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 $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding in the Euclidean space $\boldsymbol R^3$. This can be expressed as the image of an "origami map" $\Phi:D\to \boldsymbol R^3$ 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 possibility of 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.