Topological transitions in the configuration space of non-Euclidean origami

Origami structures have been proposed as a means of creating three-dimensional structures from the micro- to the macroscale and as a means of fabricating mechanical metamaterials. The design of such structures requires a deep understanding of the kinematics of origami fold patterns. Here we study the configurations of non-Euclidean origami, folding structures with Gaussian curvature concentrated on the vertices, for arbitrary origami fold patterns. The kinematics of such structures depends crucially on the sign of the Gaussian curvature. As an application of our general results, we show that the configuration space of nonintersecting, oriented vertices with positive Gaussian curvature decomposes into disconnected subspaces; there is no pathway between them without tearing the origami. In contrast, the configuration space of negative Gaussian curvature vertices remains connected. This provides a new, and only partially explored, mechanism by which the mechanics and folding of an origami structure could be controlled.