Windmill Problem: IMO 2011's Rotating Line Geometry — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The search results are dominated by pedagogical videos and media coverage of IMO problems, not raw mathematical discoveries. The only substantive mathematical artifact referenced is the 2011 IMO Problem 2 ("windmill" problem), which is a combinatorial geometry problem about rotating lines through points.
MATH: The windmill problem (2011 IMO P2): Given a finite set S of n ≥ 3 points in the plane, no three collinear, a "windmill" process rotates a line l through a point P ∈ S (pivot). When l hits another point Q ∈ S, the pivot switches to Q and rotation continues. The theorem: there exists a choice of initial line and pivot such that the windmill visits every point of S infinitely often. The proof uses a parity/invariant argument: define f(l) = number of points on one side of l. When pivot switches, f(l) changes by ± 1 or stays constant depending on whether the new pivot lies on the same side. The key invariant is that \(
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026windmill,
title = {{Windmill Problem: IMO 2011's Rotating Line Geometry — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23465256},
url = {https://zenodo.org/records/23465256},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Windmill Problem: IMO 2011's Rotating Line Geometry — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23465256
UR - https://zenodo.org/records/23465256
AB - FINDING: The search results are dominated by pedagogical videos and media coverage of IMO problems, not raw mathematical discoveries. The only substantive mathematical artifact referenced is the 2011 IMO Problem 2 ("windmill" problem), which is a combinatorial geometry problem about rotating lines through points. MATH: The windmill problem (2011 IMO P2): Given a finite set S of n ≥ 3 points in the plane, no three collinear, a "windmill" process rotates a line l through a point P ∈ S (pivot). When l hits another point Q ∈ S, the pivot switches to Q and rotation continues. The theorem: there exists a choice of initial line and pivot such that the windmill visits every point of S infinitely often. The proof uses a parity/invariant argument: define f(l) = number of points on one side of l. When pivot switches, f(l) changes by ± 1 or stays constant depending on whether the new pivot lies on the same side. The key invariant is that \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -