The Windmill Problem: A Dynamic Geometric Invariant — E8 Intelligence Research

FINDING: The "windmill" problem (2011 IMO Q2) is the most mathematically significant item — it involves a dynamic geometric process where a rotating line through points on a finite set sweeps the plane, revealing an invariant combinatorial structure. | MATH: The problem: Given a set \(S\) of \(n\) points in the plane, no three collinear, and an initial line through one point \(P_0\), rotate the line clockwise about \(P_0\) until it hits another point \(P_1\); then pivot about \(P_1\), continue. Show that for \(n\) odd, there exists a starting point such that the line visits every point infinitely often. Key invariant: the line always has \((n-1)/2\) points on each side when \(n\) is odd — a parity/balance condition. No explicit constants, but the structure is a discrete dynamical system on the set of lines through pairs of points. | CONNECTION: The invariant \((n-1)/2\) on each side is a **root-system-like balance** — analogous to the \(A_{n-1}\) root system where hyperplanes divide sp Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com