INTRODUCTION

We return to Open Problem 21.1 (p. 300): can every convex polyhedron be cut along its edges and unfolded flat into the plane to a single nonoverlapping simple polygon, a net for the polyhedron? This chapter explores the relatively meager evidence for and against a positive answer to this question, as well as several more developed, tangentially related topics.

Applications in Manufacturing

Although this problem is pursued primarily for its mathematical intrigue, it is not solely of academic interest: manufacturing parts from sheet metal (cf. Section 1.2.2, p. 13) leads directly to unfolding issues. A 3D part is approximated as a polyhedron, its surface is mapped to a collection of 2D flat patterns, each is cut from a sheet of metal and folded by a bending machine (Kim et al. 1998), and the resulting pieces assembled to form the final part. Clearly it is essential that the unfolding be nonoverlapping and great efficiency is gained if it is a single piece. The author of a Ph.D. thesis in this area laments that “Unfortunately, there is no theorem or efficient algorithm that can tell if a given 3D shape is unfoldable [without overlap] or not” (Wang 1997, p. 81). In general, those in manufacturing are most keenly interested in unfolding nonconvex polyhedra and, given the paucity of theoretical results to guide them, have relied on heuristic methods. One of the more impressive commercial products is TouchCAD by Lundströom Design, which has been used, for example, to design a one-piece vinyl cover for mobile phones (see Figure 22.1).

In this chapter we gather a few miscellaneous results and questions pertaining to “curved origami,” in either the folded shape or the creases themselves, and to its opposite, “rigid origami,” where the regions between the creases are forbidden from flexing. In general, little is known and we will merely list a few loosely related topics.

FOLDING PAPER BAGS

We have seen that essentially any origami can be folded if one allows continuous bending and folding of the paper, effectively permitting an infinite number of creases (Theorem 11.6.2). Recall this result was achieved by permitting a continuous “rolling” of the paper (Section 11.6.1, p. 189). In contrast one can explore what has been called rigid origami, which permits only a finite number of creases, between which the paper must stay rigid and flat, like a plate (Balkcom 2004; Balkcom and Mason 2004; Hull 2006, p. 222). One example of the difference between rigid and traditional origami is the inversion of a (finite) cone. Connelly (1993) shows how this can be done by continuous rolling of creases, but he has proved that such inversion is impossible with any finite set of creases.

One surprising result in this area is that the standard grocery shopping bag, which is designed to fold flat, cannot do so without bending the faces (Balkcom et al. 2004). Consider the shopping bag shown in Figure 20.1.

We believe that research in mathematical origami has been somewhat hampered by lack of clear, formal foundation. This chapter provides one such foundation, following the work of Demaine et al. (2004, 2006a). Specifically, this chapter defines three key notions: what is a piece of paper, what constitutes an individual folded state (at an instant of time) of a piece of paper, and when a continuum of these folded states (animated through time) forms a valid folding motion of a piece of paper. Each of these notions is intuitively straightforward, but the details are quite complicated, particularly for folded states and (to a lesser extent) for folding motions. In the final section (Section 11.6, p. 189), this chapter also proves a relationship between these notions: every folded state can be achieved by a folding motion. At first glance, one would not normally even distinguish between these two notions, so it is no surprise that they are equivalent. The formal equivalence is nonetheless useful, however, because it allows most of the other theorems in this book to focus on constructing folded states, knowing that such constructions can be extended to folding motions as well.

While we feel the level of formalism developed in this chapter is important, it may not be of interest to every reader. Many will be content to skip this entire chapter and follow the rest of the book using the intuitive notion that mathematical paper is just like real paper except that the paper has zero thickness.

The tree method of origami design is a general approach for “true” origami design (in contrast to the other topics that we discuss, which involve less usual forms of origami). In short, the tree method enables design of efficient and practical origami within a particular class of 3D shapes. Some components of this method, such as special cases of the constituent molecules and the idea of disk packing, as well as other methods for origami design, have been explored in the Japanese technical origami community, in particular by Jun Maekawa, Fumiaki Kawahata, and Toshiyuki Meguro. This work has led to several successful designs, but a full survey is beyond our scope (see Lang 1998, 2003). It suffices to say that the explosion in origami design over the last 20 years, during which the majority of origami models have been designed, may largely be due to an understanding of these general techniques.

