Circle Inversion Fractals

Inversion Limit Sets

An Example of Fricke and Klein

The 1897 text by Fricke and Klein contains the picture on the left, their view of the limit set of inversion in five circles.

The correct picture is shown on the right.

Click each picture to magnify in a new window.

The incorrect picture was copied by other authors, including Wilhelm Magnus' Noneulcidean Tesselations and Their Groups.

This picture appears as Figure 34 of Noneuclidean Tesselations, accompanied by this description:

"Figure 34 indicates the tesselation of the plane with replicas of the fundamental region of a group generated by the reflections in five circles, four of which form a chain ... and three of which touch in pairs. The limit points fill infinitely many limit circles and also infinitely many continuous nondifferentiable curves ... ."

Mandelbrot's algorithm for generating limit sets gave the first correct picture of this limit set, 85 years after the incorrect picture was published.

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