10 - Folding Manifolds

Summary

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.