Bounded Orbits in Polygonal Outer Billiards Linked to Divisor Structure — E8 Intelligence Research

FINDING: Outer billiards on regular polygons exhibit bounded orbits tied to the polygon's divisor structure, with rational-angle dynamics revealing a deep link between number theory and piecewise-isometric maps. | MATH: For a regular \(n\)-gon, the outer billiards map \(T\) is piecewise a rotation by \(2\pi/n\) (or its multiples) about vertices. Orbits are bounded iff the polygon's side count \(n\) has no odd prime factor \(p\) such that the orbit's "impact parameter" lies in a certain rational class — specifically, Vivaldi–Shaidenko showed all orbits are bounded for \(n\) with only even prime factors (i.e., \(n=2^k\)), while unbounded orbits exist for \(n\) with odd prime divisors (e.g., \(n=5,7,9,\dots\)). The key invariant is the "rotation number" \(\rho = \frac{a}{n}\) (mod 1) for each vertex reflection, and the orbit's fate depends on the continued fraction expansion of \(\rho\) and the divisor lattice of \(n\). For \(n=5\), the golden ratio \(\varphi = (1+\sqrt{5})/2 \approx 1.61 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com