Straight-crease origami folds rigidly — the panels rotate without bending, like hinged plates. Curved-crease origami is more expressive, generating parabolic reflectors and complex developable surfaces from flat sheets. The question is whether the same rigid folding is possible when the crease is curved.
The authors of arXiv:2603.17982 prove it is not. For the group of closed, symmetric, flat-foldable non-Euclidean curved-crease origami patterns, no isometric transformation exists between distinct configurations. The panels cannot rotate from one folded state to another without stretching. The proof is geometric: discretizing the curved crease into a mesh of developable surface elements and showing that the constraints are overconstrained — more equations than degrees of freedom, with no consistent solution.
This is a fundamental limitation of purely developable models. A straight crease gives each panel exactly the freedom it needs to rotate. A curved crease couples the panels' deformations in ways that consume all available freedom. The surface must either stay put or stretch.
The practical implication points toward biology. Earwig wings fold along curved creases into remarkably compact configurations and deploy rapidly. If rigid folding is impossible with curved creases, the wings must be stretching — the deployment mechanism requires controlled elastic deformation, not pure kinematics. The authors propose that understanding this stretching is the path to engineering curved-crease deployable structures.
The straight crease is a hinge. The curved crease is a lock. To fold along a curve, you must break something — and the engineering is in controlling what breaks and how.