In 2004, Demaine et al. showed that if P is a simply connected polygonal region in the plane, then for any piecewise- folded state  of P, there exists a folding motion from P to  [1]. We have shown that if P is not simply connected, then the conclusion of the theorem does not necessarily hold, in fact, there exists an annulus P in  such that there exists a piecewise- folded state of P which does not admit folding motions from P 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 P 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 P.

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 P in P, denoted by  with a strongly flat folding motion   from P 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

(2) For any is a good polygonal annulus.

(3) For any with we have

origami, folding motion, flat folded state, polygonal annulus.

Received: October 17, 2024; Accepted: November 25, 2024; Published: December 24, 2024

How to cite this article: Hiroko Murai, Folding motion in  that shrinks a good polygonal annulus to arbitrary small neighborhood of the inner boundary, JP Journal of Geometry and Topology 30(2) (2024), 143-154. https://doi.org/10.17654/0972415X24010

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] Erik D. Demaine, Satyan L. Devadoss, Joseph S. B. Mitchell and Joseph O’Rourke, Continuous foldability of polygonal paper, Proc. of CCCG’04, Montréal, 2004, pp. 64-67.
[2] Erik D. Demaine and Joseph O’Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, New York, 2007.
[3] Akari Iwamura and Hiroko Murai, Existence of folded states which do not admit folding motions from the unfolded state, JSIAM Letters 16 (2024), 101-104.