David Wilson writes: << I thought it was kind of neat that Borromean rings cannot be formed using circular rings (of equal or differing sizes). A related question: Given three rigid rings which are not all circular and are reasonably smooth locally, is it always possible to form Borromean rings? I was inclined to think this was true, but then I thought it might not be possible if the rings were sufficiently tangled up that any attempt to form Borromean rings would result in a more complicated linkage. Is this question easier if we require the rings to be planar?
Matthew Cook has a Borromean rings page: <http://www.paradise.caltech.edu/~cook/Workshop/Math/Borromean/Borrring.html> in which he presents several proofs of the impossibility of making the rings from perfect planar circles of any radii, and also makes the same conjecture (without any smoothness condition). --Dan