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

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.

💡 Research Summary

The paper introduces a novel framework for describing the folding kinematics of degree‑4 (four‑crease) rigid origami vertices by exploiting an analogy with 1 + 1 dimensional spacetime in special relativity. The authors first rewrite the four signed fold angles ρₓ, ρ_y, ρ_z, ρ_w in terms of the cotangent of half‑angles, r = cot(ρ_r⁄2). This transformation linearises the classic fold‑angle equations: two pairs of opposite creases (x‑z and y‑w) satisfy hyperbolic relations of the form x² − z² = a p² and y² ‑ w² = b q², where a and b are functions solely of the sector angles (α, β, γ, δ). The constants a and b play the role of spacetime intervals; they remain invariant throughout the folding motion.

The central result (Theorem 1) states that any two folded states lying on the same branch of the hyperbola (or, for flat‑foldable vertices, on the same light‑like ray) are related by a Lorentz transformation. Explicitly, \