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Amateur Hour: Paper Perfect

Robert Lang and the Science of Origami

ISSUE:  Summer 2014

Robert Lang. Alamo, CA, May 2014. (Photographs by Jason Motlagh)

On February 4, 2014, Robert Lang joined Joshua Foer onstage at the Institute Library in New Haven, Connecticut, as part of the ongoing series “Amateur Hour,” in which various tinkerers, zealots, and collectors discuss their obsessions. Lang is recognized as one of the pioneers of the marriage of origami with mathematics and technology. He is the author of numerous books, including Folding Paper: The Infinite Possibilities of Origami, coauthored with Meher McArthur. The conversation that follows was recorded live and has been edited for brevity and meaning. 

Joshua Foer: I want to start by asking you about how you made a name for yourself in the field of origami. As I understand, it all started with a single work you completed in 1987.

Yes, the cuckoo clock. It has leaves around the outside and a deer’s head on top, and a cuckoo, of course, coming out the door, and pinecone weights, and the pendulum, and all of this is folded from a single uncut sheet of paper. Oh, and it told the correct time twice a day. That got a lot of attention in the origami world, which is a very small world but it’s still got a lot of competition. After that, one thing led to another.

Just so we’re all clear about the terms here: That was folded with one uncut sheet of paper—​no glue?

No glue.

How big was the paper?

A one-​by-​ten-​foot rectangle.

How long did it take you to fold?

I worked on the design over a period of two to three months and worked on folding parts of it little by little. When I sat down to fold the display piece all at once, it took about six hours of solid folding.

Could you show us how you execute a piece of origami?

Sure. To speed this along I brought paper that already has some folds in it. This is foil-​backed paper, which is really nice for shaping. It’s basically wrapping paper, but origamists use it because it works really well when you need to fold something quickly and want to get a little bit of shaping in. 

So what are we looking at? This looks to me like a package that’s already been unwrapped.

I’ve put some of the most important creases already in this piece of paper, and I can tell you based on what the creases are what the different parts are also. The head is going to come from this little star-​shaped region here. And then we’re going to have one arm coming from the corner here, and another arm coming from the corner there, and there’s going to be a foot here and a foot there.

Where did origami start?

Well, the word origami is Japanese, and the art form as we recognize it today has its deepest roots in the Japanese craft. Paper itself was invented by the Chinese, and they took it to Japan by way of Korea around the seventh century a.d. The art began with ceremonial origami, where the folds themselves were maybe not complicated but the positions of the folds had a particular meaning. And much later there was this decorative craft as a way to make toys or decorations out of something relatively inexpensive. 

And since then there’s been some evolution in the technology of paper-​folding.

Yes. In the twentieth century, a poor metalworker named Akira Yoshizawa took up this paper-​folding art, and he began creating new figures. There were already a couple hundred simple figures, but Yoshizawa created thousands of new figures. He brought a new level of life and beauty to the art form. He established new folding techniques that allowed people to make soft organic forms and even to do some of the shaping. He was possibly the first to explore the use of foil-​based paper, the kind I’m using now. He also invented a folding language, a kind of system of dashed lines and dotted lines and arrows that allowed people to convey clearly how to fold and shape, which provided a level of communication that hadn’t existed before. This allowed people to share and build upon what they learned from each other, and that kicked off exponential growth in the world of folding that continues to this day. 

By the way, what was that intense maneuver you just did there?

Well, where a large number of creases come together and then go down flat, we call that a collapse. The next fold was just a valley fold. These are going to be mountain folds. The next fold is going to be a little more unusual. It’s called a reverse fold, where I’m actually turning the paper partially inside out. And I’m using this particular reverse fold to make something that resembles hair on the head of this man. 

So what happened to origami in the last twenty years that led to what I’ve heard called the Bug Wars?

Ah, yes, the Bug Wars. In the 1990s, we developed techniques that allowed us to do really complicated things, the most complicated of which were thought to be things that had lots of appendages—​insects and spiders and the like. If you think about how you would make an arm or appendage on a piece of paper, you’d take the paper, you could fold it in half, kind of like making a paper airplane, making your folds around a corner. Then you’d fold it in half again, and it becomes skinny. Imagine finding a way to do that simultaneously to all four corners, and then you could make something with four flaps, or four arms and legs. But if you want to make something with more, what do you do? You’ve run out of corners.

