Rainer Hoffman had a problem. His state-of-the-art computer software was being defeated by an empty bag. Hoffman, managing director of EASi Engineering near Frankfurt, Germany, was trying to work out the best way to fold an airbag into a car’s steering column. He had to find the most efficient way to pack it into the tight space available, while ensuring it still inflated instantly. His software simulation was fine for showing how the front end of a car crumples in a crash. But a folded airbag has both wrinkles and creases in it, and wrinkles are a nightmare for mathematical models: his software simply couldn’t cope.
So Hoffman began to think about other people who might need to fold complex shapes. His quest led him to Robert Lang, an independent consultant who has worked at NASA’s Jet Propulsion Laboratory in California, and who has just finished writing a book called Origami Design Secrets. Lang is perhaps the world’s first consulting engineer in origami.
Origami, the thousand-year-old Japanese art of paper folding, is usually regarded as a cute way to make delicate models of birds and flowers. But, led by Lang, a new generation of mathematicians and computer-savvy enthusiasts is expanding the craft into dazzling new territory. In the process, origami is being recast as a powerful tool for researchers facing problems as diverse as protein folding and lofting unwieldy instruments into Earth’s orbit.
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When Lang and Hoffman finally made contact, Lang handed over a copy of TreeMaker, a software program that he had written to design origami figures of previously unmanageable complexity. It has been the answer to Hoffman’s prayers. “Now,” he says, “we can fold anything.”
Once he understood how much mathematics there is behind origami, Hoffman wasn’t surprised by the technique’s ability to untangle numerical knots. Much of the maths was put there by Lang and others, and it has led to the new school of “technical origami” that has blossomed over the past 20 years.
This growth isn’t simply the result of advances in computer science. “A few people with technical backgrounds began to see a theoretical structure behind origami: that there were some principles underlying certain kinds of complicated folding patterns,” says Erik Demaine, a computer scientist at the Massachusetts Institute of Technology and one of the leaders of the new school. “Once they recognised those principles, they began building more and more complicated models.” (Demaine is interviewed on page 4o of this issue.)
Some of those principles were encoded in 1992 by the Italian-Japanese mathematician Humiaki Huzita in a set of six axioms (see Diagram). The axioms appear simple, but according to Tom Hull, a mathematician at Merrimack College in Massachusetts, they contain hidden profundities. “Origami actually allows us to do simple calculus,” Hull says. For example, axiom five doesn’t simply describe a particular fold: repeated use of it plots a parabola and defines a series of lines tangential to that parabola. Parabolas are defined by quadratic equations: that is, equations that include squared terms and no higher powers. “It may seem strange to think of an origami fold as solving an equation,” Hull admits, “but mathematically this is exactly what’s going on.”
The closer you look, the deeper the links between origami and mathematics become. In 2000, MIT graduate student Radhika Nagpal used something called origami shape language – a programming system based on Huzita’s axioms – as part of a project to create materials that change shape on command.
Why did he use origami? “You can construct a wide variety of complex shapes using a few axioms, simple fixed initial conditions, and one mechanical operation – a fold,” says Nagpal, now a computer scientist at MIT. Translated into maths, origami principles enable you to program the creation of any shape in plane geometry and, as Hull points out, solve problems in analytic geometry, such as cube doubling and angle trisections. “Origamists,” says Nagpal, “recognise few, if any, limits on origami’s power.”
With TreeMaker and another of his programs called ReferenceFinder (see “Make me a grasshopper”), Lang is pushing the boundaries of origami maths even further. The results won’t only be turning up inside car steering wheels – they could even be unfolding in space. Physicist Rod Hyde works on the Eyeglass project, a venture to make huge, lightweight telescopes that will orbit Earth. Hyde’s group at Lawrence Livermore National Laboratory is testing ways to make lenses from plastic film.
