The Group of Closed Symmetric Flat Foldable Non-Euclidean Curved Crease Origami is not Rigid Foldable: A Simple Geometric Proof

Abstract

We present a novel parabolic reflector system capable of generating a broader class of shapes beyond canonical parabolas. Using a discretized framework, we construct meshes corresponding to key families of developable surfaces, including generalized cylinders, tangent developables, and generalized cones. Both Euclidean and non-Euclidean crease patterns are examined, and we demonstrate that no isometric transformation exists between distinct configurations within this system. This result highlights a fundamental limitation of purely developable models and motivates the incorporation of controlled stretching. We propose that enabling stretch accommodation would allow transitions between configurations, laying the groundwork for a generalized theory of curved-crease stretching. Such a framework has potential applications in understanding complex biological folding systems, including the deployment mechanics of the earwig wing.