|This user page or section is currently being expanded by an editor. You are welcome to help in expanding too. If this page has not been changed in several days, please remove this template.|
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 ), 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[change | change source]
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.
Properties[change | change source]
This section needs more information.
The Mandelbrot set is compact.
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[change | change source]
- 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
|Wikimedia Commons has media related to Mandelbrot set.|