FOLDING MOTION IN THAT SHRINKS A GOOD POLYGONAL ANNULUS TO ARBITRARY SMALL NEIGHBORHOOD OF THE INNER BOUNDARY

In 2004, Demaine et al. showed that if is a simply connected polygonal region in the plane, then for any piecewise- folded state of , there exists a folding motion from to [1]. We have shown that if is not simply connected, then the conclusion of the theorem does not necessarily hold, in fact, there exists an annulus in such that there exists a piecewise- folded state of which does not admit folding motions from by using "knot and link theory". From the viewpoint of knot and link theory, it is reasonable to expect that the folded state admits a folding motion from if and only if forms a trivial link in (Conjecture). Note that the 'only if part of the Conjecture is obvious. In this paper, we give a proof of the following statement that will be used in an expected proof of the 'if part of the Conjecture for special kind of annulus . If is an annulus such that each of the boundary components bounds a polygonal convex region, then for any , there exists a flat folded state such that is contained in the -neighborhood of the inner boundary of in , denoted by with a strongly flat folding motion from to which satisfies the following: (1) For any , the number of 1 -simplices of the 1 -complex is finite, where denotes the canonical crease pattern of . (2) For any is a good polygonal annulus. (3) For any with , we have .