Farey Sequence and Fibonacci Periodicities in the Mandelbrot Set's Bulbs — E8 Intelligence Research

Abstract

FINDING: The Mandelbrot set's main cardioid angle ratios follow the Farey sequence, generating Fibonacci-related periodicities in its bulbs. | MATH: The main cardioid's internal angles are rational fractions of 2π. The largest bulb (period-2) is at angle 1/2. The next largest (period-3) bulbs are at 1/3 and 2/3. The period-4 bulbs are at 1/4 and 3/4. The period-5 bulbs at 1/5, 2/5, 3/5, 4/5. The Fibonacci sequence emerges in the ordering of these fractions via the Farey addition: (a/b) ⊕ (c/d) = (a+c)/(b+d). The golden ratio angle (1/φ² ≈ 0.382) and its complement (1/φ ≈ 0.618) appear as limiting ratios in the Farey tree, corresponding to the Feigenbaum point and the onset of chaos. | CONNECTION: The golden ratio φ = (1+√5)/2 ≈ 1.618 appears as the limiting ratio of successive Fibonacci numbers. The angle 1/φ² ≈ 0.382 and 1/φ ≈ 0.618 are the two fixed points of the Farey tree's self-similar structure. These are the same ratios found in pentagonal symmetry (crystallographic point group Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com