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.