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.

Figures (13)

• Figure 1: Caption
• Figure 2: Caption
• Figure 3: Curved Crease Parabolic Reflector (a) Top View (b) Side View (c) Closed
• Figure 4: (a) Parabolic Reflector in the Reference Configuration $\Omega$, (b) Parabolic Reflector in the Deformed Configuration $\mathcal{S}$, with two creases, the Euclidean crease, where the units have reflected symmetry, and the non-Euclidean crease where the units have rotational symmetry $D_n$, along with the symmetric boundary condition that the unit must lie in a pie wedge of $\pi/n$
• Figure 5: The process of developing the curve (i) create a $n$-gon, (ii) project it onto a parabola, (iii) copy it with $D_n$ symmetry, for (a) $n=3$, (b) $n= 6$, and (c)$n= 24$