Fibonacci Numbers in Mandelbrot Dynamics: Closed-Form Tree Computation — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Fibonacci numbers appear in Mandelbrot set dynamics (via period-doubling and Fibonacci-indexed cycles), and a closed-form tree representation exists for nth-term computation without recursion. | MATH: Fibonacci recurrence \(F_n = F_{n-1} + F_{n-2}\), Binet form \(F_n = \frac{\varphi^n - (-\varphi)^{-n}}{\sqrt{5}}\), \(\varphi = \frac{1+\sqrt{5}}{2} = 1.618...\); Mandelbrot set: \(z_{n+1} = z_n^2 + c\), with Fibonacci numbers governing counts of periodic orbits at certain \(c\) values (e.g., period-\(F_k\) cycles near the main cardioid's bifurcation cascade). | CONNECTION: Direct — \(\varphi\) and its inverse \(1/\varphi = 0.618\) are the golden ratio; the ratio \(F_{n+1}/F_n \to \varphi\) links to 0.382 = \(\varphi^{-2}\), 0.786 ≈ \(\sqrt{\varphi^{-1}}\) (used in harmonic trading); Mandelbrot's self-similarity mirrors crystallographic scaling symmetries (e.g., 5-fold Penrose tilings, which use \(\varphi\) and have no periodic lattice but exhibit quasi-crystalline order). | DEP Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212074697
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