FINDING: Continued fraction convergents of the golden ratio (φ) govern optimal spiral phyllotaxis beyond Fibonacci numbers, with higher-order convergents producing more efficient packing in non-standard phyllotactic patterns. | MATH: φ = (1+√5) /2 ≈ 1. 6180339; its continued fraction is 1;1, 1, 1,. . . . Convergents: F₍+₁/Fₙ (Fibonacci ratios: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13,. . . → φ). Beyond Fibonacci, convergents of related quadratic irrationals (e. g. , √3, √5, √7) produce spacing angles like 137. 5° (Fibonacci), 99. 5°, 77. 9°, 63. 1°, etc. , each corresponding to optimal divergence angles for different phyllotactic classes. Key constants: 0. 382 (1/φ²), 0. 618 (1/φ), 1. 618 (φ), 2. 618 (φ²). | CONNECTION: The golden ratio's convergents directly map to crystallographic-like lattice spirals in plants. The divergence angle 360°/φ² ≈ 137. 5° arises from the most irrational number (φ). Higher convergents (e. g. , 21/13, 34/21) yield finer angular increments, analogous to quasicrystal diffraction Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com
Andrew Stewart Caldin (Mon,) studied this question.