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Programming curvature using origami tessellations

Nature Materials volume 15, pages 583–588 (2016)Cite this article

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Abstract

Origami describes rules for creating folded structures from patterns on a flat sheet, but does not prescribe how patterns can be designed to fit target shapes. Here, starting from the simplest periodic origami pattern that yields one-degree-of-freedom collapsible structures—we show that scale-independent elementary geometric constructions and constrained optimization algorithms can be used to determine spatially modulated patterns that yield approximations to given surfaces of constant or varying curvature. Paper models confirm the feasibility of our calculations. We also assess the difficulty of realizing these geometric structures by quantifying the energetic barrier that separates the metastable flat and folded states. Moreover, we characterize the trade-off between the accuracy to which the pattern conforms to the target surface, and the effort associated with creating finer folds. Our approach enables the tailoring of origami patterns to drape complex surfaces independent of absolute scale, as well as the quantification of the energetic and material cost of doing so.

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Figure 1: Geometry of generalized Miura-ori.
Figure 2: Optimal calculated origami tessellations and their physical paper analogues.
Figure 3: Foldability.
Figure 4: Accuracy–effort trade-off in origami tessellations.

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Acknowledgements

We thank the Harvard Microrobotics Lab for help with laser cutting; J. Weaver and O. Ahanotu for help with measuring the stress-strain behaviour of origami hypars; and the Harvard MRSEC DMR 14-20570, NSF/JSPS EAPSI 2014 (L.H.D.), NSF DMS-1304211 (E.V.), Japan Science and Technology Agency Presto (T.T.) and the MacArthur Foundation (L.M.) for partial financial support.

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Authors and Affiliations

  1. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

    Levi H. Dudte, Etienne Vouga & L. Mahadevan

  2. Graduate School of Arts and Sciences, University of Tokyo, Tokyo 153-8902, Japan

    Tomohiro Tachi

  3. Departments of Physics, and Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138, USA

    L. Mahadevan

  4. Wyss Institute for Bio-Inspired Engineering, Harvard University, Cambridge, Massachusetts 02138, USA

    L. Mahadevan

  5. Kavli Institute for Nanobio Science and Technology, Harvard University, Cambridge, Massachusetts 02138, USA

    L. Mahadevan

Authors
  1. Levi H. Dudte
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  2. Etienne Vouga
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  3. Tomohiro Tachi
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  4. L. Mahadevan
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Contributions

L.H.D., E.V. and L.M. conceived and designed the research, with later contributions from T.T.; L.H.D. conducted the simulations and built the models; L.H.D., E.V. and L.M. analysed the results and wrote the manuscript.

Corresponding author

Correspondence to L. Mahadevan.

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Competing interests

L.H.D., E.V. and L.M. are co-inventors of the surface-fitting algorithm and design method, patent-pending.

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Dudte, L., Vouga, E., Tachi, T. et al. Programming curvature using origami tessellations. Nature Mater 15, 583–588 (2016). https://doi.org/10.1038/nmat4540

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