Circle Inversion Fractals

Mandelbrot's Algorithm

The nonlinearity of circle inversion causes some problems with the IFS approaches to generating limit sets.
We illustrate Mandelbrot's method with an example, and begin by constructing some cirlces that belong to the limit set.
Next we simplify the problem by inverting in another circle.
Now, we show that the limit set L(C1, C2, C3, C4, C5) lies in the disc bounded by S1.
Similar arguments show the limit set L(C1, C2, C3, C4, C5) lies outside the discs bounded by S2, S3, S4, and S5.
So the limit set is inside S1, and outside S2, S3, S4, and S5. We can say more.
Continuing this idea quickly generates the limit set, to any desired accuracy.

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