Mandelbrot set

This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups.

[3] Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York.

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.

The Mandelbrot set has its origin in complex dynamics, a field first investigated by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century.

The fractal was first defined and drawn in 1978 by Robert W. Brooks and Peter Matelski as part of a study of Kleinian groups.

[3] On 1 March 1980, at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York, Benoit Mandelbrot first visualized the set.

The mathematicians Heinz-Otto Peitgen and Peter Richter became well known for promoting the set with photographs, books (1986),[7] and an internationally touring exhibit of the German Goethe-Institut (1985).

[11] The work of Douady and Hubbard occurred during an increase in interest in complex dynamics and abstract mathematics,[12] and the study of the Mandelbrot set has been a centerpiece of this field ever since.

In the same way, the boundary of the Mandelbrot set can be defined as the bifurcation locus of this quadratic family, the subset of parameters near which the dynamic behavior of the polynomial (when it is iterated repeatedly) changes drastically.

This conjecture was based on computer pictures generated by programs that are unable to detect the thin filaments connecting different parts of

[17] The dynamical formula for the uniformisation of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of

These algebraic curves appear in images of the Mandelbrot set computed using the "escape time algorithm" mentioned below.

(Note that hyperbolic components intersecting the real axis correspond exactly to periodic windows in the Feigenbaum diagram.

By the work of Adrien Douady and John H. Hubbard, this conjecture would result in a simple abstract "pinched disk" model of the Mandelbrot set.

[citation needed] The Hausdorff dimension of the boundary of the Mandelbrot set equals 2 as determined by a result of Mitsuhiro Shishikura.

[27] The fact that this is greater by a whole integer than its topological dimension, which is 1, reflects the extreme fractal nature of the Mandelbrot set boundary.

Roughly speaking, Shishikura's result states that the Mandelbrot set boundary is so "wiggly" that it locally fills space as efficiently as a two-dimensional planar region.

[citation needed] It has been shown that the generalized Mandelbrot set in higher-dimensional hypercomplex number spaces (i.e. when the power

[32] The Mandelbrot Set features a fundamental cardioid shape adorned with numerous bulbs directly attached to it.

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.

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

In this context, the attracting cycle of  exhibits rotational motion around a central fixed point, completing an average of

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.

[34][38] The arrangement of bulbs within the Mandelbrot set follows a remarkable pattern governed by the Farey tree, a structure encompassing all rationals between

This ordering positions the bulbs along the boundary of the main cardioid precisely according to the rational numbers in the unit interval.

These two groups can be attributed by some metamorphosis to the two "fingers" of the "upper hand" of the Mandelbrot set; therefore, the number of "spokes" increases from one "seahorse" to the next by 2; the "hub" is a Misiurewicz point.

[citation needed] The Multibrot set is obtained by varying the value of the exponent d. The article has a video that shows the development from d = 0 to 7, at which point there are 6 i.e.

Another non-analytic generalization is the Burning Ship fractal, which is obtained by iterating the following: There exist a multitude of various algorithms for plotting the Mandelbrot set via a computing device.

Otherwise, keep iterating up to a fixed number of steps, after which we decide that our parameter is "probably" in the Mandelbrot set, or at least very close to it, and color the pixel black.

Here is the code implementing the above algorithm in Python: The value of power variable can be modified to generate an image of equivalent multibrot set (