# Post

Originally shared by John BaezThis movie shows the sense in which Julia sets are self-similar. The

f(x) = x^2 + z

and get a sequence of numbers

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

**Julia set**for a number z is the set of complex numbers that you can hit over and over with the functionf(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

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