The Art and Science of Paper Folding

A physicist with more than 40 patents to his credit would seem to have his career clearly mapped out. But Robert J. Lang’s first love is…folding paper. One of the world’s foremost artists in origami—Japanese paper folding—Lang creates creatures of such realism and complexity that it seems impossible that each is composed of a single sheet of paper, no cuts, no glue.

Inspecting Lang’s eight-inch-tall ibex, for instance, you can see its beard, ears, horns, even its cloven hooves. His grizzly bear has teeth. His insects—Lang’s favorites—have fat bodies, twiggy legs, antennae, sometimes even spread wings.

And some of Lang’s origami creations are life size, like his eight-piece orchestra commissioned by a European paper company, or the Pteranodon with a 14-foot wing span created from a single, four-meter-square piece of paper. It flies on permanent display at the Redpath Museum of Natural History in Montreal, Canada.

No wonder Lang is swamped with commissions and commercial projects. His are not your children’s flapping birds.
Putting Math to the Fold

In his pleasant garden studio, Lang uses his MacBook Pro, TreeMaker and ReferenceFinder—two freeware programs he created—and Wolfram’s Mathematica to conceptualize his creations and and map them out for rendering in paper. On a shelf above a 30-inch Apple Cinema display connected to his laptop, a zoo of origami critters stands watch.

Lang, a lanky scientist with a boyish face, salt-and-pepper beard and the inevitable Birkenstocks, says “The cool thing about origami is that it is a very mathematical art. In many arts, there’s pure artistic skill. In origami, it’s almost half and half. You can do things with pure art, you can do things with pure math, but if you put them together, you get far more satisfying results than either one alone.”

Applying mathematical principles to origami, he adds, has enormously advanced the art. Though origami is several hundred years old, Lang explains, it has been limited to simple creations such as paper cranes or boats. “The modern art form,” he says, “was born in the 20th century when a Japanese artist named Akira Yoshizawa started creating new figures of artistic beauty that inspired other origami artists to expand their horizons.” Yoshizawa also created an instructional language that then enabled them to share their creations and build off of each others’ work.

Still, the idea of creating a fully detailed insect—say, a dragonfly with all its legs—from a single sheet of paper was often dismissed as an impossibility. But the art has advanced enormously; “In fact,” says Lang, “now, not only can we do dragonflies, but mosquitoes and spiders and even centipedes.”
Computer-Aided Origami

Lang acknowledges that one of the watersheds in the development of origami as an art form came from his own work, which includes his software contributions: TreeMaker and ReferenceFinder. These two freeware programs convert simple stick figures into full-blown origami crease patterns and even detail portions of the folding sequence for artists to follow.

Using TreeMaker, for example, Lang draws a stick figure of an animal—say the legs, body, head, antlers, ears of a deer—on his Mac. Then he enters the desired length of each stick. TreeMaker constructs a set of equations—in this case about 200—to describe the graph completely, then performs an optimization and analysis and creates the crease pattern for the deer. ReferenceFinder analyzes individual points and lines in the crease pattern and spits out the folding sequence for those elements.

“If you fold the crease pattern, you get a shape, called a “base,” that is a collection of points. You then apply shaping to make the points thinner and turn them the right direction,” Lang says, “and you get the folded deer that you set out to create.”

Lang began developing TreeMaker using THINK’s Lightspeed Pascal on his Mac SE/20 and then migrated it to subsequent machines and languages as the Mac and programming languages evolved. Today he uses Xcode for his primary programming environment and C++ for his programs. “I started on a Mac,” Lang says, “and while I’ve used other OSs in my career, I love Mac. On the Mac, I can shift effortlessly and seamlessly from program to program, whether it’s designing a figure, analyzing its underlying mathematics, creating folding instructions for a book, or documenting it on my website. The Mac simply becomes an extension of my hands and mind.

“But probably the biggest reason I’ve stayed using tools on the Mac is that, in my origami work, I use a lot of programs. There’s no giant origami program. You have to use lots of programs for bits and pieces of it. And Mac programs just cooperate with each other.”
Folding Paper for Fun and Profit

For Lang, every day is different. He creates origami insects for fun, and once was commissioned to make one for an entomologist, but he confesses “there’s not much commercial demand for insects.”

The call for Lang’s commercial work comes from advertising agencies commissioning interesting images for ads, artworks for private individuals, and communities that commission bronze versions of his origami for permanent display. What Lang considers remarkable, however, is that computational origami has a surprisingly wide range of practical applications—from space telescopes and automotive applications to medicine and consumer electronics.

For a project such as the entomologist’s favorite insect, Lang begins by taking measurements from photographs: How long are the legs? How are they connected? He measure distances manually so he knows exactly how long each leg should be. In many cases he shoots his own pictures in a corner of his studio and pulls them into iPhoto, where he organizes images by subject or job for his website and client presentations.

Then Lang fires up TreeMaker and sketches a stick framework for the insect. “The first thing,” Lang says, “is to get a sense of how to allocate paper on the page and to decide whether I want the symmetries of the creases to be square, hexagon or irregularly oriented. One of those is going to be a more natural fit for the subject, and I want to use the paper efficiently.”

He works on the 30-in Cinema Display to handle the many windows he has open at one time. “I love this display,” he says. “It was my little splurge present to myself a few months ago, but it has more than paid for itself in productivity. I can show many windows that show lots of code; I can put five different projects visible at once and cut and paste and move things around. I just hadn’t realized how much time I was spending scrolling around or hiding windows in order to accommodate all of my projects and programs.”

Working in TreeMaker on his Mac, Lang can set up and specify all of the relationships for the patterns in less than 15 minutes. “I can take that exact pattern and start folding,” he says.

For other projects, Lang uses the TreeMaker pattern as a starting point. “I might refine the pattern in TreeMaker,” he says, “or I get out pencil and graph paper and sketch the pattern to adjust dimensions or proportions. Or—if the subject is really complicated—I’ll do the same drawing in Macromedia Freehand so I can precisely select shapes, move them around, and adjust their positions. That way I end up with a pattern where all the paper is used, all the parts of the insect have a corresponding region of paper, and the creases from the legs line up perfectly with the creases from the body and each other.”

For small shapes, Lang prints out the pattern and tests it. If he’s transferring a pattern to a large sheet of paper, such as the four-meter square for the Pteranodon, he exports his Freehand illustration to a generic format such as eps and imports it into Mathematica.