At the Symposium on Computational Geometry in July, Erik Demaine and Tomohiro Tachi of the University of Tokyo will announce the completion of a quest that began with a 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.
Researchers have created a universal algorithm for folding origami shapes that guarantees a minimum number of seams. Image: Christine Daniloff/MIT
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.
“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.”