Mandelbrot set
Fibonacci sequence in the Mandelbrot set

The Mandelbrot Set features a fundamental cardioid shape adorned with numerous bulbs directly attached to it. Understanding the arrangement of these bulbs requires a detailed examination of the Mandelbrot Set's boundary. As one zooms into specific portions with a geometric perspective, precise deducible information about the location within the boundary and the corresponding dynamical behavior for parameters drawn from associated bulbs emerges.

The iteration of the quadratic polynomial f c (z) = z 2 + c {\displaystyle f_{c}(z)=z^{2}+c}, where c {\displaystyle c} is a parameter drawn from one of the bulbs attached to the main cardioid within the Mandelbrot Set, gives rise to maps featuring attracting cycles of a specified period q {\displaystyle q} and a rotation number p / q {\displaystyle p/q}. In this context, the attracting cycle of exhibits rotational motion around a central fixed point, completing an average of p / q {\displaystyle p/q} revolutions at each iteration.

The bulbs within the Mandelbrot Set are distinguishable by both their attracting cycles and the geometric features of their structure. Each bulb is characterized by an antenna attached to it, emanating from a junction point and displaying a certain number of spokes indicative of its period. For instance, the 2 / 5 {\displaystyle 2/5} bulb is identified by its attracting cycle with a rotation number of 2 / 5 {\displaystyle 2/5}. Its distinctive antenna-like structure comprises a junction point from which five spokes emanate. Among these spokes, called the principal spoke is directly attached to the 2 / 5 {\displaystyle 2/5} bulb, and the 'smallest' non-principal spoke is positioned approximately 2 / 5 {\displaystyle 2/5} of a turn counterclockwise from the principal spoke, providing a distinctive identification as a 2 / 5 {\displaystyle 2/5} -bulb. This raises the question: how does one discern which among these spokes is the 'smallest'? In the theory of external rays developed by Douady and Hubbard. there are precisely two external rays landing at the root point of a satellite hyperbolic component of the Mandelbrot Set. Each of these rays possesses an external angle that undergoes doubling under the angle doubling map θ ↦ {\displaystyle \theta \mapsto } 2 θ {\displaystyle 2\theta }. According to this theorem, when two rays land at the same point, no other rays between them can intersect. Thus, the 'size' of this region is measured by determining the length of the arc between the two angles.

If the root point of the main cardioid is the cusp at c = 1 / 4 {\displaystyle c=1/4}, then the main cardioid is the 0 / 1 {\displaystyle 0/1} -bulb. The root point of any other bulb is just the point where this bulb is attached to the main cardioid. This prompts the inquiry: which is the largest bulb between the root points of the 0 / 1 {\displaystyle 0/1} and 1 / 2 {\displaystyle 1/2} -bulbs? It is clearly the 1 / 3 {\displaystyle 1/3} -bulb. And note that 1 / 3 {\displaystyle 1/3} is obtained from the previous two fractions by Farey addition, i.e., adding the numerators and adding the denominators

0 1 {\displaystyle {\frac {0}{1}}} ⊕ {\displaystyle \oplus } 1 2 {\displaystyle {\frac {1}{2}}} = {\displaystyle =} 1 3 {\displaystyle {\frac {1}{3}}}

Similarly, the largest bulb between the 1 / 3 {\displaystyle 1/3} and 1 / 2 {\displaystyle 1/2} -bulbs is the 2 / 5 {\displaystyle 2/5} -bulb, again given by Farey addition.

1 3 {\displaystyle {\frac {1}{3}}} ⊕ {\displaystyle \oplus } 1 2 {\displaystyle {\frac {1}{2}}} = {\displaystyle =} 2 5 {\displaystyle {\frac {2}{5}}}

The largest bulb between the 2 / 5 {\displaystyle 2/5} and 1 / 2 {\displaystyle 1/2} -bulb is the 3 / 7 {\displaystyle 3/7} -bulb, while the largest bulb between the 2 / 5 {\displaystyle 2/5} and 1 / 3 {\displaystyle 1/3} -bulbs is the 3 / 8 {\displaystyle 3/8} -bulb, and so on. The arrangement of bulbs within the Mandelbrot set follows a remarkable pattern governed by the Farey tree, a structure encompassing all rationals between 0 {\displaystyle 0} and 1 {\displaystyle 1}. This ordering positions the bulbs along the boundary of the main cardioid precisely according to the rational numbers in the unit interval.

Starting with the 1 / 3 {\displaystyle 1/3} bulb at the top and progressing towards the 1 / 2 {\displaystyle 1/2} circle, the sequence unfolds systematically: the largest bulb between 1 / 2 {\displaystyle 1/2} and 1 / 3 {\displaystyle 1/3} is 2 / 5 {\displaystyle 2/5}, between 1 / 3 {\displaystyle 1/3} and 2 / 5 {\displaystyle 2/5} is 3 / 8 {\displaystyle 3/8}, and so forth. Intriguingly, the denominators of the periods of circular bulbs at sequential scales in the Mandelbrot Set conform to the Fibonacci number sequence, the sequence that is made by adding the previous two terms – 1, 2, 3, 5, 8, 13, 21...

The Fibonacci sequence manifests in the number of spiral arms at a unique spot on the Mandelbrot set, mirrored both at the top and bottom. This distinctive location demands the highest number of iterations of for a detailed fractal visual, with intricate details repeating as one zooms in.
oulipo:avoid_letter[a], detectable_format:number_bullet_lists[3]