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'Mandelbrot set'
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| Title of object: |
Mandelbrot set |
| Canonical Name: |
MandelbrotSet |
| Type: |
Definition |
| Created on: |
2007-04-21 18:36:57 |
| Modified on: |
2007-04-25 19:41:56 |
| Classification: |
msc:28A80 |
Revision comment (for changes between this and next version):
| Tried to clarify per correction #11777 ('Iterated function, Julia sets'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
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Content:
The {\em Mandelbrot set} is a map on the complex plane of all possible Julia sets. Defined by the iterated function $f_c(z) = z^2 + c$, where $c$ is a complex number $a + bi$ and $z$ is the value of previous iteration, a point on the set is painted black if $c$ stabilizes and colored if $c$ escapes. $c$ is said to escape if the iterations tend to infinity, otherwise, the iterates tend to stick to a small value.
The Mandelbrot set is perhaps the most famous fractal. At every level, it has smaller copies of the overall shape that are connected to the main shape. The higher the resolution, the easier it is to see that the entire Mandelbrot set is simply connected, with no holes. Discovered by Beno\^it Mandelbrot, the first print-out of it was in black and white.
\begin{center}
\includegraphics{C:TempMandelbrotSetBW}
\end{center}
Most of the Mandelbrot set lies within the unit disk of the complex plane. The second largest `circle' (to the left of the main `circle' in the standard illustration) is centered at $-1 + 0i$. Points with real part outside the range $-2 < a < 1$ and imaginary part $(1 < |b|)i$ will certainly tend to infinity.
Nowadays, most images of the Mandelbrot set include colors, with the colors for escaping values of $c$ being assigned in some correspondence to how many iterations it took to determine that $c$ escaped.
If a point $c$ on the Mandelbrot set does not escape, then the corresponding Julia set will be connected. Some fractal programs, like FractInt, offer the option of clicking anywhere on the Mandelbrot set and then drawing the corresponding Julia set. The Julia set for $0 + 0i$ is just a circle; most of the interesting Julia sets tend to be on the edges of the Mandelbrot set. |
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