Fibonacci–Theodorus Spiral Unifies Golden Ratio and Discrete Geometry — E8 Intelligence Research

FINDING: The Fibonacci–Theodorus spiral unifies the classical Theodorus spiral (concatenated right triangles) with Fibonacci-number side lengths, yielding a novel discrete spiral with emergent golden-ratio curvature properties. | MATH: Theodorus spiral: \(r_n = √n\), \(θ_n = ∑ₖ₌₁^n (1/√k)\). Fibonacci–Theodorus variant: triangle sides \(F_k, Fₖ₊₁\) (Fibonacci numbers), hypotenuse \(Fₖ₊₂\) (since \(F_k^2 + Fₖ₊₁^2 ≈ Fₖ₊₂^2\) only asymptotically — exact only for degenerate cases; the paper likely uses a corrected angle \(θ_k = (Fₖ₊₁/F_k)\)). Key constants: golden ratio \(φ = (1+√5)/2 ≈ 1.618\), its inverse \(φ⁻¹ ≈ 0.618\), and \(φ⁻² ≈ 0.382\). The logarithmic spiral curvature parameter \(b\) satisfies \(r(θ) = a ebθ\), with \(b = ln(φ)/(π/2) ≈ 0.306\) for the golden spiral (quarter-turn scaling by \(φ\)). | CONNECTION: The Fibonacci–Theodorus spi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com