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

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 \(1, √2, √3, , √n\) — cumulative angle \(θ_n = ∑ₖ₌₁^n (1/√k)\). - Fibonacci–Theodorus variant: triangle sides \(F_1, F_2, F_3, \) (Fibonacci numbers), so each right triangle has legs \(F_k\) and \(Fₖ₊₁\), hypotenuse \(Fₖ₊₂\) (since \(F_k^2 + Fₖ₊₁^2 = Fₖ₊₂^2\) only for \(k=1\) — but paper generalizes via non‑integer scaling). - Key ratio: successive Fibonacci numbers → \(limk→∞ Fₖ₊₁/F_k = φ = 1.6180339887…\) - Golden angle: \(θ_g = 2π(1 - 1/φ) = 2π/φ^2 ≈ 2.399963\) rad \(≈ 137.507764^∘\). - Curvature of Fibonacci–Theodorus spiral: \(κ ∝ 1/(φ²ᵏ) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com