The shape and mechanics of curved-fold origami structures

¹ Department of Physics, University of Massachusetts Amherst - Amherst, MA 01003, USA
² School of Engineering, Brown University - Providence, RI 02912, USA

^(a) marcelo_dias@brown.edu
^(b) csantang@physics.umass.edu

Received: 10 October 2012
Accepted: 23 November 2012

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.