On the existence of four or more curved foldings with common creases and crease patterns
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Consider a curve Γ in a domain D in the plane R^2. Thinking of D as a piece of paper, one can make a curved folding in the Euclidean space R^3. This can be expressed as the image of an `origami map' φ:D→R^3 such that Γ is the singular set of φ, the word `origami' coming from the Japanese term for paper folding. We call the singular set image C:=φ(Γ) the crease of φ and the singular set Γ the crease pattern of φ. We are interested in the number of origami maps whose creases and crease patterns are C and Γ 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 Γ 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.
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