Topological transitions in the configuration space of non-Euclidean origami

TL;DR

This paper investigates the configuration space of non-Euclidean origami, revealing how Gaussian curvature influences the connectivity of folding configurations, which could enable new control mechanisms in origami-based structures.

Contribution

It uncovers the topological differences in the configuration space of non-Euclidean origami depending on Gaussian curvature sign, a novel insight for origami design and mechanics.

Findings

Configuration space of positive Gaussian curvature is disconnected.

Negative Gaussian curvature configurations form a connected space.

The results suggest new ways to control origami folding mechanisms.

Abstract

Origami structures have been proposed as a means of creating three-dimensional structures from the micro- to the macroscale, and as a means of fabricating mechanical metamaterials. The design of such structures requires a deep understanding of the kinematics of origami fold patterns. Here, we study the configurations of non-Euclidean origami, folding structures with Gaussian curvature concentrated on the vertices. The kinematics of such structures depends crucially on the sign of the Gaussian curvature. The configuration space of non-intersecting, oriented vertices with positive Gaussian curvature decomposes into disconnected subspaces; there is no pathway between them without tearing the origami. In contrast, the configuration space of negative Gaussian curvature vertices remain connected. This provides a new mechanism by which the mechanics and folding of an origami structure could be controlled.