Difference between revisions of "Fractal"

(New page: A fractal is defined as a figure that does not become simpler under any level of magnification. ==Mandelbrot set== Probably the most well-known example of a fractal, the Mandelbrot set is...)
 
 
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Probably the most well-known example of a fractal, the Mandelbrot set is the set of all points <math>c</math> in the [[complex plane]] for which the [[sequence]] <math>z_0=0, z_{n+1}=z_n^2+c</math> is bounded.
 
Probably the most well-known example of a fractal, the Mandelbrot set is the set of all points <math>c</math> in the [[complex plane]] for which the [[sequence]] <math>z_0=0, z_{n+1}=z_n^2+c</math> is bounded.
  
This fractal is NOT [[self-similarity|self-similar]]. However, it is almost self-similar.[[Image:MandelbrotSet.png|thumb|If one were to plot all points in the Mandelbrot set using the complex plane, it would look like this.]]
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This fractal is NOT [[self-similarity|self-similar]]. However, it is almost self-similar. If one were to plot all points in the Mandelbrot set using the complex plane, it would look like this.
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<asy>
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size(400);
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int f(pair c)
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{
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pair z=(0,0);
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int k;
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for(k=0;(k<99)&&(abs(z)<10);++k) z=z^2+c;
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return floor(k/10);
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}
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pen[] p={white,yellow,orange,blue,green,orange,magenta,red,brown,black};
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real h=0.007;
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for(int k=-350;k<70;++k)
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for(int m=0;m<200;++m)
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{
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pair P=h*((k,0)+m*dir(60));
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int n=f(P);
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if (n>0)
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{
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dot(P,linewidth(1.5)+p[n]); if(m>0) dot((P.x,-P.y),linewidth(1.5)+p[n]);
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}
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}
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</asy>

Latest revision as of 17:28, 8 January 2012

A fractal is defined as a figure that does not become simpler under any level of magnification.

Mandelbrot set

Probably the most well-known example of a fractal, the Mandelbrot set is the set of all points $c$ in the complex plane for which the sequence $z_0=0, z_{n+1}=z_n^2+c$ is bounded.

This fractal is NOT self-similar. However, it is almost self-similar. If one were to plot all points in the Mandelbrot set using the complex plane, it would look like this.

[asy] size(400);  int f(pair c) { pair z=(0,0); int k; for(k=0;(k<99)&&(abs(z)<10);++k) z=z^2+c; return floor(k/10); }  pen[] p={white,yellow,orange,blue,green,orange,magenta,red,brown,black};  real h=0.007;   for(int k=-350;k<70;++k) for(int m=0;m<200;++m) { pair P=h*((k,0)+m*dir(60)); int n=f(P); if (n>0) { dot(P,linewidth(1.5)+p[n]); if(m>0) dot((P.x,-P.y),linewidth(1.5)+p[n]); } } [/asy]

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