Bounded Orbits in Polygonal Outer Billiards Linked to Divisor Structure — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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 2π/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=2k), while unbounded orbits exist for n with odd prime divisors (e.g., n=5,7,9,…). The key invariant is the "rotation number" ρ = a/n (mod 1) for each vertex reflection, and the orbit's fate depends on the continued fraction expansion of ρ 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
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BibTeX
@misc{caldin2026bounded,
title = {{Bounded Orbits in Polygonal Outer Billiards Linked to Divisor Structure — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23446651},
url = {https://zenodo.org/records/23446651},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bounded Orbits in Polygonal Outer Billiards Linked to Divisor Structure — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23446651
UR - https://zenodo.org/records/23446651
AB - 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 2π/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,…). The key invariant is the "rotation number" ρ = a/n (mod 1) for each vertex reflection, and the orbit's fate depends on the continued fraction expansion of ρ 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
ER -