Here we concentrate on Robert Lang's work (Lang 1994a, b, 1996, 1998, 2003), which is the most extensive. Over the past decade, starting around 1993, Lang developed the tree method to the point where an algorithm and computer program have been explicitly defined and implemented: TreeMaker is freely available and runs on most platforms. Lang himself has used it to create impressively intricate origami designs that would be out of reach without his algorithm. Figure 16.1 shows one such example.

At how many points must a tangled chain in space be cut to ensure that it can be completely unraveled? No one knows. Can every paper polyhedron be squashed flat without tearing the paper? No one knows. How can an unfolded, precreased rectangular map be refolded, respecting the creases, to its original flat state? Can a single piece of paper fold to two different Platonic solids, say to a cube and to a tetrahedron, without overlapping paper? Can every convex polyhedron be cut along edges and unfolded flat in one piece without overlap? No one knows the answer to any of these questions.

These are just five of the many unsolved problems in the area of geometric folding and unfolding, the topic of this book. These problems have the unusual characteristic of being easily comprehended but they are nevertheless deep. Many also have applications to other areas of science and engineering. For example, the first question above (chain cutting) is related to computing the folded state of a protein from its amino acid sequence, the venerable “protein folding problem.” The second question (flattening) is relevant to the design of automobile airbags. A solution to the last question above (unfolding without overlap) would assist in manufacturing a three-dimensional (3D) part by cutting a metal sheet and folding it with a bending machine.

Our focus in this book is on geometric folding as it sits at the juncture between computer science and mathematics.

This second part concerns various forms of paper folding, often called origami. We start in this chapter with a historical background of paper and paper folding (Section 10.1), and of its study from mathematical and computational points of view (Section 10.2). This history can safely be skipped by the uninterested reader. Then in Section 10.3 we define several basic pieces of terminology for describing origami, before providing an overview of Part II in Section 10.4.

HISTORY OF ORIGAMI

The word “origami” comes from Japanese; it is the combination of roots “oru,” which means “fold,” and “kami,” which means “paper.” While origami was originally popularized largely by Japanese culture, its origins are believed to be pre-Japanese, roughly coinciding with the invention of paper itself. Paper, in turn, is believed to have been invented by Ts'ai Lun, a Chinese court official, in 105 a.d. The invention of paper was motivated by the then-recent invention of the camel hair brush, from 250 b.c., which could be used for writing and calligraphy.

Paper, and presumably paper folding at the same time, spread throughout the world over a long period. Buddhist monks spread paper through Korea to Japan in the sixth century a.d. Arabs occupying Samarkand, Uzbekistan, from 751 a.d. brought paper to Egypt in the 900s, and from there continued west. The Moors brought paper (and at the same time, mathematics) to Spain during their invasion in the 700s. In the 1100s, paper making became established in Jativa, Spain.

The topic of this book is the geometry of folding and unfolding, with a specific emphasis on algorithmic or computational aspects. We have partitioned the material into three parts, depending on what is being folded or unfolded: linkages (Part I, p. 7–164), paper (Part II, p. 165–296), and polyhedra (Part III, p. 297–441). Very crudely, one can view these parts as focusing on one-dimensional (1D) objects (linkages), 2D objects (paper), or 3D objects (polyhedra). The 1D–2D–3D view is neither strictly accurate nor strictly followed in the book, but it serves to place related material nearby.

One might classify according to the process. Folding starts with some unorganized generic state and ends with a more structured terminal “folded state.” Unfolding is the reverse process, but the distinction is not always so clear. Certainly we unfold polyhedra and we fold paper to create origami, but often it is more useful to view both processes as instances of “reconfiguration” between two states.

