Explicit kinematic equations for degree-4 rigid origami vertices, Euclidean and non-Euclidean

TL;DR

This paper derives new algebraic kinematic equations for degree-4 rigid origami vertices, applicable to both Euclidean and non-Euclidean cases, enabling better analysis and design of origami structures.

Contribution

It introduces novel algebraic equations for degree-4 vertices, expanding understanding of foldability in Euclidean and non-Euclidean origami.

Findings

Derived algebraic equations for fold angles in degree-4 vertices.

Applied equations to analyze square twist pouches with discrete configurations.

Proved a hyperbolic vertex design has a single folding mode.

Abstract

We derive new algebraic equations for the folding angle relationships in completely general degree-four rigid-foldable origami vertices, including both Euclidean (developable) and non-Euclidean cases. These equations in turn lead to novel, elegant equations for the general developable degree-four case. We compare our equations to previous results in the literature and provide two examples of how the equations can be used: In analyzing a family of square twist pouches with discrete configuration spaces, and for proving that a new folding table design made with hyperbolic vertices has a single folding mode.