Hidden symmetries generate rigid folding mechanisms in periodic origami
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We consider the zero-energy deformations of periodic origami sheets with generic crease patterns. Using a mapping from the linear folding motions of such sheets to force-bearing modes in conjunction with the Maxwell–Calladine index theorem we derive a relation between the number of linear folding motions and the number of rigid body modes that depends only on the average coordination number of the origami’s vertices. This supports the recent result by Tachi [T. Tachi, Origami 6, 97–108 (2015)] which shows periodic origami sheets with triangular faces exhibit two-dimensional spaces of rigidly foldable cylindrical configurations. We also find, through analytical calculation and numerical simulation, branching of this configuration space from the flat state due to geometric compatibility constraints that prohibit finite Gaussian curvature. The same counting argument leads to pairing of spatially varying modes at opposite wavenumber in triangulated origami, preventing topological polarization but permitting a family of zero-energy deformations in the bulk that may be used to reconfigure the origami sheet.
McNerney , J , Chen , B G , Theran , L S , Santangelo , C & Rocklin , Z 2020 , ' Hidden symmetries generate rigid folding mechanisms in periodic origami ' , Proceedings of the National Academy of Sciences of the United States of America , vol. 117 , no. 48 , pp. 30252-30259 . https://doi.org/10.1073/pnas.2005089117
Proceedings of the National Academy of Sciences of the United States of America
Copyright © 2020 the Author(s). This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1073/pnas.2005089117.
DescriptionFunding: NSF Award No. PHY-1554887 (B.G.C.). NSF DMR-1822638 (C.S.). Georgia Institute of Technology President’s Fellowship and the STAMI Graduate Student Fellowship ( J.M.).
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