Rigid and Flat Foldability of a Degree-Four Vertex in Origami

Rigid foldability is the property of an origami that folds continuously from an unfolded to a folded state without deformation in its facets. Although extensively researched, there exist no intrinsic conditions for the rigid foldability of a degree-four vertex, which is the simplest possible origami building block that folds nontrivially. In this paper, we derive a necessary and sufficient condition for the rigid foldability of a degree-four vertex and show that it can be reduced to a purely sufficient condition, which is equivalent to a known condition from the realm of spherical mechanisms. The implications of these conditions are discussed, which reveals the connection between rigid and flat foldability, the two most important mathematical notions in origami. In practice, this work further contributes to the design synthesis and analysis of deployable structures, in which the mechanics of degree-four vertices is omnipresent.