Another possible classification concentrates on the problems rather than the objects or the processes. A rough distinction may be drawn between design problems—given a specific folded state, design a way to fold to that state, and foldability questions—can this type of object fold to some general class of folded states. Although this classification is often a Procrustean bed, we follow it below to preview specific problem instances, providing two back-to-back minitours through the book's 1D–2D–3D organization.

OVERVIEW

In 1525 the painter and printmaker Albrecht Dürer published a book, Underweysung der Messung (later translated as “The Painter's Manual”), in which he explained the methods of perspective, which he had just learned himself on a trip to Italy (see Figure 21.1).

Dürer's book includes a description of many polyhedra, which he presented as surface unfoldings, are now called “nets.” For example, Figure 21.2 shows his net for a cuboctahedron. Even when he drew a more complex net, such as that for a truncated icosahedron (the shape of a soccer ball; Figure 21.3), he always chose an unfolding that avoided self-overlap. Although there is no evidence that Dürer distilled this property into a precise question, it is at least implicit in the practice of subsequent generations that every convex polyhedron may be so unfolded. More precisely, define an edge unfolding as a development of the surface of a polyhedron to a plane, such that the surface becomes a flat polygon bounded by segments that derive from edges of the polyhedron. One may view such an unfolding as obtained by slicing the surface along a collection of edges. We would like an unfolding to possess three characteristics, enjoyed by all Dürer's drawings:

1. The unfolding is a single, simply connected piece.

2. The boundary of the unfolding is composed of (whole) edges of the polyhedron, that is, the unfolding is a union of polyhedron faces.

3. The unfolding does not self-overlap, that is, it is a “simple polygon.”

TRISECTION

“Trisecting an angle” is one of the problems inherited from Greek antiquity. It asks for a series of constructions by straight edge and compass that trisects a given angle. This problem remained unsolved for 2,000 years and was finally shown to be algebraically impossible in the nineteenth century by Wantzel (1836). We saw earlier, in Figure 3.6 (p. 33), that it has been known for over a century that a linkage can trisect an angle. More recently it was established that it is possible to trisect an angle via origami folds; Figure 19.1 illustrates the elegant construction of Abe.

The reader may sense that it is not clear that this construction actually works, nor what are the exact rules of the game. We will attempt to elucidate both these issues.

HUZITA'S AXIOMS AND HATORI'S ADDITION

What is “constructible” by origami folds was greatly clarified by Humiaki Huzita in 1985, who presented a set of six axioms of origami construction (Hull 1996; Huzita and Scimemi 1989; Murakami 1987).

These axioms intend to capture what can be constructed from origami “points” and “lines” via a single fold. A line is a crease in a (finite) piece of paper or the boundary of the paper. A point is an intersection of two lines. Initially, the (traditionally square) paper has lines determined by the boundary edges. Crease the paper in half and you construct two points where the medial crease hits the paper boundary (see Figure 19.2).

Having seen in the previous sections that chains can only lock in 3D, it is natural to investigate the conditions that permitchains to lock in 3D. Sections 5.3.3 and 6.9 showed that chains with non-self-intersecting projections cannot lock. In this section we report on the beginnings of an exploration of when chains can lock and, in particular, when pairs of chains can “interlock.” This line of investigation was prompted by a question posed by Anna Lubiw (Demaine and O'Rourke 2000): Into how many pieces must a chain be cut (at vertices) so that the pieces can be separated and straightened? In a sense, this question crystallizes the vague issue of the degree of “lockedness” of a chain—how tangled it is—in the form of a precise problem. The “knitting needles” example (Figure 6.2, p. 88) is only slightly locked, in that removal of one vertex suffices to unlock it (recall that Cantarella and Johnson proved that no chain of fewer than 5 links can be locked).Concatenating many copies of the knitting needles leads to a lower bound of ⌊ (n − 1)/4⌋ on the number of cuts needed to separate a chain of n links, as illustrated in Figure 7.1: each copy of the 5-link chain must have one of its four interior vertices cut. An upper bound of ⌊ (n − 3)/2⌋ has been obtained, but otherwise Lubiw's problem remains unsolved.

