  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.
