Mandelbrot fractal: How is it possible? I'm a programmer and have recently played around a bit with rendering Mandelbrot fractals / zooming into them.
What I can't grasp: How can such infinite, complex shapes come out of somewhat 10 lines of deterministic code?
How is it possible that when zooming ever deeper and deeper, there are still completely new shapes coming up, while the algorithm remains the same?
Does the set maybe give us some deep insight into our universe or even other dimensions, as it involves complex numbers?

Let $m(z,c) = z^2 + c$, consider the sequence of polynomials: $m(z,z),\,$ $m(m(z,z),z)$, ... , which are $$z^2+z,\quad z^4 + 2z^3 + z^2 + z,\quad z^8 + 4z^7 + 6z^6 + 6z^5 + 5z^4 + 2z^3 + z^2 + z,\quad...\quad.$$
Note that in terms of complex numbers the transformation $z \mapsto m(z,c)$ can be seen a way of twisting and squashing the sphere over itself.. if you keep kneading something you are bound to get tearing and crumpled filaments and such like.
Here are graphs of the first seven (produced by the software here):







