Propagation of curved folding: The folded annulus with multiple creases exists

TL;DR

This paper investigates the geometric propagation of curved folds in developable surfaces, providing new formulae and insights into the existence of complex folded annuli with multiple creases.

Contribution

It introduces novel relations between geometric descriptors at folds and demonstrates how proper folds can be propagated to multiple creases in developable surfaces.

Findings

Derived symmetric formulae relating curvature and torsion at folds

Proved proper fold propagation to finite multiple creases

Identified challenges in extending results to infinitely many foldlines

Abstract

In this paper we consider developable surfaces which are isometric to planar domains and which are piecewise differentiable, exhibiting folds along curves. The paper revolves around the longstanding problem of existence of the so-called folded annulus with multiple creases, which we partially settle by building upon a deeper understanding of how a curved fold propagates to additional prescribed foldlines. After recalling some crucial properties of developables, we describe the local behaviour of curved folding employing normal curvature and relative torsion as parameters and then compute the very general relation between such geometric descriptors at consecutive folds, obtaining novel formulae enjoying a nice degree of symmetry. We make use of these formulae to prove that any proper fold can be propagated to an arbitrary finite number of rescaled copies of the first foldline and to give reasons why problems involving infinitely many foldlines are harder to solve.