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.

Chapter 10 introduces a more abstract approach to studying origami by considering how we might fold a Riemannian manifold in arbitrary dimension.This generalizes origami in several ways:First, instead of folding flat paper we may consider folding two-dimensional sheets that possess curvature, like the surface of a sphere or a torus.Second, instead of folding flat paper along straight line creases that, when folded flat, reflect one side of the paper onto the other, we may consider folding a three-dimensional manifold along crease planes which reflect one side of space onto the other, or fold n-dimensional space along crease hyperplanes of dimension (n?1).Work in this area by Robertson (1977) and Lawrence and Spingarn (1989) is presented along with more decent additions, such as generalizations of Maekawa’s Theorem and the sufficient direction of Kawasaki's Theorem in higher dimensions.