User:PiRSquared17/Mandelbrot set

File:Mandel zoom 00 mandelbrot set.jpg
An image of the Mandelbrot set; the color shows how quickly the sequence (see below) for that value c grew (black means in the set).

The Mandelbrot set, named after Benoît Mandelbrot, is a famous example of a fractal. It is based on the equation zn+1 = zn2 + c, where c and z are complex numbers and n is zero or a positive integer (natural number). Starting with z0=0, c is in the Mandelbrot set if the absolute value of zn does not go off to infinity however large n gets. An equivalent definition of the Mandelbrot set is that it is the set of all complex numbers c such that the Julia set (2) of c is connected (i.e., 0 is a member of that Julia set).

For example, if c = 1 then the sequence is 0, 1, 2, 5, 26,…, which goes to infinity. Therefore, 1 is not an element of the Mandelbrot set.

On the other hand, if c is equal to the imaginary unit, i, (i is defined by <math>i^2 = -1</math>), then the sequence is 0, i, (−1 + i), −i, (−1 + i), −i…, which does not go to infinity (it stays bounded). This means that i is in the Mandelbrot set.

When graphed, the Mandelbrot set is very pretty[source?] and recognizable.

A generalization of Mandelbrot sets allows any exponent: zn+1 = znd + c. These sets are called Multibrot sets. The Multibrot set for d = 2 is the Mandelbrot set.

History

File:Mandel.png
Picture of Mandelbrot set made by Robert W. Brooks and Peter Matelski in 1978. This is the first picture of the Mandelbrot set.

The Mandelbrot set is related to complex dynamics, which was first studied by Gaston Julia and Pierre Fatou. The first pictures of this fractal were drawn in 1978 by Robert W. Brooks and Peter Matelski.[1][2][3]

Properties

The Mandelbrot set is compact.

Self-similarity

File:Mandelbrot zoom.gif
Animation showing self-similarity in the Mandelbrot set
File:Mandelzoom1.jpg
Self-similarity around Misiurewicz point −0.1011 + 0.9563i.
File:Blue Mandelbrot Zoom.jpg
Quasi-self-similarity: a close copy of the Mandelbrot self found inside the Mandelbrot set

The Mandelbrot set is self-similar. It contains an infinite number of little copies of itself. These small copies are slightly different (mostly because of the threads connecting them to the main set).

References

  1. Robert Brooks and Peter Matelski, The dynamics of 2-generator subgroups of PSL(2,C), in "Riemann Surfaces and Related Topics", ed. Kra and Maskit, Ann. Math. Stud. 97, 65–71, ISBN 0-691-08264-2
  2. http://www.math.harvard.edu/archive/118r_spring_05/handouts/mandelbrot.pdf
  3. http://books.google.com/books?lr=&id=S9fCX9j_JPwC&pg=PA6
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