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 [T. Tachi, Origami 6, 97–108 (2015)] 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.

我们考虑具有折痕模式的周期性折纸的零能变形。结合Maxwell–Calladine指数定理，使用从此类薄板的线性折叠运动到受力模式的映射，我们得出线性折叠运动的数量与刚体模式的数量之间的关系，该关系仅取决于平均配位折纸顶点的数量。这支持了Tachi [T. 大溪，折纸6，6，97–108（2015）]，其中显示了具有三角形面的周期性折纸片，其二维空间具有可刚性折叠的圆柱形状。通过分析计算和数值模拟，我们还发现由于几何相容性约束（限制了有限的高斯曲率），该配置空间从平坦状态分支出来。相同的计数论点导致在三角折纸中以相反波数的空间变化模式配对，防止了拓扑极化，但允许本体中的零能量变形族可用于重新构造折纸。