# Day #21: Mandelbrot Set and the Fractal Culture

Do you actually know what is the dimension of the boundary of the Mandelbrot set??…

A: The boundary of the Mandelbrot set has Hausdorf dimension… 2. Source: List of Fractals by Hausdorf Dimension and http://math.bu.edu/DYSYS/chaos-game/node6.html

### Mandelbrot Set

##### Mandelbrot set detail

The Mandelbrot set is the set of complex numbersc for which the function f c ( z ) = z 2 + c {\displaystyle f_{c}(z)=z^{2}+c} does not diverge when iterated from z = 0 {\displaystyle z=0} , i.e., for which the sequence f c ( 0 ) {\displaystyle f_{c}(0)} , f c ( f c ( 0 ) ) {\displaystyle f_{c}(f_{c}(0))} , etc., remains bounded in absolute value.

The set is closely related to the idea of Julia sets, which produce similarly complex shapes. Its definition and name are due to Adrien Douady, in tribute to the mathematicianBenoit Mandelbrot.[1]

Mandelbrot set images may be created by sampling the complex numbers and determining, for each sample point c, whether the result of iterating the above function goes to infinity. Treating the real and imaginary parts of each number c as image coordinates, pixels may then be colored according to how rapidly the sequence diverges, with the color 0 (black) usually used for points where the sequence does not diverge.

Images of the Mandelbrot set exhibit an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The “style” of this repeating detail depends on the region of the set being examined. The set’s boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.

The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of mathematical visualization.

#### History

The first published picture of the Mandelbrot set, by Robert W. Brooks and Peter Matelski in 1978
##### The work of Douady and Hubbard coincided with a huge increase in interest in complex dynamics and abstract mathematics, and the study of the Mandelbrot set has been a centerpiece of this field ever since. An exhaustive list of all the mathematicians who have contributed to the understanding of this set since then is beyond the scope of this article, but such a list would notably include Mikhail Lyubich,[11][12]Curt McMullen, John Milnor, Mitsuhiro Shishikura, and Jean-Christophe Yoccoz.

Source:

Benoit Mandelbrot

https://en.wikipedia.org/wiki/Mandelbrot_set

http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html

https://de.wikipedia.org/wiki/Mandelbrot-Menge