In this chapter, geometric construction possibilities that go beyond traditional origami are described.Multifolds, which are origami operations that allow for the simultaneous creation of more than one crease, are the main focus of the chapter.Multifolds provide more algebraic power to origami constructions, and multifold methods of performing angle quintisections and 11-gons are described.Alperin and Lang’s proof that multifold origami can find roots of arbitrary polynomials is given.Origami constructions that include curved creases are also explored, showing that they can allow the construction of some transcendental numbers.

Chapter 6 covers global flat foldability.This includes determining how we can tell if a crease pattern with multiple vertices will fold flat without forcing the paper to self-intersect, as well as discovering properties that all such crease patterns have, beyond what was covered in the previous chapter.Justin’s Theorem, which is a generalization of Kawasaki’s and Maekawa’s Theorems and whose proof uses elements of basic knot theory, is covered, as are Justin’s non-crossing conditions that provide necessary and sufficient conditions for a general crease pattern to fold flat.The matrix model from Chapter 5 is developed further to create a formal folding map for flat origami.Finally, Bern and Hayes’ seminal proof that determining flat foldability of a given crease pattern is NP-Hard is presented and updated with more recent results on box pleating.

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 8 covers a variety of other mathematical problems that arise from flat origami.For example, how many times can you fold a piece of paper in half?This old question has a more interesting answer, discovered by a high-school student, than is commonly known.Can any shape be folded?Origami artists want to claim that the answer to this question is, “Yes,” but how should the question be formalized mathematically?And if we really can fold anything, how do we do so?The process of origami design can be quantified, and the ways in which complex origami designers do so to, say, create an origami insect are summarized.Another origami puzzle with a surprising answer is:If a square piece of paper is folded into a flat object, what is the biggest perimeter that object can have?Finally, we summarize the theory behind, and proof of, the famous Fold-and-One-Cut Theorem.

In this chapter, proofs are given that origami can solve quadratic and cubic equations.For example, when folding a point onto a line the resultingcrease line is tangent to a parabola.Methods used include Lill’s Method for geometrically finding real roots of arbitrary polynomials and Margherita Beloch’s solution to solving cubics from the 1930s.

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.