The Mandelbrot set is made by iteration of the complex map (function)
z_{n+1} = z_{n}^{2} + C for different C
and z_{o} = 0.
Points on the parameter plane C with bounded z_{n}
form the Mandelbrot set (the black region on these pictures). Color outside
the Mandelbrot set shows, how fast z_{n} go to infinity.
The Mandelbrot set contains small copies of the "main cardioid"
(in the white square) shown on the right picture. These small copies are
connected with the main cardioid by filaments which are formed by other
tiny cardioids. These strucrures are called the "Mandelbrot hair"
or filaments.
Each point C in the parameter space specifies the geometric structure of the corresponding Julia set J(C) in the dynamical plane z. If C_{1} is in the Mandelbrot set, the Julia set is connected. If C_{2} is not in the Mandelbrot set, the Julia set is a Cantor dust. You see connected 'Douady's rabbit' and 'Cantor dust' below.
You see that every J set is self-similar. But the most amazing thing is that the M's filaments and corresponding J set are similar too (see the right top and bottom images). And two more similar M and J sets below. |
More fractals
Now we can go to a Julia orbit trip. 600x600 Mandelbrot set.
See also the Mandelbrot Set in Eric W. Weisstein's "World of Mathematics"