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

Abstract

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 \(z_n = \sum_{k=1}^n i \sqrt{k}\) (or similar), with side lengths \(\sqrt{k}\). Fibonacci–Theodorus variant: side lengths \(F_k\) (Fibonacci numbers), so the \(n\)-th triangle has legs \(F_n, F_{n+1}\), hypotenuse \(F_{n+2}\) (since \(F_{n+1}^2 + F_n^2 \approx F_{n+2}^2\) only asymptotically — exact only for \(n=1\): \(1^2+1^2=2 \neq 2^2\); the paper likely uses a modified closure). Golden angle: \(\theta_g = 2\pi(1-\phi^{-1}) = 2\pi(2-\phi) \approx 137.507764^\circ \approx 2.399963\) rad. Golden ratio: \(\phi = (1+\sqrt{5})/2 \approx 1.6180339887\). Related constants: \(\phi^{-1} = \phi-1 \approx 0.6180339887\), \(\phi^{-2} \approx 0.381966\), \(\phi^{-3} \approx 0.236068\), \(\phi^{-4} \approx 0.145898\). | CONNECTION: The golden angle is the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com