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Programming curvature using origami tessellations


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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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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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.

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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).

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