The Windmill Invariant: Rotational Symmetry in Finite Point Sets — E8 Intelligence Research

FINDING: The 2011 IMO windmill problem (Q2) is the most structurally significant finding — it encodes a rotational symmetry invariant over finite point sets, revealing a hidden combinatorial-geometric conservation law. | MATH: The problem: Given a finite set of \(n\) points in general position, a "windmill" process rotates a line \(l\) through a pivot point \(p\), switching pivot to the next point hit as \(l\) rotates. Key invariant: for \(n\) odd, every point is a pivot exactly once per full \(180^∘\) rotation; for \(n\) even, each point is a pivot exactly twice. This yields a periodic orbit in the configuration space of (line, pivot) pairs — a discrete dynamical system with period \(n\) (odd) or \(2n\) (even). The proof uses a parity argument on the number of points on each side of the rotating line, which is invariant under the pivot switch. | CONNECTION: The windmill process is a discrete analogue of a rotation in the plane — the line's angle sweeps through \(180^∘\), and t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com