Rigid folding equations of degree-6 origami vertices

TL;DR

This paper extends the mathematical understanding of degree-6 origami vertices with equal sector angles, providing explicit folding equations, configuration spaces, and insights into rigid origami mechanisms for engineering applications.

Contribution

It introduces new algebraic folding angle relationships and configuration space computations for symmetric degree-6 vertices, expanding the theoretical framework of rigid origami.

Findings

Enumerated viable rigid folding modes of degree-6 vertices

Derived algebraic relationships between folding angles

Uncovered new insights into Weierstrass substitutions in origami modeling

Abstract

Rigid origami, with applications ranging from nano-robots to unfolding solar sails in space, describes when a material is folded along straight crease line segments while keeping the regions between the creases planar. Prior work has found explicit equations for the folding angles of a flat-foldable degree-4 origami vertex and some cases of degree-6 vertices. We extend this work to generalized symmetries of the degree-6 vertex where all sector angles equal $60^\circ$. We enumerate the different viable rigid folding modes of these degree-6 crease patterns and then use $2^{nd}$-order Taylor expansions and prior rigid folding techniques to find algebraic folding angle relationships between the creases. This allows us to explicitly compute the configuration space of these degree-6 vertices, and in the process we uncover new explanations for the effectiveness of Weierstrass substitutions in modeling rigid origami. These results expand the toolbox of rigid origami mechanisms that engineers and materials scientists may use in origami-inspired designs.