Inversion Limit Sets

A collection of circles, C1, ... , CN can generate a set of points, often a fractal, by methods similar to the familiar Iterated Function Systems (IFS) of fractal geometry. For simplicity, here we consider only circles C1, ... , CN bounding discs D1, ... , DN having disjoint interiors.

Starting with a point z0 outside the discs, we pick a circle at random and invert the starting point, obtaining a new point z1. Then we pick another circle at random, with the restriction that it cannot be the same circle across which we just inverted (property (v)), and invert z1, obtaining a new point z2. We keep repeating the process. The sequence of points zi generated converges to a set called the limit set of inversion across C1, ... , CN. The limit set is characterized by the property that arbitrarily close to each of its points lies a some zi.

If we have two circles, the limit set is just two points.

In the case of four disjoint circles, the limit set is a Cantor set wrapped around a circle.

On this page, we see a group of five circles. Now the limit set consists of a much richer set of points, and the image is more complex. The fractal nature of this limit set is evident, although to be sure it is a more complicated fractal than the familiar Sierpinski gasket. Here the self-similarity is nonlinear: the shape is made of distorted smaller copies of itself.

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