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

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-13
DOI
https://doi.org/10.5281/zenodo.22732542
Primary Topic
Computational Geometry and Mesh Generation
Type
preprint
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preprint

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

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Computational Geometry and Mesh Generation
preprint

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

Andrew Stewart Caldin
preprint en

Abstract

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

Zenodo (CERN European Organization for Nuclear Research)
Computational Geometry and Mesh Generation
DOIOpen Access PDFOpenAlex
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The Windmill Problem: A Dynamic Geometric Invariant — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS