(This is kind of an explain-like-I'm-five question. "This looks interesting, but I don't get it at all. Explain it to me.")
In 1893 Felix Klein wrote (1, 2) — or maybe "said," because this was a lecture? —
Let any number of non-intersecting circles 1,2,3,4,..., be given (Fig. 10), and let every circle be reflected (i.e. transformed by inver- sion, or reciprocal radii vectores) upon every other circle; then repeat this operation again and again, ad infinitum. The question is, what will be the configuration formed by the totality of all the cir- cles, and in particular what will be the position of the limiting points. There is no difficulty in answering these questions by purely logical reasoning; but the imagination seems to fail utterly when we try to form a mental image of the result.
I'm afraid my own logical reasoning isn't up to Klein's standard. In fact, I'm not even sure what we're doing here, and the original accompanying drawing (left) doesn't really clarify things.
I used GeoGebra (which is quite cool!) to draw out a bunch of inversions; I see that the centers of the circle images kind of converge to a path-ish thing, but I don't see what Klein means by "the limiting points", and I certainly can't describe what I see using "purely logical reasoning." So, uh... what am I looking at here? And why does Klein find it interesting?
This GeoGebra workspace can be seen and interacted-with here.
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The "limiting points" refers to the points of the plane that lie inside infinitely many circles. They form a sort of "fractal dust" which I won't try to describe here, but you can search "Schottky fractals" for some pictures.
If you want to see lots of pretty pictures related to limit sets of groups generated by reflecting circles, and also learn more about the associated theory, see the book "Indra's Pearls". I highly recommend it.