Fibonacci vs. Theodorus Spirals: Distinct Triangle-Based Square-Root Link — E8 Intelligence Research

FINDING: Fibonacci spiral is a logarithmic spiral approximation with ratio φ, but the Fibonacci–Theodorus spiral offers a distinct, triangle-based construction linking Fibonacci numbers to square-root spirals. | MATH: Golden ratio φ = (1+√5)/2 ≈ 1.6180339887; reciprocal φ⁻¹ = φ−1 ≈ 0.6180339887; φ² = φ+1 ≈ 2.6180339887. Fibonacci recurrence Fₙ = Fₙ₋₁ + Fₙ₋₂, F₀=0, F₁=1. Logarithmic spiral polar equation r(θ) = a·e^(bθ), where for Fibonacci approximation b = (2·ln φ)/π ≈ 0.30635 (per quarter-turn). Fibonacci–Theodorus spiral: concatenated right triangles with legs of lengths Fₙ and Fₙ₊₁, hypotenuse √(Fₙ² + Fₙ₊₁²) — not a pure logarithmic spiral but a polygonal approximation whose curvature varies with n. | CONNECTION: The golden ratio appears as the limit of Fₙ₊₁/Fₙ → φ. The ratio 0.618 (φ⁻¹) is the self-similar scaling factor in the spiral's successive quarter-turns. The Fibonacci–Theodorus construction involves √(Fₙ² + Fₙ₊₁²) — for large n, this hypotenuse ≈ Fₙ₊₁·√(1+φ⁻²) = Fₙ₊₁·√(2−φ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com