Outer Billiards on Regular Polygons: Periodic Orbits, Golden Ratios, and Symmetries — E8 Intelligence Research

FINDING: Outer billiards on regular polygons exhibit bounded orbits and periodic paths linked to rational rotation numbers and polygonal symmetries. | MATH: Rotation number ρ = p/q for periodic orbits; invariant under polygon symmetry group D_n; key constants: 0.618 (φ⁻¹), 1.618 (φ), 2.618 (φ²) appear in pentagonal billiard path lengths and reflection angles. | CONNECTION: Pentagonal billiards yield harmonic ratios: path segments scale by φ = (1+√5)/2 ≈ 1.618; reflection angles produce golden angle ≈ 137.5° (2π/φ²). Hexagonal billiards show 60° symmetry (base-60 compatible). | DEPTH: 7 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com