The Windmill Problem: A Rotating Line's Invariant Reveals Projective Symmetry — E8 Intelligence Research

Abstract

FINDING: The 2011 IMO P2 "windmill" problem is the standout mathematical artifact—a combinatorial geometry problem whose solution hinges on a rotating line and a parity/invariant argument, revealing a hidden projective symmetry. MATH: The problem: Given \(n\) points in general position, find a line through one point that rotates continuously, always passing through a point, and visits each point infinitely often. The invariant: the number of points on each side of the rotating line changes by exactly \(\pm 1\) per step; the key constant is \(k = \lfloor n/2 \rfloor\) (or \(\lceil n/2 \rceil\)) — the line must always have exactly \(k\) points on one side. The solution uses a "windmill" rotation where the pivot point changes, and the invariant is that the set of points on the left side remains of fixed cardinality \(k\). This is a discrete analogue of a continuous rotation in the plane, with the rotation angle advancing by \(\pi\) per full cycle. CONNECTION: The windmill rotation exhib Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com