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鈥檚 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鈥檛 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鈥檚 Jet Propulsion Laboratory in California, and who has just finished writing a book called Origami Design Secrets. Lang is perhaps the world鈥檚 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鈥檚 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鈥檚 prayers. 鈥淣ow,鈥 he says, 鈥渨e can fold anything.鈥
Once he understood how much mathematics there is behind origami, Hoffman wasn鈥檛 surprised by the technique鈥檚 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 鈥渢echnical origami鈥 that has blossomed over the past 20 years.
This growth isn鈥檛 simply the result of advances in computer science. 鈥淎 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,鈥 says Erik Demaine, a computer scientist at the Massachusetts Institute of Technology and one of the leaders of the new school. 鈥淥nce they recognised those principles, they began building more and more complicated models.鈥 (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. 鈥淥rigami actually allows us to do simple calculus,鈥 Hull says. For example, axiom five doesn鈥檛 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. 鈥淚t may seem strange to think of an origami fold as solving an equation,鈥 Hull admits, 鈥渂ut mathematically this is exactly what鈥檚 going on.鈥
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 鈥 a programming system based on Huzita鈥檚 axioms 鈥 as part of a project to create materials that change shape on command.
Why did he use origami? 鈥淵ou can construct a wide variety of complex shapes using a few axioms, simple fixed initial conditions, and one mechanical operation 鈥 a fold,鈥 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. 鈥淥rigamists,鈥 says Nagpal, 鈥渞ecognise few, if any, limits on origami鈥檚 power.鈥
With TreeMaker and another of his programs called ReferenceFinder (see 鈥淢ake me a grasshopper鈥), Lang is pushing the boundaries of origami maths even further. The results won鈥檛 only be turning up inside car steering wheels 鈥 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鈥檚 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鈥檛 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鈥檛 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. 鈥淲e need to minimise creasing and put the creases in known places so we can compensate for their effects,鈥 says Hyde.
More than a decade ago, Hyde heard physicist and origami artist Koryo Miura of Tokyo University鈥檚 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. 鈥淭hat stuck with me,鈥 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鈥檚. It had the same property of many planes folding into a single cell, Hyde recalls. 鈥淏ut when we tried to fold it ourselves, we found we couldn鈥檛 unfold it. It was nice mathematically, but useless in the real world.鈥 Lang then helped them 鈥渢hink outside the plane鈥, 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鈥檚 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 鈥渇old-and-cut鈥 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鈥檛 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. 鈥淲e had developed our intuition to the point that we could see some principles emerging,鈥 he says. 鈥淓ventually, 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.鈥
Erik Demaine鈥檚 solution and Lang鈥檚 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. 鈥淐ommon to both is the idea of starting with a straight skeleton,鈥 says Demaine. Lang applies the ideas to convex polygons 鈥 shapes whose interior angles are all less than 180 degrees 鈥 while Demaine generalises the skeleton to more complex shapes. 鈥淣ow our idea is to combine them into one new thing,鈥 he says.
To achieve this, the two got together in Demaine鈥檚 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. 鈥淥ccasionally, we鈥檇 be using the whiteboard or writing down algorithms in words and Robert [Lang] would say, 鈥楴o, what I mean is that you fold it this way鈥 and he鈥檇 tear a sheet off the pad and make some incredible move. His visualisation for folding is truly impressive.鈥
And so is their result. 鈥淲e have been able to show that, for any surface on which you can move from one point to another without leaving the surface, it鈥檚 possible to fold any three-dimensional shape from one uncut piece of paper,鈥 says Demaine. 鈥淪o you can find, in a computationally feasible way, pretty efficient solutions that don鈥檛 have lots of wasted paper.鈥 Good news for engineers and artists alike.
Within three years, the pair expect to encode their insights in software. 鈥淭he kinds of problems we鈥檙e addressing 鈥 folding polyhedra and then folding one polyhedron to other polyhedra 鈥 are perhaps less artistically interesting, but they could be powerful in engineering applications,鈥 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. 鈥淲e might even be able to design the most efficient folding patterns for synthesised proteins in order to make specific molecules,鈥 suggests Demaine.
Yet the soul of this ancient art 鈥 or of this new engineering tool 鈥 isn鈥檛 going to be synthesised on a disc. 鈥淭he computer is a tool,鈥 Lang emphasises. 鈥淚t鈥檚 efficient for laying down a pattern or framework. But human skill will always be needed to implement the design. It鈥檚 the person that makes the difference between creating something awkward and lifeless or something elegant and beautiful.鈥
Make me a grasshopper
Robert Lang鈥檚 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. 鈥淭hat鈥, says Lang, 鈥渋s the key.鈥
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. 鈥淏eing able to plan a design with eight legs, or six legs and wings, or two sets of wings plus antennae, was almost inconceivable,鈥 says Lang.
But with TreeMaker it鈥檚 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鈥檚 insight was to view each of a design鈥檚 鈥渇laps鈥 鈥 such as each leg of a grasshopper 鈥 as occupying a circular region of a piece of paper, with the circle鈥檚 radius equal to the length of the flap. To create a figure, none of the circles can be allowed to overlap, as you can鈥檛 fold the same area of the paper to make two different things. 鈥淵ou can鈥檛 tell a computer to design an origami base with eight flaps,鈥 Lang says, 鈥渂ut you can tell it to find a way to pack eight circles with all of their centres inside a square of a given size.鈥
The design software works for any origami figure with a uniaxial base, meaning that all of the figure鈥檚 flaps lie along the stick figure鈥檚 single spine. Designs with multiple segments 鈥 such as an insect with separate head, thorax, and tail sections 鈥 require an additional element that Lang calls 鈥渞ivers鈥. These are blank channels of constant width that separate the circles.
Where do you start?
But having a pattern of creases to fold doesn鈥檛 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. 鈥淣ow what are all the different ways that you could fold any of those nine known points to any other of the nine points?鈥 Lang muses. 鈥淭hen what are all the ways you could join any of those new points to any other?鈥
The program joins points to points, lines to lines, or any combination of the two by implementing four of Huzita鈥檚 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鈥檚 location at the intersection of the folds. 鈥淚t鈥檚 almost spooky sometimes how simple a sequence can turn out to be,鈥 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 鈥渕olecules鈥 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鈥檚 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 鈥渦niversal molecule鈥 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鈥檚 website is at