  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.
