Julia sets

Consider the quadratic map $f_c(z)=z^2+c$ and define

$z_{n+1}=f_c(z_n)=z_n^2+c$

where $z_n$ and $c$ are complex numbers. Given an initial condition $z_0$, we can compute its image $z_1=f_c(z_0)$, next the image of its image, namely $z_2=f_c(z_1)=f_c(f_c(z_0))$, and so forth and so on. The infinite sequence $(z_0,z_1,...)$ gives the trajectory of $z_0$ under the map $f_c$ in the complex plane.

The Julia set associated with a quadratic polynomial $f_c$ is, informally speaking, the set of initial conditions $z_0$ whose trajectories do not tend to infinity. Depending on the value of $c$, you get different Julia sets, the beauty of which is hard to deny.