## Paper folding patterns to produce any 3 D structure.

In a 1999 paper, Erik Demaine — now an MIT professor of electrical engineering and computer science, but then an 18-year-old PhD student at the University of Waterloo, in Canada — described an algorithm that could determine how to fold a piece of paper into any conceivable 3-D shape.

It was a milestone paper in the field of computational origami, but the algorithm didn’t yield very practical folding patterns. Essentially, it took a very long strip of paper and wound it into the desired shape. The resulting structures tended to have lots of seams where the strip doubled back on itself, so they weren’t very sturdy.

At the Symposium on Computational Geometry in July, Demaine and Tomohiro Tachi of the University of Tokyo will announce the completion of a quest that began with that 1999 paper: a universal algorithm for folding origami shapes that guarantees a minimum number of seams.

“In 1999, we proved that you could fold any polyhedron, but the way that we showed how to do it was very inefficient,” Demaine says. “It’s efficient if your initial piece of paper is super-long and skinny. But if you were going to start with a square piece of paper, then that old method would basically fold the square paper down to a thin strip, wasting almost all the material. The new result promises to be much more efficient. It’s a totally different strategy for thinking about how to make a polyhedron.”

Demaine and Tachi are also working to implement the algorithm in a new version of Origamizer, the free software for generating origami crease patterns whose first version Tachi released in 2008.

Maintaining boundaries

The researchers’ algorithm designs crease patterns for producing any polyhedron — that is, a 3-D surface made up of many flat facets. Computer graphics software, for instance, models 3-D objects as polyhedra consisting of many tiny triangles. “Any curved shape you could approximate with lots of little flat sides,” Demaine explains.

Technically speaking, the guarantee that the folding will involve the minimum number of seams means that it preserves the “boundaries” of the original piece of paper. Suppose, for instance, that you have a circular piece of paper and want to fold it into a cup. Leaving a smaller circle at the center of the piece of paper flat, you could bunch the sides together in a pleated pattern; in fact, some water-cooler cups are manufactured on this exact design.

In this case, the boundary of the cup — its rim — is the same as that of the unfolded circle — its outer edge. The same would not be true with the folding produced by Demaine and his colleagues’ earlier algorithm. There, the cup would consist of a thin strip of paper wrapped round and round in a coil — and it probably wouldn’t hold water.

“The new algorithm is supposed to give you much better, more practical foldings,” Demaine says. “We don’t know how to quantify that mathematically, exactly, other than it seems to work much better in practice. But we do have one mathematical property that nicely distinguishes the two methods. The new method keeps the boundary of the original piece of paper on the boundary of the surface you’re trying to make. We call this watertightness.”

A closed surface — such as a sphere — doesn’t have a boundary, so an origami approximation of it will require a seam where boundaries meet. But “the user gets to choose where to put that boundary,” Demaine says. “You can’t get an entire closed surface to be watertight, because the boundary has to be somewhere, but you get to choose where that is.”