Consider a curve \Gamma in a domain D in the plane \boldsymbol  ^2. Thinking of D as a piece of paper, one can make a curved folding in the Euclidean space \boldsymbol  ^3. This can be expressed as the image of an `origami map' \varphi :D\to \boldsymbol  ^3 such that \Gamma is the singular set of \varphi , the word `origami' coming from the Japanese term for paper folding. We call the singular set image C:= \varphi (\Gamma) the crease of \varphi and the singular set \Gamma the crease pattern of \varphi . 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 previous authors' work, two other new possibilities and an explicit example with four such non-congruent distinct curved foldings were established. In this paper, we show that the case of four mutually non-congruent curved foldings with the same crease and crease pattern occurs if and only if  \Gamma and C do not admit any symmetries. Moreover , when C is a closed curve, we show that there are infinitely many distinct possibilities for curved foldings with the same crease and crease pattern, in general.
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.
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