Windmill Problem: Parity-Stable Partition Invariant in Point Sets — E8 Intelligence Research

FINDING: The windmill problem (IMO 2011 P2) is a combinatorial-geometric invariant about rotating lines through point sets, revealing a parity-stable partition structure. | MATH: For \(n\) points in general position, a "windmill" line \(l\) rotates about a pivot point; after \(180^\circ\) rotation, the line visits each point as pivot exactly once. Key invariant: the number of points on each side of \(l\) remains constant (specifically \((n-3)/2\) and \((n-1)/2\) for odd \(n\)). This forces a periodic orbit of pivots — a cyclic permutation of the point set. | CONNECTION: The pivot sequence forms a Hamiltonian cycle on the complete graph of points — a discrete analogue of a closed geodesic on a sphere. The constant side-count ratio \((n-3):(n-1)\) approaches 1:1 as \(n\to\infty\), but for small \(n\) (e.g., \(n=5\)) gives 1:2 — reminiscent of the golden ratio's continued fraction truncations (1/2, 2/3, 3/5...). The rotation by \(180^\circ\) is a half-turn symmetry, linking to dihedral gr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com