Jigsaw puzzle design of pluripotent origami

Abstract

Origami is rapidly transforming the design of robots1,2, deployable structures3,4,5,6 and metamaterials7,8,9,10,11,12,13,14. 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.

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Fig. 1: Rigidly foldable tiles.
Fig. 2: Jigsaw origami tilings.
Fig. 3: Rational design of pluripotent crease patterns.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon request.

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Acknowledgements

We thank B.G.-g. Chen, C. Coulais, Y. Shokef and P.-R. ten Wolde for fruitful discussions, D. Ursem and R. Struik for technical support, and the Netherlands Organization for Scientific Research for funding through grants NWO 680-47-609, NWO-680-47-453 and FOM-12CMA02.

Author information

M.v.H. conceived of the project. P.D. carried out the experiments. P.D., N.V., S.W. and M.v.H. developed the theoretical framework. P.D., S.W. and M.v.H., wrote the manuscript.

Correspondence to Scott Waitukaitis.

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Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Zeyuan He and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary text, Tables 1–3, Figs. 1–21 and refs. 1 and 2.

Supplementary Video 1

A video showing each of the 14 possible folding branches for a 3 × 3 3D-printed, class 2 crease pattern. For details on construction, see the Supplementary Information.

Supplementary Video 2

A video showing each of the 14 possible folding branches for a second 3D-printed, class 2 crease pattern. For details on construction, see the Supplementary Information.

Supplementary Video 3

A video showing a single class 1 crease pattern that folds into two predetermined shapes, the Greek letters α and ω. The 36 × 36-tile pattern is designed as described in the main text and Methods. Folding simulations were made in Blender and with the aid of the Rigid Origami Simulator by Tachi.

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Dieleman, P., Vasmel, N., Waitukaitis, S. et al. Jigsaw puzzle design of pluripotent origami. Nat. Phys. 16, 63–68 (2020). https://doi.org/10.1038/s41567-019-0677-3

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