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. The kinematics of such structures depends crucially on the sign of the Gaussian curvature. The configuration space of non-intersecting, 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 remain connected. This provides a new mechanism by which the mechanics and folding of an origami structure could be controlled.

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