School of Physics, Georgia Institute of Technology, Atlanta, GA 30332.
Department of Physics, Syracuse University, Syracuse, NY 13244.
Proc Natl Acad Sci U S A. 2020 Dec 1;117(48):30252-30259. doi: 10.1073/pnas.2005089117. Epub 2020 Nov 16.
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, 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.
我们研究了具有一般折痕模式的周期性折纸的零能变形。利用将这种折纸的线性折叠运动映射到承载力模式的方法，以及麦克斯韦-卡拉迪恩指标定理，我们推导出了线性折叠运动的数量和刚体模式的数量之间的关系，该关系仅取决于折纸的顶点的平均配位数。这支持了 Tachi 的最新结果，即具有三角形面的周期性折纸表现出二维刚性可折叠圆柱配置的空间。我们还通过分析计算和数值模拟发现，由于几何相容性约束，该配置空间会从平面状态发生分支，这些约束禁止有限高斯曲率。相同的计数论点导致在三角折纸中在相反波数处的空间变化模式的配对，这阻止了拓扑极化，但允许在体积中存在一组零能变形，这些变形可用于重新配置折纸。
