Golden Angle and Fibonacci–Theodorus Spiral Unify Plant Phyllotaxis — E8 Intelligence Research

FINDING: The golden angle (≈137.507764°), derived from the golden ratio, governs phyllotaxis (leaf/flower arrangement) and produces Fibonacci spiral patterns in plants; a new Fibonacci–Theodorus spiral extends this to concatenated right triangles. MATH: - Golden angle: \( \theta = 360^\circ \times (1 - 1/\phi) = 360^\circ \times (2 - \phi) = 360^\circ / \phi^2 \approx 137.507764^\circ \), where \( \phi = (1+\sqrt{5})/2 \approx 1.6180339887 \). - Equivalent: \( \theta = 2\pi / \phi^2 \) radians. - Fibonacci recurrence: \( F_{n+1} = F_n + F_{n-1} \), with \( F_1 = F_2 = 1 \). - Phyllotaxis condition: successive primordia are placed at angle \( \theta \) from previous; the continued fraction of \( \theta/360^\circ = 1/\phi^2 = [0;2,1,1,1,\ldots] \) ensures maximal packing efficiency. - Fibonacci–Theodorus spiral (arXiv:2407.07109): triangle side lengths \( F_n \), hypotenuse \( \sqrt{F_n^2 + F_{n+1}^2} \), spiral angle increments \( \arctan(F_{n+1}/F_n) \to \arctan(\phi) \appr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com