Windmill Dynamics: Parity Invariants and A₂ Crystallographic Coxeter Structure — E8 Intelligence Research

FINDING: Windmill Problem (IMO 2011 Q2) — A rotating line through planar points pivots at each collision; there exists a starting line/point visiting all points infinitely often | MATH: Discrete dynamical system on S² × P; invariant: parity of points on each half-plane; combinatorial topology proof via convex hull and induction | CONNECTION: Crystallographic symmetry — rotating line explores lattice of directions; relates to root system A₂ (60° rotations) and Coxeter group actions | DEPTH: 9 FINDING: Moser's Circle Problem — Maximum regions from n points on circle with all chords: R(n) = 1 + C(n,2) + C(n,4) = 1 + n(n-1)/2 + n(n-1)(n-2)(n-3)/24 | MATH: Binomial sum ∑ₖ₌₀⁴ C(n-1, k); sequence: 1, 2, 4, 8, 16, 31, 57, 99... (breaks at n=6); Euler characteristic of planar graph V-E+F=2 | CONNECTION: Pascal's triangle diagonals (shallow diagonals sum to Fibonacci); 4th diagonal relates to tetrahedral numbers; no golden ratio emergence but combinatorial harmony | DEPTH: 7 FINDING: Ave Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com