Branches of Triangulated Origami Near the Unfolded State

Origami structures are characterized by a network of folds and vertices joining unbendable plates. For applications to mechanical design and self-folding structures, it is essential to understand the interplay between the set of folds in the unfolded origami and the possible 3D folded configurations. When deforming a structure that has been folded, one can often linearize the geometric constraints, but the degeneracy of the unfolded state makes a linear approach impossible there. We derive a theory for the second-order infinitesimal rigidity of an initially unfolded triangulated origami structure and use it to study the set of nearly unfolded configurations of origami with four boundary vertices. We find that locally, this set consists of a number of distinct "branches" which intersect at the unfolded state, and that the number of these branches is exponential in the number of vertices. We find numerical and analytical evidence that suggests that the branches are characterized by choosing each internal vertex to either "pop up" or "pop down." The large number of pathways along which one can fold an initially unfolded origami structure strongly indicates that a generic structure is likely to become trapped in a "misfolded" state. Thus, new techniques for creating self-folding origami are likely necessary; controlling the popping state of the vertices may be one possibility.