The Inverse Iteration Method (IIM) is a very fast and efficient way of generating Julia sets. Yet, it does not give you complete colorful pictures like the pixel method does. Instead, it generates outlines of the Julia sets, which can still be very beautiful. Below are two pictures of the same Julia set, which was generated using different methods:

The IIM is based on changing the function that generates the Julia set into its inverse. For example, the function z = z^2 + c will be changed into z = sqr(z – c). Now, instead of the points being repelled from the border of the Julia Set, they will be attracted to it! If we start with some point and iterate using the inverse function the same way as it is done in strange attractors, we will eventually hit a point on the border of the corresponding Julia set. Since the inverse function sqr(z – c) uses a square root, at every iteration we have a choice of either the negative or the positive square root. This can be chosen at random.

Optionally, we can make the program keep track of the number of times every pixel was visited. Once some point was hit too many times, we can restart iteration with a different initial point. This will keep us from redrawing the same parts of the set too many times without drawing all parts of the fractal. Often, this method turns out much more efficient than the regular IIM. It is called the Modified Inverse Iteration Method (MIIM).