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. | + | 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. |
| + | |||
| + | <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> | ||
Latest revision as of 18:28, 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 
in the complex plane for which the sequence 
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]](http://latex.artofproblemsolving.com/f/a/1/fa104cf432d5df3ce6a0111f9f9a747a1170761a.png)
