This movie shows the sense in which Julia sets are self-similar.   The Julia set for a number z is the set of complex numbers that you can hit over and over with the function

f(x) = x^2 + z

and get a sequence of numbers that remains bounded.  By definition, the Julia set gets mapped to itself by this function f. 

In this movie, when it's not wiggling around, the black stuff is the Julia set for a number z equal to roughly 0.8 + 0.2 i.  But it's animated: at time t, we see what happens when you take the Julia set and apply the function

f(x,t) = x^{2^t} + tz. 

When t = 0 this function does nothing.  By the time t = 1, this function equals

f(x) = x^2 + z

so it maps the Julia set into itself.  And then the animation loops around!

Anders Kaseorg put this animated gif on Quora:

http://www.quora.com/Fractals/Why-are-Julia-sets-fractals

but I don't know where he got it.  And by the way, what I'm calling the Julia set for the number z is technically called the filled Julia set for the function f(x) = x^2 + z.  For more definitions and pictures, see:

http://en.wikipedia.org/wiki/Julia_set

#fractals