Unifying Fibonacci and Theodorus Spirals via Golden Angle Curvature — E8 Intelligence Research

FINDING: The Fibonacci–Theodorus spiral unifies the classical Theodorus spiral (concatenated right triangles) with Fibonacci-number side lengths, revealing a new curvature relation tied to the golden angle. | MATH: Theodorus spiral: vertices at \(z_n = ∑ₖ₌₁^n i √k\) (or similar), with side lengths \(√k\). Fibonacci–Theodorus variant: side lengths \(F_k\) (Fibonacci numbers), so the \(n\)-th triangle has legs \(F_n, Fₙ₊₁\), hypotenuse \(Fₙ₊₂\) (since \(Fₙ₊₁^2 + F_n^2 ≈ Fₙ₊₂^2\) only asymptotically — exact only for \(n=1\): \(1^2+1^2=2 ≠ 2^2\); the paper likely uses a modified closure). Golden angle: \(θ_g = 2π(1-φ⁻¹) = 2π(2-φ) ≈ 137.507764^∘ ≈ 2.399963\) rad. Golden ratio: \(φ = (1+√5)/2 ≈ 1.6180339887\). Related constants: \(φ⁻¹ = φ-1 ≈ 0.6180339887\), \(φ⁻² ≈ 0.381966\), \(φ⁻³ ≈ 0.236068\), \(φ⁻⁴ ≈ 0.145898\). | CONNECTION: The golden angle is the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com