After 15 years in the laser-optoelectronics industry, Robert Lang left his job as an engineering manager at a company in Silicon Valley to focus on writing a book on origami design. That was in 2001. He has never looked back.

Lang got hooked on origami as a small child and stuck with it. In college, the academic demands ratcheted up a few notches compared with high school, and as an outlet he revved up his origami, too. “Every time you crack a folding sequence or come up with a new design, you solve a problem,” he says. “I think the satisfaction—the jolt of dopamine when something activates the brain’s reward center—is the same thing that motivates a lot of scientists. Every time you discover something new, it’s pleasurable. You want to do more of it.”

Today Lang is a freelance artist and consultant in northern California’s Bay Area. He makes amazingly intricate animals, plants, and other figures, and he works on applications like telescopes, solar sails, camping gear, automotive accessories, and medical devices. “If you have stowage and deployment,” he says, “origami plays a role.”

Lang earned his bachelor’s degree at Caltech and his master’s degree at Stanford University, both in electrical engineering. He returned to Caltech to earn a PhD in applied physics, with a focus on semiconductor lasers and optics. “When I look back,” says Lang, “many of the electrical engineering courses are no longer useful—there is not too much call for being able to program in 8-bit assembly language anymore. But the physics and math I still call upon every day.” That’s especially the case when he’s applying origami to engineering problems. “Physics is what allows me to connect origami to many different fields, the thing that lets me make that bridge.”

**PT:** What was your career path after graduate school?

**LANG:** I did a postdoc for a year at a company in Germany. Then I came to the Jet Propulsion Lab, where I worked for about four and a half years, until being lured to a smallish company in Silicon Valley. I worked in lasers and optics. In 2000 the company was bought. And by 2001, the siren song of origami had grown too strong; I quit my job in lasers.

What I found was that I had enough origami-related opportunities to keep me occupied and gainfully employed. And it was interesting and exciting, especially when I started getting involved in technological applications and the theory of origami. The mathematics is as deep and rich and challenging as the mathematics in lasers and optoelectronics had been. So it kind of scratched all the itches, and I’ve never gone back.

**PT:** What kind of work do you get as an origami artist and consultant?

**LANG:** There has always been paid teaching and lecturing, workshops, origami art sales—I create artworks and sell them to collectors. And there is a fair amount of commercial art, for advertising. And consulting on applications for companies, where a company says we are making a product that involves folding in some way; can you help us design a fold pattern? Or, can you come up with fold patterns that make our product work better?

**PT:** What are the most interesting applications you have worked on?

**LANG:** Some of the most interesting are still under nondisclosure agreements. But the space projects are interesting. I worked on the Eyeglass telescope for Lawrence Livermore National Laboratory, and I did some analysis for the Starshade project for NASA—it is an occulter that will be used to look for planets orbiting faraway stars from a space-based telescope. One of the things that make these projects interesting is that they are big. Starshade is 30 meters across. The folding patterns are called flasher patterns, and they sort of wind up in an interesting way. It’s both useful and beautiful. I like the combination.

In a collaborative project with university scientists, we developed a technology for microscopic folds. The thing being folded was only 800 microns across. Obviously that’s too small to do by hand. You can’t even see the finished thing with your naked eye—it has to happen under a microscope. We developed a technology for self-folding that can be used to explore the properties of mechanical metamaterials.

**PT:** Are there challenges in terms of materials?

**LANG:** I don’t get deeply into the materials, but some material factors affect what I do. Coming up with fold patterns that work with a material of non-negligible thickness calls for different and more challenging mathematics than fold patterns that work with paper or a material that is thin enough that you can ignore the thickness.

**PT:** What tools do you use?

**LANG:** I use tweezers for tiny things. I use a laser cutter to mark and score paper and other materials. For some patterns, folding by hand is very hard to do precisely. For some applications I have used polymers, metal, wood, laminates. In many cases the folds have to be created by some mechanical process.

