Origami is rapidly transforming the design of robots(1,2), deployable structures(3-6) and metamaterials(7-14). However, as foldability requires a large number of complex compatibility conditions that are difficult to satisfy, the design of crease patterns is limited to heuristics and computer optimization. Here we introduce a systematic strategy that enables intuitive and effective design of complex crease patterns that are guaranteed to fold. First, we exploit symmetries to construct 140 distinct foldable motifs, and represent these as jigsaw puzzle pieces. We then show that when these pieces are fitted together they encode foldable crease patterns. This maps origami design to solving combinatorial problems, which allows us to systematically create, count and classify a vast number of crease patterns. We show that all of these crease patterns are pluripotent-capable of folding into multiple shapes-and solve exactly for the number of possible shapes for each pattern. Finally, we employ our framework to rationally design a crease pattern that folds into two independently defined target shapes, and fabricate such pluripotent origami. Our results provide physicists, mathematicians and engineers with a powerful new design strategy.
The crease patterns for origami-based mechanical metamaterials can fold into myriad 3D shapes, but predicting foldability is no simple task. A framework for designing foldable patterns offers a neat alternative to extensive computer optimization.