Circle Inversion Fractals

Mandelbrot's Algorithm

For some special types of limit sets, Apollonian packings among them, Benoit Mandelbrot found a much quicker method, which we illustrate with this example, inversion in the five circles C₁, C₂, C₃, C₄, and C₅.

(This is similar to another example of inversion in five circles.)

Now S₁ is orthogonal to C₁, C₂, C₃, and C₄.

Because these circles are tangent in pairs at their intersections with S₁, as we have seen, the limit set L(C₁, C₂, C₃, C₄) = S₁.

Similarly, L(C₁, C₂, C₅) = S₂, L(C₂, C₃, C₅) = S₃, L(C₃, C₄, C₅) = S₄, and L(C₄, C₁, C₅) = S₅.

Consequently, L(C₁, C₂, C₃, C₄, C₅) contains S₁, S₂, S₃, S₄, and S₅.

What else can we say about the limit set?

Return to Mandelbrot's method.