It is simply not possible to fit such large telescopes into spacecraft. But Hyde can’t simply cut his huge lens up, because when it is reassembled all the pieces need to come back into precise registration so the lens can form distortion-free images. And he can’t simply fold it anyhow: creases are bad news too. Even if there were a way to smooth the material after unfolding it in space, the creases would distort the images captured by the lens. “We need to minimise creasing and put the creases in known places so we can compensate for their effects,” says Hyde.
More than a decade ago, Hyde heard physicist and origami artist Koryo Miura of Tokyo University’s Institute of Space and Aeronautical Science explain how to fold a membrane as a series of congruent parallelograms. Instead of layering them on top of each other like pages in a newspaper, he folded sections to create a ridged structure that opens easily when tugged at one corner, much like a concertina. “That stuck with me,” says Hyde. But the structure that Miura described was square. Hyde needed something round.
Then he met Lang, who pointed him to some Japanese work similar to Miura’s. It had the same property of many planes folding into a single cell, Hyde recalls. “But when we tried to fold it ourselves, we found we couldn’t unfold it. It was nice mathematically, but useless in the real world.” Lang then helped them “think outside the plane”, as Hyde puts it. The result is a design that can fold a 5-metre membrane of plastic film into a box 1.5 metres across, just the right size to fit in the space shuttle’s hold.
Lang is now pushing origami to reveal even deeper secrets, in collaboration with Demaine. As Lang was perfecting TreeMaker, Demaine was trying to solve another long-standing origami puzzle known as the “fold-and-cut” problem. He was trying to find out how many different shapes he could make by folding a piece of paper and then making one straight cut. Can you create any shape or are there some things that just can’t be made that way?
Over two years, Demaine, his father Martin, and Anna Lubiw of the computer science department at the University of Waterloo in Ontario folded and cut increasingly intricate shapes. “We had developed our intuition to the point that we could see some principles emerging,” he says. “Eventually, we were able to formulate a theorem that if you choose your folds and your cut precisely, you can make the silhouette of a bird, a heart or any polygon.”
Erik Demaine’s solution and Lang’s software turned out to complement each other. Both show that it is possible to create designs previously thought impossible, but each takes a distinct route to a slightly different result. “Common to both is the idea of starting with a straight skeleton,” says Demaine. Lang applies the ideas to convex polygons – shapes whose interior angles are all less than 180 degrees – while Demaine generalises the skeleton to more complex shapes. “Now our idea is to combine them into one new thing,” he says.
To achieve this, the two got together in Demaine’s office at MIT last July. For a week, Demaine recalls, they sat together, folding paper and talking nodes, edges, figures of merit, bisectors and algebraic functions. “Occasionally, we’d be using the whiteboard or writing down algorithms in words and Robert [Lang] would say, ‘No, what I mean is that you fold it this way’ and he’d tear a sheet off the pad and make some incredible move. His visualisation for folding is truly impressive.”
And so is their result. “We have been able to show that, for any surface on which you can move from one point to another without leaving the surface, it’s possible to fold any three-dimensional shape from one uncut piece of paper,” says Demaine. “So you can find, in a computationally feasible way, pretty efficient solutions that don’t have lots of wasted paper.” Good news for engineers and artists alike.
Within three years, the pair expect to encode their insights in software. “The kinds of problems we’re addressing – folding polyhedra and then folding one polyhedron to other polyhedra – are perhaps less artistically interesting, but they could be powerful in engineering applications,” Demaine says. It would make possible a more flexible approach to designing complex shapes, especially for things like airbags, solar sails or parachutes that need to be compacted, or nested or collapsed, or for self-assembly, to build complex structures from small units. “We might even be able to design the most efficient folding patterns for synthesised proteins in order to make specific molecules,” suggests Demaine.
Yet the soul of this ancient art – or of this new engineering tool – isn’t going to be synthesised on a disc. “The computer is a tool,” Lang emphasises. “It’s efficient for laying down a pattern or framework. But human skill will always be needed to implement the design. It’s the person that makes the difference between creating something awkward and lifeless or something elegant and beautiful.”
