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: \(r_n = \sqrt{n}\), \(\theta_n = \sum_{k=1}^n \arctan(1/\sqrt{k})\). Fibonacci–Theodorus variant: triangle sides \(F_k, F_{k+1}\) (Fibonacci numbers), hypotenuse \(F_{k+2}\) (since \(F_k^2 + F_{k+1}^2 \approx F_{k+2}^2\) only asymptotically — exact only for degenerate cases; the paper likely uses a corrected angle \(\theta_k = \arctan(F_{k+1}/F_k)\)). Key constants: golden ratio \(\varphi = (1+\sqrt{5})/2 \approx 1.618\), its inverse \(\varphi^{-1} \approx 0.618\), and \(\varphi^{-2} \approx 0.382\). The logarithmic spiral curvature parameter \(b\) satisfies \(r(\theta) = a e^{b\theta}\), with \(b = \ln(\varphi)/(\pi/2) \approx 0.306\) for the golden spiral (quarter-turn scaling by \(\varphi\)). | CONNECTION: The Fibonacci–Theodorus spi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com