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Julia Sets are fractals very closely related to the iconic Mandelbrot set, possibly the most famous image in mathematics. The Julia set for a given constant ** c** is the boundary of the set of all (complex) points for which the polynomial:

*z ^{2} + c*

converges under iteration. To show what this means it's best to consider an example, say if we had *c = -1*. Then we consider any complex point, say *0* and iterate the equation above.

*z _{0} = 0^{2} – 1 = -1*

*z _{1} = (-1)^{2} – 1 = 0*

*z _{2} = 0^{2} – 1 = -1*

We can see that this will continue forever and not reach infinity, so we say that *z = 0* converges, and this is a part of the **prisoner set** (in black below). Points that do not converge are said to be in the **escape set** (in colour below; red diverges quickly whereas green takes longer to reach infinity).

The Julia set is all the points that are neither in the escape set nor in the prisoner set, and is represented by the (almost invisible) boundary between the two areas.

Well, that's the science. How about the pictures?

The above image is the Julia set for the value *c = -1*. It was generated by a simple algorithm, the source code of which is viewable here. You can play about with the tool using the form below.