Re: [math-fun] Borromean Rings
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
Dan wrote
Matthew Cook has a Borromean rings page: <http://www.paradise.caltech.edu/~cook/Workshop/Math/Borromean/Borrring.html> For a drawing by the little-known David Borromeo, see http://www.tweedledum.com/rwg/borrostar.htm (click to clarify). The puzzle here, if there is one, is to trick Macsyma's naive hidden surface algorithm into drawing this, despite a belief in planar faces.
I am "desperately" trying to find another copy of a Borromean puzzle sold briefly at the close of G4G5. Six identical (but for colors: two red, two blue, two yellow) injection-molded plastic half-hoops, with notches in the inside corners of their ends. Assembly was a serious dexterity challenge. --rwg PS: the puzzle sold for $5.
On Wed, 11 Jun 2003, R. William Gosper wrote:
Dan wrote
Matthew Cook has a Borromean rings page: <http://www.paradise.caltech.edu/~cook/Workshop/Math/Borromean/Borrring.html> For a drawing by the little-known David Borromeo, see http://www.tweedledum.com/rwg/borrostar.htm (click to clarify). The puzzle here, if there is one, is to trick Macsyma's naive hidden surface algorithm into drawing this, despite a belief in planar faces.
I am "desperately" trying to find another copy of a Borromean puzzle sold briefly at the close of G4G5. Six identical (but for colors: two red, two blue, two yellow) injection-molded plastic half-hoops, with notches in the inside corners of their ends. Assembly was a serious dexterity challenge. --rwg PS: the puzzle sold for $5.
Yes, I had a copy of this, and agree. On a related topic; I once studied the links you can make with exactly circular rings, and classified the "doubly-transitive" ones. These are 1) the n-string unlink 2) the n-string Hopf link 3) the pentalink 4) the hexalink The last two are really quite intriguing, the pentalink being obtained by deleting any ring from the hexalink. I have a feeling that if the latter were made with colored solid rubber tori of just the right thickness, it would have just 5 stable configurations (all isomorphic, but with different colors in different places) that one could "click between". However, I don't know how to get such tori. Any ideas? John Conway
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