During AMC testing, the AoPS Wiki is in read-only mode. No edits can be made.

Fractal

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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