# 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 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 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.