In this chapter, tools from analysis are brought to bear on flat foldings of high-dimensional Euclidean space. The exposition follows the work of Dacorogna, Marcellini, and Paolini from 2008, who discovered that high-dimensional flat folding maps, which they call rigid maps, can be solutions to certain Dirichlet partial differential equations. This approach offers a different proof of the Recovery Theorem from Lawrence and Spingarn (1989), and the folding maps that result from Dirichlet problems can sometimes have crease patterns that exhibit interesting self-similarity.

The final chapter considers more theoretical aspects of rigid origami.The first section outlines a proof that deciding whether or not an origami crease pattern can be rigidly folded from the unfolded state using some subset of the creases is NP-hard.Then configuration spaces of rigid origami crease patterns are discussed in more depth than in the previous chapter, including a proof that the germ of single-vertex rigid origami configuration spaces is isomorphic to the germ of a quadratic form.Examples of disconnected rigid origami configuration spaces are also included.The chapter, and book, ends with an introduction to the theory of self-folding, where we imagine that a crease pattern is rigidly folded using actuators on the creases, and we wish for these actuators to fold the crease pattern to a target state and not to some other rigid origami state.The aim is to characterize when simple actuating forces can do this, and we present the current theory behind this as well as its limitations.

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.