Projective Duality Transforms Windmill Process into Billiard with Parity Invariant Ensuring Recurrence — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Windmill Problem (2011 IMO P2) — A line rotating through a finite planar point set (no three collinear) pivots at each encountered point; there exists a starting configuration visiting all points infinitely often | MATH: Projective duality maps points↔lines; windmill process becomes a billiard on the dual arrangement; parity invariant on half-plane point counts forces global recurrence | CONNECTION: Crystallographic root system A₂ symmetry in the dual line arrangement; pivot sequence generates a discrete dynamical system on the projective plane with 120° rotational symmetry (2π/3) | DEPTH: 9 FINDING: Average Cube Shadow Area — Cauchy's surface area formula: mean projection area of convex body = ¼ × surface area; for unit cube: 6/4 = 1.5 | MATH: ⟨A_proj⟩ = (1/4) ∫_∂K dS = S/4; cube: S=6 → ⟨A⟩=3/2; generalizes to n-dimensions: ⟨V Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225467680