This applet draws standard Julia Sets in a non standard, but quicker way. To understand this method, first lets look at the standard method. Using the standard method, if the point you are calculating the colour for is inside the set, then it will stay inside the set, no matter how many times you apply the iterative formula. In other words, that point is 'attracted' to the Julia Set. Points outside the set, however, rapidly increase in magnitude after subsequent iterations. That is, they are 'attracted' to infinity. Either way, by applying the iterative formula, the point is 'repelled' from the edge of the set.
But what if we used the inverse of the formula? Then no matter where we started, this inverse formula would move towards the edge of the Julia Set, and away from the interior of the set, and infinity. This in effect gives us a tracing of the boundary of the Julia Set.
Unfortunately, the inverse of the Julia Set function is z = plus or minus square root of (z - c). (I will use real HTML when MATH tags are properly supported) As you can see this is not a true function, since it has two values, but instead forms a binary tree. That is, after n iterations, we have 2n values of z. By using a recursive function, it is easy to walk through this binary tree and plot the function.
If you want to plot a different Julia Set, wait till the current one is finished (a copyright message apears on the status line), then click somewhere on the applet. To make the selection of starting point easier, The coordinates of the mouse are displayed on the status bar. Some values you might want to try are (-0.74543, 0.11301) and (0, 1).
Note: The algorithm for this applet was swiped from fractint, as a lot are, but the code is all my own.