On the Gaussian Curvature of Creased Tubes

Abstract

We calculate the Gaussian curvature of a curved, twisted crease in terms of the rate of change of solid angle along its length; we find that this depends on the fold angle across the crease and on the curvature along it but is independent of twist. We use this result to resolve a paradox concerning the geometry of a creased, twisted-prismatic tube; that there can be no Gaussian curvature overall despite the surface being doubly-curved.