You need more corners—or paper that’s not square?

Or paper that’s not square. If you want to use a square, you’ve got to figure out a way to make a flap out of the middle of an edge, rather than the corner. If I take my square and fold it in half, I’ve just turned the middle of an edge into a corner. So maybe that’s a way of getting a new corner. But that’s a very inefficient way. You could use that approach to get maybe five or six flaps. So to make a human being with four legs plus a head, that’s five flaps, so that would work. But to make a beetle with six legs and antennae and jaws, that would be ten flaps, and there’s no way to get that from a square. Or so people thought. These new design techniques showed us how to allocate paper in such a way that there was enough paper for every part that we needed. 

And this was actually a mathematical problem, a geometrical problem.

Yes. Now, when you hear the term math, you think of equations. But math is much broader than that. It’s the study of patterns and relationships. When origami artists figured out the relationship between an appendage and the pattern of folds that you needed to give rise to that arm—​that’s math, geometry. And so what are the relationships between the regions of paper you need to make a certain shape, and what are the relationships between the patterns of folds that actually collapse the paper down into the shape that you’re after? That’s all mathematical.

And I gather you developed an algorithm and a piece of computer software that allowed you to map out the crease lines on a piece of paper that would map onto any object you could imagine wanting to create. 

That’s a pretty accurate description of it. The computer program was the proof of the algorithm, but the important thing was the algorithm itself. The one I came up with was one of the first, but people have built upon it and come up with more. I called my algorithm tree theory, or circle packing. And I should mention that it was almost the exact same algorithm that a Japanese artist developed independently at almost the exact same time. 

How do you account for that?

It’s one of those things: If the field is fertile, the seeds sprout at the same time. I discovered this when I went to Japan in 1992, and I was sitting during a lull in the activities and I started drawing. I was developing a design, and I was doodling these circles, because circles are an important part of the algorithm. And someone walked over and said, “I know what you’re doing. Meguro Toshiyuki does that.” Who’s Meguro Toshiyuki? A couple days later they got us together, and even though I spoke no Japanese, and he spoke no English, we were doing the same thing. Our patterns, our diagrams, our designs were exactly the same. So it was just—​the time was ripe.

What kinds of things were suddenly possible with this new algorithm?

Just about everything. We did tons of bugs, of course. The first guy would come up with a beetle, and someone would say, “That’s a nice six-​legged beetle, but here’s my spider, and it’s got eight legs.” And the next person would say, “Well, here’s my dragonfly, which has six legs and four wings, which makes ten! And here’s my beetle that has horns.” And so on. As an example, I brought one of my insect designs, a very recent one. This is a yellow jacket.

This is one sheet of paper, no glue, no cuts.

And it’s made from origami paper made by Michael LaFosse. I daresay it wouldn’t have been possible to fold something like this thirty years ago, because the paper didn’t exist. As we developed the new techniques, there were also people developing papers that could withstand the rigors, the stresses and forces and challenges, that we put on our paper with the new designs. 

Why did it take a new kind of paper to make this possible?

Well, one of the things you can see here is that the legs are really thin. There might be thirty or forty layers of paper in that tiny little back leg. To get the thin legs, there have to be a lot of folds, a lot of pressure when I’m folding to get them to clamp down and stay and not rip. The paper has to be really thin and really strong, but it also has to accept a crease. That need to take a crease yet remain strong, those are conflicting objectives. That was the challenge.

How many moves did it take to create the yellow jacket?

A very wild guess would be many hundreds. The design for this was probably about a week, and the folding was a couple of days. Not solid folding the whole day. I’d fold some bits and then put it aside, then fold a little more. One of the techniques Yoshizawa invented is called wet folding, where, as we’re folding, we will very selectively dampen the paper. That’s how the legs stay thin. As I fold it, I dampen it ever so slightly and then set it aside to dry before folding more.

Are there objections to the intrusion of all this math and computer science into this gentle, dexterous art?

I think the math and computer science are fully compatible with the gentility of the art—​but there are certainly people who eschew the use of mathematical techniques. I haven’t heard of anyone saying there’s something wrong with it—​it’s more that some people choose to use this set of techniques, and others choose to use different ones. It’s like the difference between oil and acrylic in painting. It’s all painting, but some people prefer to do watercolors, and that’s fine.

