone started on Mac, and most people who do origami diagrams today use Macs. The creative tools on a Mac are there, and they work together, and things work intuitively so they don’t get in your way.”

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