Generalization of Rigid Foldable Quadrilateral Mesh Origami

In general, a quadrilateral mesh surface does not enable a continuous rigid motion because an overconstrained system, in which the number of constraints around degree-4 vertices (three for each vertex) exceeds the number of variables (the number of hinges), is constructed. However, it is known that the developable double corrugation surface, called Miura-ori, produces a rigid deployment mechanism. The rigid-foldability of Miura-ori is due to the singularity in its pattern, where a single vertex is repeated. We generalize the geometric condition for enabling rigid motion in general quadrilateral mesh origami without the trivial repeating symmetry. To ensure the existence of a finite motion, we derive the identity of functions from the formula for degree-4 single-vertex origami. This yields a variety of unexplored generalized shapes of quadrilateral mesh origami that preserve finite rigid-foldability in addition to developability and flat-foldability.