On the existence of four or more curved foldings with common creases and crease patterns (Q2094266)

Given a curve \(\Gamma\) within a domain \(D\) of the Euclidean plane \(\mathbb{R}^2\), one may consider \(D\) as a piece of paper and then ``fold it along \(\Gamma\)'' in order to obtain a developable surface in the Euclidean space \(\mathbb{R}^3\). Using the authors' terminology, this process of \textit{curved folding} is based upon an \textit{origami map} \(\Phi\colon D\to \mathbb{R}^3\) such that the \textit{crease pattern} \(\Gamma\) comprises all singular points of \(\Phi\), whereas the \textit{crease} \(C:=\Phi(\Gamma)\) is a space curve in \(\mathbb{R}^3\). The article's main results provide the possible values for the number \(N\) of congruence classes of curved foldings with the same crease pattern \(\Gamma\) and the same crease \(C\). It turns out that, depending on the crease \(C\), there are two essentially different cases. Firstly, it is supposed that \(C\) is a non-closed simple arc. If, furthermore, neither \(\Gamma\) nor \(C\) admit any non-identical Euclidean isometries, then \(N=4\); otherwise \(N\leq 2\). In addition, an explicit characterisation of all instances with \(N=1\) is provided. Secondly, \(C\) is assumed to be a closed curve without self-intersections. Then \(N\) turns out infinite unless the following applies: \(C\) is a circle and \(\Gamma\) is a subset of a circle.