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.