We can simplify understanding some aspects of the limit set by inverting in another circle, selected to convert some of the inversions to reflections across lines.
Let S be the circle with center the point of tangency of C1 and C2, and passing through the point of tangency of C3 and C4.
To simplify the labeling, we denote Ci by i, S1 by A, and S2 by B.
By property (vii) of inversion, inversion in S takes circles 1 and 2 to the vertical lines 1' and 2', and takes the circles A and B to the horizontal lines A' and B'.
By Property (vi) of inversion, inversion in S takes the circles A, B, and 5 to the circles A', B', and 5'.
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Moreover, inversion in S takes inversion in the circles 1, 2, A and B to reflection across the lines 1', 2', A', and B'.
Return to Mandelbrot's method.