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We consider the zero-energy deformations of periodic origami sheets withgeneric crease patterns. Using a mapping from the linear folding motions ofsuch sheets to force-bearing modes in conjunction with the Maxwell-Calladineindex theorem we derive a relation between the number of linear folding motionsand the number of rigid body modes that depends only on the averagecoordination number of the origami's vertices. This supports the recent resultby Tachi which shows periodic origami sheets with triangular faces exhibittwo-dimensional spaces of rigidly foldable cylindrical configurations. We alsofind, through analytical calculation and numerical simulation, branching ofthis configuration space from the flat state due to geometric compatibilityconstraints that prohibit finite Gaussian curvature. The same counting argumentleads to pairing of spatially varying modes at opposite wavenumber intriangulated origami, preventing topological polarization but permitting afamily of zero energy deformations in the bulk that may be used to reconfigurethe origami sheet.

Hidden symmetries generate rigid folding mechanisms in periodic origami