Lorentz transformation for the kinematics of degree-4 rigid origami vertices and compatibility of rigid-foldable polygons

TL;DR

This paper introduces a Lorentz transformation framework for understanding the kinematics of degree-4 rigid origami vertices, generalizing fold-angle multipliers and establishing compatibility conditions for rigid-foldable polygons.

Contribution

It develops a novel Lorentz transformation analogy for origami vertex kinematics, extending fold-angle multipliers to general vertices and deriving a compatibility theorem for rigid-foldable polygons.

Findings

Lorentz transformations relate folded states of degree-4 vertices.

Generalized fold-angle multipliers for non-flat vertices.

Compatibility conditions for rigid-foldable polygons.

Abstract

We offer new insight into the folding kinematics of degree-4 rigid origami vertices by drawing an analogy to spacetime in special relativity. Specifically, folded states of the vertex, described by pairs of fold angles in terms of cotangent of half-angles, are related through Lorentz transformations in $1+1$ dimensions. Linear ordinary differential equations are derived for the tangent vectors on two-dimensional fold-angle planes, with the coefficient matrix depending exclusively on the sector angles. By taking the limit to the flat state, we generalize the fold-angle multipliers previously defined for flat-foldable vertices to general and collinear developable degree-4 vertices, and obtain a compatibility theorem on the rigid-foldability of polygons with $n$ developable degree-4 vertices. We further explore the rigid-foldable polygons of equimodular type and compose tangent vectors involving fold angles at the creases of the central polygon.