https://en.wikipedia.org/wiki/Borromean_rings is now a Good Article on Wikipedia. You can't make the Borromean rings with geometric circles: as Tverberg observed, an inversion would make one of them a line. But then each of the other two circles would span an angle less than π as viewed from the line, leaving an unspanned direction along which the line could escape to infinity, contradicting the inseparability of the rings. #proofinatoot
Same proof works for four pairwise-unlinked circles: move them until two circles touch, and invert at the touching point to get two parallel lines and two circles. The two circles each block a range of directions less than π and the other line only blocks a single direction, so each line still has a direction along which it can escape to infinity, separating the two lines from the two circles.
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