For many of my geometric tessellations, the coordinates of the vertices only come about by solving a set of simultaneous algebraic nonlinear equations. You have to do a numerical solution; there is no closed-form formula. I write the code, and my computer crunches and crunches and then spits out the crease pattern. Then I export the crease pattern to the laser cutter, and it scores the lines. For certain designs, I can import a drawing that defines the desired cross section and then, using my code, compute the crease pattern for a folded shape that is determined by the drawing.

**PT:** What fields of math does origami involve?

**LANG:** Mainly computational geometry, but origami pulls in aspects from other fields of math: combinatorics, group theory, graph theory, number theory.

Early on, when I was still in lasers and working on origami and the math of origami on the side, I developed a theory of how to design origami figures that were fundamentally tree graphs. For any shape of which a stick figure is a good approximation, I can follow a very specific mathematical algorithm to create the crease pattern that folds into the shape. The tree theory work showed not only that you could fold any stick figure–like shape, but that the result would be the largest possible for a given starting sheet of paper. A friend encouraged me to submit the paper to a computational geometry conference. That got me into the technical literature, and the early jobs came about because some engineer did a literature search and found my paper.

**PT:** Have there been surprises in your origami practice?

**LANG:** The biggest surprise was the tree theory that I did for that first computer science paper, because it showed that any shape—any number of legs or limbs—was doable, and doable efficiently. By efficiently, I mean that the ratio between the size of the finished object and the paper you started from is reasonably close to the theoretical limit.

Prior to that point, people thought that insects and spiders and things with lots of legs were impossible, or only a few would be possible. Up until that point, I was solving toy problems with math—problems for which I already knew the answer. But once I got the math developed, I could solve problems that I didn’t know the answer to in advance.

**PT:** What about the napkin problem? You have solved a version of it, correct?

**LANG:** Yes. The napkin problem asks if you can increase the perimeter of a paper by folding it into a flat shape. So, take a square. If I fold it once, I make the perimeter smaller. Another fold may make the perimeter larger, but not as large as it was originally. But what if you make more folds? The problem asked one to prove that no matter how you fold a square, you could never get the perimeter to be larger than the original square. The problem showed up on mailing lists in the 1990s.

I already knew from the mathematical work I’d done on stick figures that it was false—that, in fact, it’s pretty easy to fold a shape that gives a larger perimeter.

That raised the next question: How big could the perimeter go? And the same theory that gives rise to those insects can be employed to show a very surprising result: There is no upper bound. So, from a square of mathematically ideal paper, I could fold a shape whose perimeter is the radius of the known universe. The number of folds you need grows quickly, and the shape you would get is a microscopically tiny star. If I make a star for which the length of its points is 1/M the size of the square, with M^{2} points, the perimeter is M^{2 }times 1/M, which is M. So by picking larger values of M, you make more and more points, but the star is smaller and smaller with denser and denser points. That’s the napkin problem.

**PT:** How do you design?

**LANG:** Historical origami used cuts and multiple sheets. But there came to be a focus on one uncut square. There is an elegance, a purity, to it.

I am not a professional computer scientist, but computer scientists have taken this on and pushed those questions. One of the things they have shown is that literally any shape that is connected can be folded from a single uncut square.

For animals I use the relationships between shapes and find a packing arrangement of shapes that correspond to the parts of the animal. For more geometric shapes, it’s pure computation. And for applications, all those rules about no cuts and single squares go out the window—if cuts help, if multiple sheets help, if a weird shape helps, that is what we use.

One of the differences between origami and almost every other form of sculpture is that origami is strongly conservative—it’s neither additive nor subtractive. Think about clay―it can be both. Painting is only additive, and marble sculpture is purely subtractive. With origami, what you start with is what you end with. There are very strong mathematical constraints.

**PT:** What’s the most challenging thing you have made?

**LANG:** A cactus with spines. It took me seven years from start to finish. It has about 400 spines. It’s just a lot of intricate and tedious folding. That made it a challenge.

**PT:** What do you plan to do next?

**LANG:** I have a book that I’ve been working on, *Twists, Tilings, and Tessellations: Mathematical Methods for Geometric Origami*. It’s all about the mathematics for geometric origami. It’s my 17th book. And one of my interests right now is curved folding. I missed a lot of the math in college that I would need for that, so I’ve been reading up on differential geometry. You need it to design things with curves.