A cereal box may be flattened in the familiar manner illustrated in Figure 18.1: by pushing in the two sides of the box (with dashed lines), the front and back of the box pop out and the whole box squashes flat.

This process leads to a natural mathematical problem: which polyhedra can be flattened, that is, folded to lie in a plane? This problem is a different kind of paper folding problem than we have encountered before, because now our piece of paper is a polyhedron, not a flat sheet. Our goal is merely to find some flat folding of the piece of paper, whereas normally our piece of paper is flat to begin with!

CONNECTION TO PART III: MODELS OF FOLDING

In Part III we will address the rigidity or flexibility of polyhedra from first principles (Sections 23.1 and 23.2, p. 341ff). In particular, Cauchy's rigidity theorem establishes that all convex polyhedra–so in particular a box, or a box with additional creases–cannot be flexed at all. So how is it that we are able to flatten the box? Even for nonconvex polyhedra, any flattening of a polyhedron necessarily decreases its volume to zero; yet the Bellows theorem (Section 23.2.4, p. 348) says that the volume of a polyhedron is constant throughout any flexing.

This seeming contradiction highlights an important aspect of our model of flattening: while Cauchy's rigidity theorem and the Bellows theorem require the faces to remain rigid plates, here we allow faces to curve and flex.

FOLDING POLYGONS: PRELIMINARIES

The question

Was first explicitly posed in 1996 (Lubiw and O'Rourke 1996).Here we mean fold without overlap (in constrast to the wrapping permitted in Theorem15.2.1, p. 236) and of course without leaving gaps. We have seen in Section 23.3 (p. 348) that Alexandrov's Theorem provides an answer: whenever a polygon has an Alexandrov gluing. (Recall that, in this context, a “polyhedron” could be a flat, doubly covered polygon.) This then reorients the question to

Before turning to an algorithmic answer to this question, we first show that not all polygons have an Alexandrov gluing, that is, not all polygons are foldable, and indeed the foldable polygons are rare.

Not-Foldable Polygons

Lemma 25.1.1. Some polygons cannot be folded to any convex polyhedron.

Proof: Consider the polygon P shown in Figure 25.1. P has three consecutive reflex vertices (a, b, c), with the exterior angle β at b small. All other vertices are convex, with interior angles strictly larger than β.

Either the gluing zipsat b, gluing edge ba to edge bc, or some other point(s) of ∂P glue to b. The first possibility forces a to glue to c, exceeding 2π there; so this gluing is not Alexandrov. The second possibility cannot occur with P, because no point of ∂P has small enough internal angle to fit at b. Thus there is no Alexandrov gluing of P.

Our focus in this first part is on one-dimensional (1D) linkages, and mostly on especially simple linkages we call “chains.” Linkages are useful models for robot arms and for folding proteins; these and other applications will be detailed in Section 1.2. After defining linkages and setting some terminology, we quickly review the contents of this first part.

Linkage definitions. A linkage is a collection of fixed-length 1D segments joined at their endpoints to form a graph. A segment endpoint is also called a vertex. The segments are often called links or bars, and the shared endpoints are called joints or vertices. The bars correspond to graph edges and the joints to graph nodes. Some joints may be pinned to be fixed to specific locations. Although telescoping links and sliding joints are of considerable interest in mechanics, we only explore fixed-length links and joints fixed at endpoints. (We'll use the term mechanism to loosely indicate any collection of rigid bodies connected by joints, hinges, sliders, etc.) An example of a linkage is shown in Figure 1.1.

Overview. After classifying problems in this chapter, we turn to presenting some of the basic upper and lower complexity bounds obtained in the past 20 years in Chapter 2.We then explore in Chapter 3 classical mechanisms, particularly the pursuit of straight-line linkage motion. In contrast to these linkages, whose whole purpose is motion, we next study in Chapter 4 when a linkage is rigid, that is, can move at all. Most of the remainder of Part I concentrates on chains, starting with reconfiguring chains under various constraints in Chapter 5.