This chapter takes the concept of rigid origami and puts it in motion, studying how a crease pattern flexes from the unfolded state to a continuum of rigid origami states.The treatment presented starts with the more general theory of flexible polyhedral surfaces, then moves to the special case of origami.The configuration space of the rigid foldings of a crease pattern is introduced, and the tools of reciprocal-parallel and reciprocal diagrams are used to establish conditions for infinitesimal and second-order rigid foldability.This is used to prove that a single-vertex origami crease pattern has a rigid folding from the unfolded state if and only if it has a nontrivial zero-area reciprocal diagram.These results are then used to establish equations for the folding angles of a degree-4 flat-foldable vertex that are linear when parameterized by the tangent of half the folding angles, also known as the Weierstrass transformation.An intrinsic condition for an origami vertex crease pattern to be rigidly foldable from the unfolded state is also given.

Rigid origami describes origami where each face of the crease pattern is flat, as if made from stiff metal.Modeling rigid origami with matrices allows one to describe materials that have been folded into a three-dimensional shape, as opposed to flat origami.This chapter describes this matrix model and proves its key features.In addition, a generalization of Maekawa’s Theorem for three-dimensional rigid origami is introduced, as is modeling rigid origami with the Gauss map from differential geometry.The latter turns out to be a useful tool for the remainder of the book.