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

TL;DR

This paper proves that a specific class of non-Euclidean curved crease origami structures cannot be rigidly folded, using a simple geometric argument, highlighting fundamental limitations in their mobility.

Contribution

It provides a straightforward geometric proof that certain symmetric flat foldable non-Euclidean origami structures are not rigidly foldable, revealing inherent constraints in their deformation capabilities.

Findings

No isometric transformation exists between distinct configurations.

Certain developable surface meshes cannot be rigidly folded.

The result emphasizes the need for stretch in foldable models.

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.