  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.
