Marginal elasticity of periodic triangulated origami

Origami, the classical art of folding paper, has inspired much recent work on assembling complex 3D structures from planar sheets. Origami, and more generally hinged structures with rigid panels, where all faces are triangles have special properties due to having a bulk balance of mechanical degrees of freedom and constraints. We study two families of periodic triangulated origami structures, one based on the Miura ori and one based on a kagome-like pattern due to Ron Resch. We point out the consequences of the balance of degrees of freedom and constraints for these "metamaterial plates" and show how the elasticity can be tuned by changing the unit cell geometry.