Fibonacci–Theodorus Spiral: Golden-Ratio Harmonics in Curvature — E8 Intelligence Research

Abstract

FINDING: The Fibonacci–Theodorus spiral unifies the classical Theodorus spiral (concatenated right triangles) with Fibonacci-number side lengths, yielding a novel logarithmic-like spiral whose curvature encodes golden-ratio harmonics. MATH: - Classical Theodorus spiral: triangles with sides \(1, \sqrt{2}, \sqrt{3}, \dots, \sqrt{n}\) — cumulative angle \(\theta_n = \sum_{k=1}^n \arctan(1/\sqrt{k})\). - Fibonacci–Theodorus variant: triangle sides \(F_1, F_2, F_3, \dots\) (Fibonacci numbers), so each right triangle has legs \(F_k\) and \(F_{k+1}\), hypotenuse \(F_{k+2}\) (since \(F_k^2 + F_{k+1}^2 = F_{k+2}^2\) only for \(k=1\) — but paper generalizes via non‑integer scaling). - Key ratio: successive Fibonacci numbers → \(\lim_{k\to\infty} F_{k+1}/F_k = \varphi = 1.6180339887\ldots\) - Golden angle: \(\theta_g = 2\pi(1 - 1/\varphi) = 2\pi/\varphi^2 \approx 2.399963\) rad \(\approx 137.507764^\circ\). - Curvature of Fibonacci–Theodorus spiral: \(\kappa \propto 1/(\varphi^{2k}) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com