Voronoi Tessellation of Fermat Spirals Bridges Lattice Regularity — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Voronoi tessellation of Fermat spiral phyllotaxis yields a hexagonal regularity index that interpolates between square and honeycomb lattices, with the Poisson-Voronoi limit as the generic case. | MATH: Fermat spiral: \( r = c\sqrt{\theta} \) (golden angle \(\theta \approx 137.5077^\circ = 2\pi/\phi^2\)); Voronoi cell area distribution for Poisson-Voronoi: \( f(A) = \frac{343}{15}\sqrt{\frac{7}{2\pi}} A^{5/2} e^{-7A/2} \) (mean area normalized to 1); hexagonal regularity index \( H = \frac{6A}{\sqrt{3}P^2} \) (H=1 for perfect hexagon, H→0 for degenerate cells); symmetry-break parameter \( \lambda \) interpolates from square (\(H \approx 0.785\)) to honeycomb (\(H=1\)) to Poisson (\(H \approx 0.907\) — the known universal value). | CONNECTION: **Golden ratio \(\phi = 1.618\)** appears directly in the Fermat spiral divergence angle \(2\pi/\phi^2 \approx 2.39996\) rad — the phyllotaxis packing that maximizes Voronoi cell regularity. The Poisson-Voronoi hexagonality \(H \approx 0. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204491223
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