Geometric compatibility constraints dictate the mechanical response of soft systems that can be utilized for the design of mechanical metamaterials such as the negative Poisson ratio Miura-ori origami crease pattern. We examine the broad family of cr
ease patterns composed of unit cells with four generic parallelogram faces, expanding upon the family of Morph patterns, and characterize the familys low-energy modes via a permutation symmetry between vertices. We map these modes to the resulting strains and curvatures at the intercellular level where the same symmetries elucidate a geometric relationship between the strains of the systems rigid planar mode and the curvatures of its semi-rigid bend mode. Our formalism for the analysis of low-energy modes generalizes to arbitrary numbers of quadrilateral---not necessarily parallelogram---faces where symmetries may play an important role in the design of origami metamaterials.
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 origamis 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.
