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Origami structures with a critical transition to bistability arising from hidden degrees of freedom

A Corrigendum to this article was published on 22 April 2015

This article has been updated


Origami is used beyond purely aesthetic pursuits to design responsive and customizable mechanical metamaterials1,2,3,4,5,6,7,8. However, a generalized physical understanding of origami remains elusive, owing to the challenge of determining whether local kinematic constraints are globally compatible and to an incomplete understanding of how the folded sheet’s material properties contribute to the overall mechanical response9,10,11,12,13,14. Here, we show that the traditional square twist, whose crease pattern has zero degrees of freedom (DOF) and therefore should not be foldable, can nevertheless be folded by accessing bending deformations that are not explicit in the crease pattern. These hidden bending DOF are separated from the crease DOF by an energy gap that gives rise to a geometrically driven critical bifurcation between mono- and bistability. Noting its potential utility for fabricating mechanical switches, we use a temperature-responsive polymer-gel version of the square twist to demonstrate hysteretic folding dynamics at the sub-millimetre scale.

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Figure 1: Schematics and photographs introducing the square twist’s essential geometric properties and mechanical characteristics.
Figure 2: Experimental strain-controlled mechanical data studying the transition between mono- and bistability in square twists.
Figure 3: Simulation results for the square twist with non-rigid facets.
Figure 4: A sub-millimetre-scale self-folding polymer-gel version of the square twist is used to verify the geometric nature of bistability in stress-controlled conditions.

Change history

  • 31 March 2015

    In the version of this Letter originally published, the authors Jesse L. Silverberg and Jun-Hee Na should have been denoted as having contributed equally to this work. This has now been corrected in the online versions of the Letter.


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The authors thank J. Mosely, U. Nguyen, B. Johnson, B. Parker and M. Schneider for artistic inspiration, as well as O. Vincent, N. Bende, C-K. Tung, S. Waitukaitis and the Cohen lab for useful discussions. We also thank F. Parish for assistance with the laser cutter. This work was funded by the National Science Foundation through award EFRI ODISSEI-1240441.

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Authors and Affiliations



J.L.S., J-H.N., R.C.H. and I.C. designed the research; J.L.S., J-H.N. and A.A.E. conducted the research and interpreted the results; B.L., T.C.H., C.D.S., R.J.L., R.C.H. and I.C. supervised the research and interpreted the results; J.L.S., J-H.N., A.A.E., T.C.H., R.J.L. and I.C. prepared the manuscript.

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Correspondence to Jesse L. Silverberg.

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The authors declare no competing financial interests.

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Silverberg, J., Na, JH., Evans, A. et al. Origami structures with a critical transition to bistability arising from hidden degrees of freedom. Nature Mater 14, 389–393 (2015).

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