Windmill Dynamics: Rotational Symmetry in Combinatorial Geometry — E8 Intelligence Research

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-18
DOI
https://doi.org/10.5281/zenodo.22824399
Primary Topic
Computability, Logic, AI Algorithms
Type
preprint
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preprint

Windmill Dynamics: Rotational Symmetry in Combinatorial Geometry — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Computability, Logic, AI Algorithms
preprint

Windmill Dynamics: Rotational Symmetry in Combinatorial Geometry — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: The 2011 IMO Problem 2 ("windmill") is a combinatorial geometry problem whose solution hinges on a discrete rotation group acting on a finite point set, with an invariant line through a pivot point. | MATH: The core structure: Given \(n\) points in general position (no three collinear), a "windmill" is a line \(L\) through one point \(P\) (the pivot) that rotates continuously; when \(L\) passes through another point \(Q\), the pivot switches to \(Q\) (the line now rotates about \(Q\)). The invariant: the number of points on each side of \(L\) remains constant throughout the process. For \(n\) odd, the line always has \((n-1)/2\) points on each side; for \(n\) even, the counts alternate between \((n-2)/2\) and \(n/2\) on one side. The proof uses a parity argument: each pivot switch changes the side-count by exactly \(\pm 1\) for the old and new pivot, but the total imbalance is preserved modulo 2. The rotation is effectively a cyclic action of \(\mathbb{Z}_n\) on the set of "ba Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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