Pairing global symmetries with folding mechanics to transform all periodically triangulated origami

Thin sheets restricted to folding at designated creases, as in the traditional Japanese art of origami, have been engineered to deploy devices from the atomic to the macroscopic scale. However, the relation between the crease pattern and the paths to accessible structures is highly nontrivial. We investigate the entire class of periodically triangulated origami, revealing a hidden symmetry between global motions and linear folding mechanisms. Such periodic patterns always admit a two-dimensional manifold of cylindrical configurations as previously shown by Tomohiro Tachi. Adding a single quadrilateral face to the unit cell restricts the system to a single degree of freedom without fine-tuning the geometry. By transforming along these paths, we can change the mechanical response at the boundary. Our analysis can be extended to similar systems with balanced constraints and degrees of freedom such as kirigami, continuum sheets, and magnetic systems.