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?

I would resume it in one sentence:
ITERATION (OR FEEDBACK) CREATES COMPLEXITY

Julia sets and the Mandelbrot set all come from the iteration of the family of functions $q_c(z)=z^2+c$. What could be simpler that such a second degree polynomial? The complexity comes from the iteration
$$
z_{n+1}=q_c(z_n).
$$
Even if the function $q_c$ is simple, the behaviour of the sequence $\{z_n\}$ may be very complex. In fact, it may be chaotic.
We are in the presence of sensitive dependence on initial conditions (a.k.a. the butterfly effect.) Iterations that begin at two very close initial points, may separate after a sufficiently large number of iterations. The Mandelbrot set is the set of complex nmbers $c$ such that the sequence
$$
0,\ q_c(0)=c,\ q_c(q_c(0))=c^2+c,\ q_c(q_c(q_c(0)))=(c^2+c)^2+c,\dots
$$
remains bounded. The behaviour of that sequence may me very different for two very close values of $c$. That is the reason why you see that complex behaviour when zooming up on a small region.
