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Origami transformed

Computational origami expands the scope of paper folding

Origami is the metamorphosis of paper through folding. According to Japanese tradition, one must begin with a single sheet of square paper, and without cutting or gluing, fold that paper into its final form. The familiar origami icons – the crane and the frog – possess a certain elegance in their simplicity and abstractness.

But modern-day folders aim far beyond these basic forms. While adhering to the minimalist Japanese guidelines of a single square sheet, they fold fantastically complex shapes. Brian Chan, a PhD student in fluid dynamics at MIT, is an origami folder who designs intricate forms out of uncut paper.

“There really is nothing continuous that cannot be folded,” declares Chan, who has folded myriad different types of bugs, Japanese Manga characters, and, for the 2006 Origami USA challenge, a giant squid attacking a pirate ship.

Discovering the pattern of creases on flat paper that will give rise to a three-dimensional figure is the primary challenge in origami. Traditionally, crease patterns were designed by hand, but today, pattern-generating computer programs are redefining the boundaries of the ancient art.

Robert J. Lang, a physicist and origami theorist, has developed a computer program called TreeMaker which uses “tree theory” to find the crease pattern for the “base” of any object, which can be represented as a stick figure or “tree.” The base of the piece is a folding structure containing a flap for each future appendage, which must then be shaped to form the finished origami.

Lang claims that his program can design bases that could never be conceived using traditional methods.

“It’s now possible, with TreeMaker, to solve for origami bases that are truly more complicated than anything a person could design by hand,” he says.

In addition to creating elegant works of art, Lang’s mathematical origami theories have also had more practical applications. He has used his origami concepts to help the Lawrence Livermore Lab develop a telescope that can be shipped into space in its folded form, and then unfold in space without damaging the instrument. Lang has also helped design optimal ways to fold airbags for rapid inflation.

According to Lang’s tree theory, each appendage of a figure – like an arm or a leg, or sea monster tentacle – can be mapped onto a flat sheet of paper as a circle, where the radius of the circle is the length of the appendage. “Circle packing” – or making each circle as large as possible without overlapping others – is then used to make the most of the space on the page, because unused space reduces the size of the folded origami.

A distance between the circles, known as a “river”, then specifies the amount of space in between each appendage. Chan describes how the circles defined using tree theory then “fold up like an umbrella” to create each limb of the final figure. Folds that connect the centres of the circles become the edges of the base to be used for the finished product. With the edges defined, more complex figures can then be broken down into their basic quadrilaterals and triangles – simple shapes that are easy to produce.

Lang is not alone in his quest to understand the mathematics behind folding. There is a vibrant community of folders enamoured with the increasingly complex and beautiful forms that computers make possible. The subject also has a place in academia; Erik Demaine, a professor specializing in computational origami at MIT, offers a course titled “Geometric Folding Algorithms: Linkages, Origami, Polyhedra,” which focuses on the geometry of the folding and unfolding of complex structures. Dr. Demaine has applied his theories to other disciplines such as the complex folding structure of biological proteins.

Chan, who doesn’t use software to design his origami, sells his pieces online at