Mandelbrot Set

In order to understand how the Fibonacci sets are found inside the Mandelbrot Set, you must first be able to recognize the fractal. Here is a brief explanation from Wikipedia:

In mathematics, the Mandelbrot set, named after Benoît Mandelbrot, is a set of points in the complex plane, the boundary of which forms a fractal. Mathematically, the Mandelbrot set can be defined as the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded. That is, a complex number, c, is in the Mandelbrot set if, when starting with z0=0 and applying the iteration repeatedly, the absolute value of zn never exceeds a certain number (that number depends on c) however large n gets.

For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26… and so on. This sequence goes towards infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set.

On the other hand, c = i gives the sequence 0, i, (-1 + i), -i, (-1 + i), -i…, which is bounded, and so i belongs to the Mandelbrot set.

When computed and graphed on the complex plane, the Mandelbrot Set is seen to have an elaborate boundary, which does not simplify at any given magnification. This qualifies the boundary as a fractal.

The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and for being a complicated structure arising from a simple definition. Benoît Mandelbrot and others worked hard to communicate this area of mathematics to the public.

 

Here are a few labeled pictures of the Mandelbrot Set explaining its parts:

Mandelbrot1

Mandelbrot2

Mandelbrot3

Mandelbrot4


 

The period of a primary bulb is determined by counting the number of “spokes” on their main antenna.

This is where Fibonacci can be found again!

 

Mandelbrot5

 

 

 

This phenomenon is unexplainable via notions of how the math works, and it is still unknown why the Fibonacci sequence appears in the Mandelbrot Set.