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Jigsaw puzzle design of pluripotent origami

Nature Physics volume 16, pages 63–68 (2020)Cite this article

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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.

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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.

References

  1. Felton, S., Tolley, M., Demaine, E., Rus, D. & Wood, R. A method for building self-folding machines. Science 345, 644–646 (2014).

    Article  ADS  Google Scholar 

  2. Miskin, M. Z. et al. Graphene-based bimorphs for micron-sized, autonomous origami machines. Proc. Natl Acad. Sci. USA 115, 466–470 (2018).

    Article  ADS  Google Scholar 

  3. Miura, K. Method of Packaging and Deployment of Large Membranes Report No. 618 (Institute of Space and Astronautical Science, 1985).

  4. Kuribayashi, K. et al. Self-deployable origami stent grafts as a biomedical application of Ni-rich TiNi shape memory alloy foil. Mat. Sci. Eng. A 419, 131–137 (2006).

    Article  Google Scholar 

  5. Wilson, L., Pellegrino, S., & Danner, R. Origami inspired concepts for space telescopes. In 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (2013).

  6. Evgueni, T. P., Tachi, T. & Paulino, G. H. Origami tubes assembled into stiff, yet reconfigurable structures and metamaterials. Proc. Natl Acad. Sci. USA 112, 12321–12326 (2015).

    Article  ADS  Google Scholar 

  7. Schenk, M. & Guest, S. D. Geometry of Miura-folded metamaterials. Proc. Natl Acad. Sci. USA 110, 3276–3281 (2013).

    Article  ADS  Google Scholar 

  8. Silverberg, J. L., Evans, A. A., McLeod, L. & Hayward, R. C. Using origami design principles to fold reprogrammable mechanical metamaterials. Science 345, 647–650 (2014).

    Article  ADS  Google Scholar 

  9. Waitukaitis, S., Menaut, R., Chen, B. G. & van Hecke, M. Origami multistability: from single vertices to metasheets. Phys. Rev. Lett. 114, 055503 (2015).

    Article  ADS  Google Scholar 

  10. Silverberg, J. L. et al. Origami structures with a critical transition to bistability arising from hidden degrees of freedom. Nat. Mater. 14, 389–393 (2015).

    Article  ADS  Google Scholar 

  11. Chen, B. G. et al. Topological mechanics of origami and kirigami. Phys. Rev. Lett. 116, 113501 (2016).

    Google Scholar 

  12. Dudte, L. H., Vouga, E., Tachi, T. & Mahadevan, L. Programming curvature using origami tessellations. Nat. Mater. 15, 583–588 (2016).

    Article  ADS  Google Scholar 

  13. Overvelde, J. T. B., Weaver, J. C., Hoberman, C. & Bertoldi, K. Rational design of reconfigurable prismatic architected materials. Nature 541, 347–352 (2017).

    Article  ADS  Google Scholar 

  14. Bertoldi, K., Vitelli, V., Christensen, J. & van Hecke, M. Flexible mechanical metamaterials. Nat. Rev. Mat. 2, 17066 (2017).

    Article  Google Scholar 

  15. Lang, R. J. Origami Design Secrets: Mathematical Methods for an Ancient Art 2nd edn (Taylor and Francis, 2011).

  16. Ginepro, J. & Hull, T. C. Counting Miura-ori foldings. J. Int. Seq. 17, 14108 (2014).

    MathSciNet  MATH  Google Scholar 

  17. Arkin, E. M. et al. When can you fold a map? Comp. Geom. 29, 23–46 (2004).

    Article  MathSciNet  Google Scholar 

  18. Waitukaitis, S. & van Hecke, M. Origami building blocks: generic and special four-vertices. Phys. Rev. E 93, 023003 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  19. Chen, B. G. & Santangelo, C. D. Branches of triangulated origami near the unfolded state. Phys. Rev. X 8, 011034 (2018).

    Google Scholar 

  20. Evans, A. A., Silverberg, J. L. & Santangelo, C. D. Lattice mechanics of origami tessellations. Phys. Rev. E 92, 013205 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  21. Barreto, P. T. Lines meeting on a surface: the ‘Mars’ paperfolding. In Proc. 2nd International Meeting of Origami Science and Scientific Origami (ed. Miura, K.) 323–331 (Tokyo Seian University of Art and Design, 1997).

  22. Huffman, D. A. Curvature and creases: a primer on paper. IEEE Trans. Comp. C 25, 1010–1019 (1976).

    Article  Google Scholar 

  23. Tachi, T. Generalization of rigid foldable quadrilateral mesh origami. J. Int. Assoc. Shell Spat. Struct. 50, 173–179 (2009).

    Google Scholar 

  24. Kokotsakis, A. Uber bewegliche Polyeder. Math. Ann. 107, 627–647 (1933).

    Article  MathSciNet  Google Scholar 

  25. Stern, M., Pinson, M. B. & Murugan, A. The complexity of folding self-folding origami. Phys. Rev. X 7, 041070 (2017).

    Google Scholar 

  26. belcastro, S.-M. & Hull, T. C. Modeling the folding of paper into three dimensions using affine transformations. Lin. Alg. Appl. 348, 273–282 (2002).

    Article  Google Scholar 

  27. Izmestiev, I. Classification of flexible kokotsakis polyhedra with quadrangular base. Int. Math. Res. Not. 3, 715–808 (2017).

    MathSciNet  MATH  Google Scholar 

  28. Stachel, H. Flexible polyhedral surfaces with two flat poses. Symmetry 7, 774–787 (2015).

    Article  MathSciNet  Google Scholar 

  29. Demaine, E. & Orourke, J. Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Cambridge Univ. Press, 2007).

  30. He, Z. & Guest, S. D. Approximating a target surface with 1-DOF rigid origami. In Origami 7: Seventh International Meeting of Origami Science, Mathematics, and Education (ed. Lang, R.) 505–520 (Tarquin Publications, 2018).

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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.

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Authors and Affiliations

  1. Huygens-Kamerling Onnes Laboratories, University of Leiden, Leiden, the Netherlands

    Peter Dieleman, Niek Vasmel, Scott Waitukaitis & Martin van Hecke

  2. AMOLF, Amsterdam, the Netherlands

    Peter Dieleman, Scott Waitukaitis & Martin van Hecke

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  1. Peter Dieleman
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  2. Niek Vasmel
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  3. Scott Waitukaitis
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  4. Martin van Hecke
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Contributions

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.

Corresponding author

Correspondence to Scott Waitukaitis.

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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.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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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