On the Gaussian Curvature of Creased Tubes

TL;DR

This paper derives a formula for Gaussian curvature in creased tubes, revealing its dependence on fold angle and curvature but not twist, and resolves a paradox about their overall curvature.

Contribution

It introduces a new calculation method for Gaussian curvature in creased, twisted structures and clarifies their geometric properties.

Findings

Gaussian curvature depends on fold angle and curvature, not twist

No overall Gaussian curvature in creased, twisted-prismatic tubes

Resolves a paradox in the geometry 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.