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From Julia Sets you can also connect to the Higher Dimensions theme

Julia Sets animation
by Nick Mee

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Gaston Julia studied the convergence properties of sequences of numbers. These sequences have the following form: take a number c, then for each complex value of x the sequence consists of the terms

and so on, each term being equal to the square of the previous term plus c. For some starting numbers x this sequence will diverge, the terms will increase in size indefinitely, whilst for other starting numbers every term in the sequence remains finite. Julia was interested in those values of x that produce sequences that remain finite. These values of x constitutre the Julia set of c.

By generating a sequence of Julia Sets for values of the parameter c corresponding to closely spaced points on a closed path in the plane it is possible to create animations such as the one above. These animations illustrate just how complicated the form of the Julia Sets can be and also how their structure is dramatically dependent on the value of the constant c. The Julia Sets fall into two classes: they are either connected or they consist of isolated points.