The shape and mechanics of curved fold origami structures

TL;DR

This paper develops mathematical models to describe the three-dimensional shapes of curved fold origami structures, revealing their geometric properties and how they form complex surfaces like helicoids.

Contribution

It introduces recursion and continuum equations to predict the shape of curved fold origami, linking fold geometry to resulting surface curvature.

Findings

The surface has negative Gaussian curvature proportional to the square of torsion.

A series of open folds with constant angle form a helicoid.

Models accurately describe the 3D shape of curved fold origami structures.

Abstract

We develop recursion equations to describe the three-dimensional shape of a sheet upon which a series of concentric curved folds have been inscribed. In the case of no stretching outside the fold, the three-dimensional shape of a single fold prescribes the shape of the entire origami structure. To better explore these structures, we derive continuum equations, valid in the limit of vanishing spacing between folds, to describe the smooth surface intersecting all the mountain folds. We find that this surface has negative Gaussian curvature with magnitude equal to the square of the fold's torsion. A series of open folds with constant fold angle generate a helicoid.