Parity Invariants and Periodic Orbits in the IMO Windmill Problem — E8 Intelligence Research

Abstract

FINDING: The 2011 IMO windmill problem (Q2) is a combinatorial geometry problem whose solution hinges on a parity-invariant pivot rotation, revealing a hidden periodic orbit structure. | MATH: Given n points in general position, a "windmill" is a line rotating about a pivot point p_i, switching pivot to the point it just passed. The key invariant: the number of points on each side of the line changes by exactly ±1 per switch, and the total "winding" over a full 180° rotation cycles through all points exactly once if n is odd, twice if n is even. Formal: For n points, the process is periodic with period n (odd) or 2n (even), and the pivot sequence forms a Hamiltonian cycle on the point set's convex hull layers. | CONNECTION: The pivot-switching rule is isomorphic to a root system of type A_{n-1} — the sequence of pivots traces a path through the weight lattice, with the parity of n determining whether the orbit is a single cycle (odd) or a double cover (even). This mirrors the crystallo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com