On the existence of four or more curved foldings with common creases and crease patterns

Abstract

Consider an oriented curve Gamma in a domain D in the plane {varvec{R}}^2. Thinking of D as a piece of paper, one can make a curved folding in the Euclidean space {varvec{R}}^3. This can be expressed as the image of an “origami map” Phi :Drightarrow {varvec{R}}^3 such that Gamma is the singular set of Phi , the word “origami” coming from the Japanese term for paper folding. We call the singular set image C:=Phi (Gamma ) the crease of Phi and the singular set Gamma the crease pattern of Phi . We are interested in the number of origami maps whose creases and crease patterns are C and Gamma , respectively. Two such possibilities have been known. In the authors’ previous work, two other new possibilities and an explicit example with four such non-congruent distinct curved foldings were established. In this paper, we determine the possible values for the number N of congruence classes of curved foldings with the same crease and crease pattern. As a consequence, if C is a non-closed simple arc, then N=4 if and only if both Gamma and C do not admit any symmetries. On the other hand, when C is a closed curve, there are infinitely many distinct possibilities for curved foldings with the same crease and crease pattern, in general.