Make me a grasshopper
Robert Lang’s TreeMaker program laysout and prints a pattern of creases that an adventurous folder can use to create a fantastic variety of shapes. Lang has written a second program, called ReferenceFinder, that shows which folds to make and in which order to make them. It also pinpoints exactly where on the paper to make the first few folds from which all others follow. “That”, says Lang, “is the key.”
Artists and authors of origami pattern books are already using his programs to create pieces vastly more complex than ever before. Twenty years ago, for example, insects were considered almost impossible to fold. “Being able to plan a design with eight legs, or six legs and wings, or two sets of wings plus antennae, was almost inconceivable,” says Lang.
But with TreeMaker it’s relatively easy. Say you want to create a grasshopper. First, you use the software to draw a stick-figure representation of the shape you want to fold: two long lines for the back legs, a medium line representing the abdomen, two short lines for the thorax and head, and more short lines for the other legs and antennae. Then you specify how big you want the body to be. Finally the program computes and prints a pattern of folds, using the least amount of material, that the artist or engineer should follow to create the figure.
Circles of influence
Lang’s insight was to view each of a design’s “flaps” – such as each leg of a grasshopper – as occupying a circular region of a piece of paper, with the circle’s radius equal to the length of the flap. To create a figure, none of the circles can be allowed to overlap, as you can’t fold the same area of the paper to make two different things. “You can’t tell a computer to design an origami base with eight flaps,” Lang says, “but you can tell it to find a way to pack eight circles with all of their centres inside a square of a given size.”
The design software works for any origami figure with a uniaxial base, meaning that all of the figure’s flaps lie along the stick figure’s single spine. Designs with multiple segments – such as an insect with separate head, thorax, and tail sections – require an additional element that Lang calls “rivers”. These are blank channels of constant width that separate the circles.
Where do you start?
But having a pattern of creases to fold doesn’t do you a lot of good unless you know where on the paper to start folding and in which order to fold the creases. If you start folding anywhere but at the right place, you can wind up with too little material to make some part of the figure or too much left over in some other part. TreeMaker finds that correct spot automatically and prints out the creases on a sheet of paper. And for purists who insist on using a blank sheet of paper instead of a computer-generated blueprint, ReferenceFinder uses folding alone to locate the right starting point.
It works like this: there are only four points you can locate precisely on a blank sheet of paper: the four corners. If you fold the sheet in half horizontally and vertically, you have five additional points: the four midpoints of the edges, and the centre of the paper where all the folds meet. “Now what are all the different ways that you could fold any of those nine known points to any other of the nine points?” Lang muses. “Then what are all the ways you could join any of those new points to any other?”
The program joins points to points, lines to lines, or any combination of the two by implementing four of Huzita’s six axioms. It turns out that by making just five or six folds, you can specify the position of more than 3000 creases and 20,000 points.
Choose any spot on a sheet of paper and the program searches through about 400,000 points to find the closest match and shows you the sequence of folds that will identify the point’s location at the intersection of the folds. “It’s almost spooky sometimes how simple a sequence can turn out to be,” says Lang.
The pattern of major creases that the software lays out forms a pattern of large polygons. The second stage of design is to fill in those polygons with all of the creases that make up the details of the design, the details that distinguish a cat from a lion, for example. And this is where “molecules” come in.
Pattern libraries
Toshiyuki Meguro, a Japanese biochemist and origami artist, coined the term to describe the constellation of smaller folds that lie inside the big polygons and make up the design’s structural details. He, Lang and others have managed to lay out a library of molecule patterns that can be used to fill in any shape of polygon mapped by large creases. And now Lang has created a “universal molecule” that designers can use to map the crease pattern for any polygon, no matter how many sides it has.
- Origami Design Secrets: Mathematical methods for an ancient art by Robert Lang, to be published in 2003
- Tom Hull’s website is at