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Helical Miura Origami

We characterize the phase-space of all Helical Miura Origami. These structures are obtained by taking a partially folded Miura parallelogram as the unit cell, applying a generic helical or rod group to the cell, and characterizing all the parameters that lead to a globally compatible origami structure. When such compatibility is achieved, the result is cylindrical-type origami that can be manufactured from a suitably designed flat tessellation and "rolled-up" by a rigidly foldable motion into a cylinder. We find that the closed Helical Miura Origami are generically rigid to deformations that preserve cylindrical symmetry, but multistable. We are inspired by the ways atomic structures deform [1] to develop two broad strategies for reconfigurability: motion by slip, which involves relaxing the closure condition; and motion by phase transformation, which exploits multistability. Taken together, these results provide a comprehensive description of the phase-space of cylindrical origami, as well as quantitative design guidance for their use as actuators or metamaterials that exploit twist, axial extension, radial expansion, and symmetry.