Hidden symmetries generate rigid folding mechanisms in periodic origami

Abstract

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 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.

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• Omnimodal topological polarization of bilayer networks: analysis in the Maxwell limit and experiments on a 3D-printed prototype
• Duality,Hidden Symmetry and Dynamic Isomerism in 2D Hinge Structures
• Chirality Effects in Molecular Chainmail
• Symmetry-driven Phononic Metamaterials
• Prospecting for Pluripotency in Metamaterial Design
• Nonlinear Kinematics of Recursive Origami Inspired by the Spidron
• Lagrangian approach to origami vertex analysis: Kinematics