New wrinkle on origami turns designing folding structures into child’s play

Most people associate origami with colorful cranes and decorative frogs, but the ancient Asian art of folding paper may be a whole lot more useful than that. Scientists have used it to make tiny robots and other self-folding 3D devices, for example. Now, a team of soft-matter physicists has invented a method for designing origami by essentially assembling puzzle pieces that encode the various points or vertices where folds meet. The approach could make designing folding robots much easier.

“It’s a big advance, and I’m pretty excited about it,” says Christian Santangelo, a theoretical physicist at Syracuse University in New York who was not involved in the work.

Wadding up a sheet of paper is easy; making it fold is much harder. Imagine you dot the sheet with random points and connect them with straight lines to create quadrilaterals—shapes with four sides, potentially all of different lengths. That pattern is a potential origami, with each line representing a foldable crease. Meanwhile, each point represents a vertex at which four creases meet, the minimum needed for folding. However, most of those random patterns won’t fold. If fact, finding one that will is among the hardest computational problems, says Martin van Hecke, a physicist at Leiden University in the Netherlands and the research institute AMOLF in Amsterdam. “The chances are basically zero” that any random pattern will fold, he says.

So van Hecke, Scott Waitukaitis, a physicist at the Institute of Science and Technology Austria in Klosterneuburg, and colleagues invented a way to generate origami that are sure to fold. They started with a single vertex defined by the four angles between its creases. Then they generated three related vertices by reversing the order of the angles; replacing each angle with a number given by subtracting the original from 180°; or applying both steps. Finally, the scientists created quadrilaterals with different combinations of vertices. Of the more than 65,000 possible quadrilaterals, only 140 of them would fold, the researchers found.

However, from that lexicon of foldable quadrilaterals, the researchers could assemble much larger origamis. The quadrilaterals fit together only in certain combinations—for example, so that the sum of the four angles at any vertex equaled 360°, and so that three of the creases fold up and one folded down or vice versa. The researchers found that they could represent quadrilaterals as colored, basically square puzzle pieces that fit together only in certain ways and according to color rules. Devising an origami then becomes as easy as assembling puzzle pieces, the researchers report today in Nature Physics. “This allows us to design a pattern that’s guaranteed to fold,” Waitukaitis says.

But the team went even further. It classified the puzzle pieces into eight categories that would reveal how many different ways an origami would fold. That’s important, Santangelo says, because if a structure folds in too many ways then its final shape becomes difficult to control. In the new work, some classes produce structures that fold in just two ways. What’s more, the different classes produce origamis with different curvatures. Mixing classes, Waitukaitis and van Hecke designed a 36–by 36–puzzle piece origami that could switch between the shapes of the Greek letters alpha and omega (see video). They then fashioned the origami out of a plastic sheet to prove it works.

Scientists already have computer algorithms that aid design by searching for legitimate origami patterns, Santangelo notes, but the new work is promising because those programs are relatively inefficient. Still, the puzzle origami approach may not be quite as simple as it at first appears, he says. “It’s elegant,” Santangelo says, “but personally I find the rules hard to understand.”

To make the whole thing more concrete, Waitukaitis and van Hecke actually made the puzzle pieces they describe in the paper. Van Hecke says he even uses them during talks to help explain what they’re doing. “We just thought that if we ever had to explain this to average humans this would be a good way to do it.”