The field of flat-foldable origami is introduced, which involves a mix of geometry and combinantorics.This chapter focuses on local properties of flat origami, meaning the study of how and when a single vertex in an origami crease pattern will be able to fold flat.The classic theorems of Kawasaki and Maekawa are proved and generalizations are made to folding vertices on cone-shaped (i.e., non-developable) paper.The problem of counting valid mountain-valley assignments of flat-foldable vertices is solved, and the configuration space of flat-foldable vertices of a fixed degree is characterized.A matrix model for formalizing flat-vertex folds is introduced, and the chapter ends with historical notes on this topic.

The field of origami numbers in the complex plane that are constructible by straight-line, one-crease-at-a-time origami is characterized to be the smallest subfield of the complex numbers that can be obtained by 2-3 towers of extension fields.Other ways to describe this field are also discussed.

Chapter 7 delves into a handful of combinatorial problems in flat origami theory that are more general than the single-vertex problems considered in Chapter 5. First, we count the number of locally-valid mountain-valley assignments of certain origami tessellations, like the square twist and Miura-ori tessellations. Then the stamp-folding problem is discussed, where the crease pattern is a grid of squares and we want to fold them into a one-stamp pile in as many ways as possible.Then the tethered membrane model of polymer folding is considered from soft-matter physics, which translates into origami as counting the number of flat-foldable crease patterns that can be made as a subset of edges from the regular triangle lattice.Many of these problems establish connections between flat foldings and graph colorings and statistical mechanics.

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 introduction discusses the intricacies of origami art, how origami has become popular in science and engineering applications in the 2000s, and the author's motivation for writing this book.

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.

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.

The idea of using origami to perform geometric constructions (as opposed to the classic tools of straightedge and compass) is introduced.Origami construction methods are presented for folding an equilateral triangle, dividing a line segment into n equal pieces, trisecting angles, and folding a regular heptagon.The basic origami operations are also presented and proved to be the complete list of basic construction moves that can be made in origami by folding one straight crease at a time.Historical remarks are also included.

In Chapter 9 we explore the work of Kawasaki and Yoshida from1988, where group theory is used to relate the symmetries of a flat origami crease pattern to the symmetries of its folded image.This is then applied to origami tessellations to show that if the tessellation’s symmetries form a crystallographic group of the plane, then the symmetry group of the folded paper must be isomorphic to the symmetry group of the crease pattern.