Perhaps the simplest geometric formulation of origami design is to fold a desired 2D or 3D shape from a specified shape of paper (typically square). For example, we may be given the polygonal region shown in Figure 15.1(a) representing the silhouette of a horse, and our goal is to fold a square piece of paper into a flat origami with this silhouette. The origami literature provides countless examples of 2D and 3D shapes foldable from a square piece of paper. Folding a 3D shape can also be thought of as wrapping a general-shape gift.

The desired shape should be made of flat sides–polygonal (in 2D) or polyhedral (in 3D)–to be achievable by finitely many folds of a polygonal piece of paper, but other than this basic constraint, it is conceivable that any shape is foldable from a sufficiently large piece of paper. Although this problem is implicit throughout the origami literature, the problem was not formally posed until 1999 by Bern and Hayes (1996) and that too only in the 2D case. The 3D version–a kind of “gift-wrapping” for complex polyhedral gifts–was implicitly studied as early as 1960, for example, by Gardner(1990, 1995a), but the general problem appears not to have been formally posed in the literature.

A further variation on these problems, introduced by Demaine et al. (2000c), is to suppose that the original piece of paper is bicolored: a different color on each side.