PIMS Chats with Robert Lang

Origami, and the math behind the folds

Robert Lang became interested in Origami at the age of six. He’d found a book with instructions on how to fold traditional designs and was hooked. “All you needed was paper, which there was an endless supply of. You could fold what you wanted over and over again.”

He found more books and designs and kept on folding, soon finding that there were things he wanted to make that weren’t in the books. He started creating his own designs using the techniques that were visible from the others’ books and thus marked the first decade of his origami life.

In high school and his early teens, Lang encountered the books of Martin Gardner (who wrote the Mathematical Games column for Scientific American for 30+ years). “He wrote tremendously excitingly about math” says Lang. “There is a whole generation of mathematicians who were inspired and turned on to math by him. He turned me on to math, which led me to go down a mathematical and scientific avenue for my education and career.”

Lang attended Caltech to study electrical engineering and optics. He found, over the course of his undergrad and graduate study, that there is a model often followed in engineering and sciences: “whatever the phenomenon is, you try to understand the mathematical underpinnings– what are the basic laws? If you know those, you can use the tools of mathematics to learn more or build what you want. And in a way it comes for free, because you are accomplishing all of these new things using old math.”

He was still doing origami as a hobby, and was starting to see parallels with engineering in that there are very clearly fixed laws that govern what can and can’t be done. “They are really geometric laws,” says Lang. “For example, I realized that if I wanted to create something that had a lot of legs, I needed to allocate circular sections of paper for each leg to accomplish that.” He used math to understand and improve his origami. And it’s been paying off quite well!

Lang operates as a professional origami artist and consultant. “My life is a series of small projects, and they’re very diverse, which keeps it interesting and exciting.” He travels, giving lectures and workshops, some academic, but others to companies and organizations teaching folding techniques that are applicable to industrial design. “I’ll consult with companies for which some aspect of their product involves folding, usually because it needs to collapse in some way.”

He receives both artistic and commercial commissions for his origami. An art collector might want a piece of display origami, whereas an advertising project might require a specific folded shape. “Often, he says, “a company wants to express money savings and show all the things you can do with the money you save. I fold the money into the shapes of the items.” Lang has also written a number of books (14 to be exact) and has two more in progress.

The largest size sheet of paper he’s worked with was five by six metres, for a trade show display in which he folded several life-size instrumentalists. The bass player was the largest and took one full day. “It’s not the size so much that matters, but the complexity” he says. “The longest single folding project I worked on was a flying pteranodon (which is at McGill’s Redpath Museum). The paper for that was only about 15 foot square, but it took five days of folding and building the support framework (wires and rods).” He explains that for large projects the paper becomes very flimsy, and a skeleton is required to support the structure.

As the origami becomes more complex, Lang sometimes uses computer programs that he has developed to map out the crease patterns. “There is a huge diversity in the type and style of origami” he says. “Some can only be done by computer, while others can only be designed by hand because no one has come up with a computer algorithm to produce them. However I like equal amounts of human interaction and choice and computer assistance. There are things that we don’t know how to program into a computer program (creativity for example), but at the same time working out certain aspects of a design entirely by hand would be forbiddingly tedious.”

One of Lang’s most interesting and high profile projects involved developing a folding pattern for a space telescope. Using a folding pattern he developed, a five-metre diffractive lens cut into panels with metal flexure hinges was built. Another required incorporating an origami algorithm into a company’s software designs for determining the folding pattern of airbags.

Currently, Lang is working on a paper that describes an aperiodic origami surface. Aperiodic tilings are interesting to mathematicians for many reasons, to physicists because they are found in the real world in so-called quasicrystals, and to recreational mathematicians because they make beautiful patterns. One of the popular genres of origami is modular origami, which uses many sheets of paper folded as identical units to make polyhedral structures (cube, octahedron, etc). “No one, to my knowledge, has used origami to make an aperiodic 3D form,” says Lang. But recently, he developed a method to do so and is utilizing it as a tool for creatively wrapping a number of math and origami concepts together.

Lang is recognized as one of the foremost origami artists in the world as well as a pioneer in computational origami and the development of formal design algorithms for folding. He holds a Ph.D. in Applied Physics from Caltech, has worked at NASA/Jet Propulsion Laboratory, Spectra
Diode Laboratories, and JDS Uniphase and has authored or co-authored over 80 papers and  45 patents in lasers and optoelectronics as well as authoring, co-authoring, or editing 12 books and a CD-ROM on origami. He is a full-time artist and consultant on origami and its applications to engineering problems. He received Caltech's highest honor, the Distinguished Alumni Award, in 2009.

Visit Robert Lang’s website at: www.langorigami.com

On April 19, 2012, Robert Lang delivered the PIMS Public Lecture, “From Flapping Birds to Space Telescopes: The Modern Science of Origami” at the UBC site.