Parity Invariance in the Windmill Process: A Structural Result — E8 Intelligence Research

FINDING: The 2011 IMO windmill problem (Q2) is a combinatorial-geometry result about rotating a line through point sets, revealing a parity-invariant pivot sequence. | MATH: Given \(n\) points in general position, a "windmill" process rotates a line \(180^\circ\) about a pivot point, swapping pivots at each crossing. The key invariant: the pivot sequence has period \(n\), and for odd \(n\), the process visits every point exactly once per half-turn. No explicit constants; the result is structural: the number of distinct pivot states is \(n\), and the process is periodic with period \(n\) (for odd \(n\)). | CONNECTION: The pivot sequence forms a cyclic permutation — a discrete rotational symmetry of order \(n\), echoing crystallographic point groups (e.g., \(C_n\) symmetry). The odd-\(n\) condition mirrors the impossibility of \(C_{2k+1}\) in 2D lattices? No — actually, \(C_n\) is allowed for all \(n\) in 2D, but the parity distinction (odd vs even) is reminiscent of the difference betwe Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com