So why do this? Why does this move you? 

The thing that originally attracted me forty years ago as a small child was that I could create all these different shapes with nothing more than a sheet of paper. There’s something beautiful and elegant about that—​and something practical, too. As a kid, it was a way to make toys, and I didn’t need to buy anything. But underneath all that, there’s this elegance. All the diversity of these figures is inherent in that plain sheet of paper. When you’re looking at one of these, your eyes may say you’re looking at a yellow jacket, but you’re still looking at a square of paper. And there’s a certain integrity that comes from the fact that we could take one of these and very carefully take it apart and get back to the square that we started with.

You have, I believe, two degrees from Caltech—​is that correct?

Yes, a bachelor’s and Ph.D.

You’ve got fifty patents to your name, you had a job at a high-​tech engineering firm. You were an editor at the IEEE Journal of Quantum Electronics. Then in 2001 you decided you were going to become a professional paper folder. That sounds like a risky career move.

Well, there was a book that I had been struggling with for about ten years. I wanted to write a book that was more than a collection of recipes, which is what most of my previous origami books had been. I wanted to write a book about how you design origami, so that everyone who wanted to could design their own origami, not just have to follow step-​by-​step instructions. I came to the conclusion that to write that kind of a book I needed for it to become my sole focus. If I wasn’t doing it full-​time, it would never get written. So what was more important to me—​engineering and management, or writing this book? I decided that there were a lot of other physicists in the world, and that anything I could’ve done in physics or management, someone else could do. But—​he says with great modesty—​I was the only one who could write the book that I had in mind. 

I was surprised to learn from your book that origami has, or will, affect all of our lives. How so?

Well, it may affect you in medicine. It could affect your consumer purchases. I’ve done consulting for a variety of companies that want to use folding in their products. If the object you’re building needs to exist in two states—​big and flat in an unfolded state, and small in a collapsed state—​there’s a good chance that origami can provide a solution, either because the folding patterns already exist or because we can use our techniques to do industrial design.

What’s an example of how it influences medicine?

One example is a product I consulted on, a heart implant for people with congestive heart failure. Essentially, it’s a bag that wraps around the heart that provides a kind of restoring force. The heart wants to swell up, and the bag tries to nudge it back to size. The company that came up with this concept designed it so that the bag is injected through a tube between your ribs. Normally, heart surgery means you pry open the chest. They wanted something less invasive. So this sheet-​like bag needed to be collapsed in a very precise, controllable way in order to fit inside the tube. I came up with some folding patterns, and they used some of them in their product. So if you have congestive heart failure in your future, and you don’t have a scar down your chest, it might be because of origami.

Likewise, we know some things about the world beyond planet Earth because of origami.

Indeed! One of the places where you have these two requirements—​big and flat and yet small—​is anything big and flat that goes into space. Because the only way anything gets into space is to go up in a rocket. Space is at a premium in rockets. So what are big, flat things that go into space? Lenses, reflectors, telescopes, solar arrays, antennae—​and these are all structures that various space agencies have either looked at or adopted origami for the solution. The Jet Propulsion Lab is developing a satellite that will use a solar array. But it’s a planetary mission—​it’s going to go quite far from Earth. When you go far from Earth, the solar power is very weak, and you need a really big array. So they’re looking at making an eighty-​foot solar array, which has to fold down into a few meters in diameter. They’re using an origami pattern to fold this thing down into a nice tube that fits into a rocket.

The same thing could be said of air bags.

Air bags are an even better example of the unexpected spin-​off that can happen from origami. Let me make a slight digression. To make these insects, one of the things we came up with was the realization that there were certain patterns of folds that were necessary, that would allow you to fold any number of arms and legs. And those folds were called molecules. Meguro came up with that term to describe them. So we use molecules all the time to design bugs. Now, air-​bag manufacturers wanted to do simulations of their air-​bag deployment without having to crash so many cars. You can just do a simulation on a computer to tell you whether the air bag works. But in their simulation, they had a problem: The air bag entered the simulation inflated. The first thing they needed to do to simulate an air bag was to break up the surface into lots of tiny triangles and flatten it, and all of the flattening had to happen along the edges of the triangles. So they needed to know where the folds would go and then align the triangles to those folds before doing any simulation. But they didn’t know that. When they came to the world of origami—​because we know how to flatten things using folds—​we showed that the folds needed to flatten an air bag were those molecules, those crease patterns. They were the same patterns we came up with to design bugs, and the air-​bag manufacturers could use them in their simulations.

