Now, he designs and folds pieces for collectors and art galleries, while contributing to a growing field of practical applications in space research, medical technology and other fields.
“There’s been a whole series of spinoff benefits,” Lang says. “When you have a mathematical description of origami, of folding, it’s not only useful in the creation of art, but it’s also useful in the solution of other technological problems.”
In origami, shapes must be folded from a single uncut sheet of paper. Thirty years ago, Lang says, people thought it was nearly impossible to make origami insects, for example, because the forms have too many thin appendages. But Lang found that the artistic problem could be turned into a mathematical one, using geometry to find the crease pattern that could produce the desired shape.
In the early 1990s he wrote a software program called “TreeMaker” that embedded the necessary math in the program, so that non-mathematician origami designers could use it as well.
That program, along with others’ work, has allowed origami designs to become exponentially more complex, according to Lang.
“We’ve seen the equivalent of Moore’s law in origami,” he says, referring to the computer law that has held steady for the past five decades, in which the amount of computer memory that can fit on a chip has doubled about every two years. “At any given point in time the things that people are doing would have been thought impossible ten years earlier.”
Two photographs on Lang’s Web site, for example, chronicle the evolution of the origami stag beetle. The first beetle, which Lang designed in 1989, has six legs, two antennae and a body divided into sections. That was state of the art origami in 1989 — it was as intricate a design as origami experts could imagine. But the second beetle, designed just last year, is much more detailed, with each leg ending in two tiny pincers.
“The field of origami art has gone through something of a renaissance in the past 20 or so years,” says Tom Hull, a math professor at Western New England College. “Greater understanding of the math of origami leads to more complex origami design.”
Hull, another lifelong origami enthusiast, has made studying the math of origami the focus of his career. He says that when he first began to research it in graduate school, in the early 1990s, very little work was available.
“There were some papers in Japanese,” he said. “Nowadays there are a lot more people interested.”
In fact, over the past 20 years there have been four international conferences on the math and science of origami — the last, in 2006, was held at CalTech and attracted 165 attendees. In February, Lang and Hull joined other scientists to talk about their work at the annual meeting of the American Association for the Advancement of Science.
The research has led to practical as well as artistic advances.
Lang himself consults on various projects that involve the math and science of folding. In 2001, he worked with engineers at Lawrence Livermore National Laboratory to design a folding telescope that could deploy in space. The researchers wanted to launch a telescope lens 100 meters wide, but had to figure out how to fold it so that it could fit in a three-meter wide rocket, then unfold in space without snagging. The researchers built a working prototype, although the project later lost funding and the design was never used in space, Lang says.
But other origami-based designs have made it into orbit. In 1995, Japanese scientists used an origami folding pattern to pack a solar panel array on a vessel called the Space Flight Unit.
Lang has also worked with a medical technology company called Paracor to develop a new device to treat congestive heart failure — a kind of mesh net that fits around the heart to keep it from enlarging. Lang worked with the company to develop a strategy to pack the net into a tube so that it could be inserted into the body between two ribs, rather than through open heart surgery. The device is now in clinical trials.
Even some of the more esoteric-seeming origami math research may have practical applications. Tom Hull’s research involves trying to figure out the number of different flat objects that can be folded from a piece of paper with a particular crease pattern.
“Typically there’s a lot of ways to do it and figuring out how many is non-trivial — that’s what mathematicians say for hard,” Hull explains.
The question might sound abstract, but it turns out that physicists who study membranes — like blood cell walls — are interested in the same thing, because some membranes tend to crumple along regular lines and it can be important to figure out how many ways they can collapse.
“I’m trying to learn the techniques [the physicists] are using, and maybe my work will influence them as well,” Hull says.