The Hyperbolic, Eisenstein–Lattice, and FRLT Interpretation of the Figure‑Eight Choreography in the Three‑Body Problem

Abstract

The figure‑eight three‑body choreography is reformulated using a complex shape parameter matching the hyperbolic structure of the figure‑eight knot complement. Its action becomes a hyperbolic geodesic functional, and the Eisenstein lattice’s unique fixed point serves as the FRLT pivot generating the orbit.