Now, the world has caught on to the fact that origami has practical applications. The National Science Foundation, in the last two years, put out a series of grant requests for science projects that involve origami. I think you mentioned that they require an artist be a part of the team. You ended up finding yourself on basically a third of the winning team projects. Can you tell us about the cutting-​edge science that you are applying origami toward?

One of the projects is at Brigham Young University, led by Larry Howell and Spencer Magleby, who are in the engineering department. They’re developing what are called compliant mechanisms—​machines that flex, that don’t have separate physical moving parts but where all the activity happens by bending the shape. And they’ve already made things like artificial joints and medical devices, retractors for operations, that make use of this concept of compliant mechanisms. And, in fact, for the medical retractor, the device that doctors use to move an organ out of the way, they’re currently developing an origami-​based solution.

Another team is working on something that has the wonderful name of photomorphon fibers. These are fibers that, when you shine light in them, will bend and flex and can be controlled. So you can embed those fibers in a sheet, and thereby make the sheet bend upward, but you can also bend the fibers by themselves. One of their first projects will be to make a brain probe, where you have this optical fiber threaded into the brain, and by shining a light into the fiber, you can control it so that it makes its way to the spot where it’s needed. 

Can you tell us a little bit about materials that take advantage of origami at the molecular level?

There’s another group that I’m involved with, from the University of Massachusetts at Amherst and Cornell University. They’re making what are called mechanical metamaterials, trying to use folds to impart physical properties to materials that couldn’t otherwise exist in the real world. So what might one of those properties be? One is a thing that’s got the technical name “negative Poisson’s ratio,” which means something that when you stretch it, it gets larger in all dimensions. Normally, if you take a rubber band, you stretch it one way, it gets skinnier in the other two directions. But by using origami, you can actually make a material that when you stretch it, it expands in all directions. And that has advantages in packing materials, in structural panels for aerospace and the like. There’s another Caltech group which is doing what they call DNA origami. So rather than folding sheets of things, they’re folding individual DNA molecules. Now, it looks pretty different from your traditional flapping-​bird origami, because these are long chains of molecules rather than sheets. But scientists are actually folding up little machines that can shuttle around molecules to make containers for anticancer drugs, things that can latch onto cancerous cells and kill them, that do microanalysis in situ, in the body—​really amazing things.

So that’s the state of the art in the application of origami. What’s the state of the art of the art? 

It was unchanging for hundreds of years, but when it started changing in the middle of last century, it took off. And there’s no sign of it stopping. So while things like this yellow jacket were state of the art five years ago, the state of the art is way beyond what I’ve brought here. There are pieces that require hours and hours of folding—​I can think of one example that took forty hours of solid folding, though not by me.

What was it? What did it look like?

It’s a famous design called the Eastern Dragon, by Satoshi Kamiya, one of the great Japanese folders. It’s an incredibly ornate little dragon, that’s sort of coiled up. It has claws, a dragon’s head. One of the things that’s quite remarkable is that it has probably several hundred scales, all folded from a single square of paper.

You’ve obviously constructed a really interesting life for yourself, and an unconventional career trajectory. I wonder what advice you would give to people who are just starting to figure out what they’re going to do with their lives.

Well, there’s no way I would have ever guessed or predicted where I would end up. So one bit of advice is: Don’t stress about trying to plan the rest of your life or career, because it’s going to be different no matter what you plan for. But the most important thing is—​well, two things: One is to pursue your passion and not really worry about whether there’s a practical payoff. Many things you wouldn’t have thought would have a practical payoff, like paper-​folding, turn out to have one. And things you might think would be practical might never come to pass. So the enjoyment and satisfaction you get from pursuing your passion: You know you’re going to have that. And the second piece of advice is based on one of my favorite sayings, by Louis Pasteur: “Chance favors the prepared mind.” And that just means: Be curious. Learn about lots of stuff. Innovation happens not in the mainstream areas of research—​it’s always happening at the fringes, in interdisciplinary ways, combining two fields that don’t seem as if they go together. If you learn a lot about a lot of interesting things, you’ll find those magical coincidental surprises where two fields come together and something beautiful happens. 


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