A rigid origami elliptic-hyperbolic vertex duality

TL;DR

This paper uncovers a duality between elliptic and hyperbolic degree-4 rigid origami vertices, revealing a mathematical structure that can be used to design flexible 3D metamaterial structures.

Contribution

It introduces a novel duality between elliptic and hyperbolic vertices in rigid origami, enhancing understanding of their kinematics and potential applications.

Findings

Elliptic and hyperbolic vertices have equivalent kinematics.

Duality simplifies analysis of complex origami structures.

Potential for designing metamaterials with flexible properties.

Abstract

The field of rigid origami concerns the folding of stiff, inelastic plates of material along crease lines that act like hinges and form a straight-line planar graph, called the crease pattern of the origami. Crease pattern vertices in the interior of the folded material and that are adjacent to four crease lines, i.e. degree-4 vertices, have a single degree of freedom and can be chained together to make flexible polyhedral surfaces. Degree-4 vertices that can fold to a completely flat state have folding kinematics that are very well-understood, and thus they have been used in many engineering and physics applications. However, degree-4 vertices that are not flat-foldable or not folded from flat paper so that the vertex forms either an elliptic or hyperbolic cone, have folding angles at the creases that follow more complicated kinematic equations. In this work we present a new duality between general degree-4 rigid origami vertices, where dual vertices come in elliptic-hyperbolic pairs that have essentially equivalent kinematics. This reveals a mathematical structure in the space of degree-4 rigid origami vertices that can be leveraged in applications, for example in the construction of flexible 3D structures that possess metamaterial properties.