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.