Talk:Borromean rings

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Can this arrangement be twisted about into a Mobius Strip? THAT would be weird. Same surface continuity, no two rings connected, yet "one" complex. I can't get my mind around that! bt -- unsigned comment by anonymous IP 68.102.13.50, 06:48, 26 August 2006

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[edit] clarification

Why wouldn't you be able to form this figure from regular circles? Or should it say from two-dimensional figures, as the linkages require overlap? -- nae'blis (talk) 20:27, 1 February 2006 (UTC)

No -- you can't do it with exact geometric circles. Take a close look at the picture. Michael Hardy 21:19, 1 February 2006 (UTC)
I can assure you that I designed my tattoo [1] using geometric circles as the basis - can you explain that to me again? Is the 2.ε-dimensional overlap the 'impossibility', or the shape of the ellipse/circle? -- nae'blis (talk) 23:28, 1 February 2006 (UTC)
I didn't mean you can't make 2-dimensional pictures of it with exact circles. In the first place, any two of the circles would have to be in two different planes; otherwise they would have common points rather than being linked. That means we need to embed them in a three-dimensional space. I was speaking of the actual circles, not of pictures of them. So is this article. Michael Hardy 23:46, 1 February 2006 (UTC)
Thanks, I'll make a small clarification to the opening paragraph for those who aren't as adept at topology. -- nae'blis (talk) 15:12, 2 February 2006 (UTC)
OK, I've found a reference: B. Lindström, "Borromean Circles are Impossible", American Mathematical Monthly, volume 98 (1991), pages 340—341. I'm going to add that to the article.
But anyway, it seems you had in mind 2-dimensional pictures in which the circles are perfect circles. No one has said that those are impossible, and you already see those in the article. Michael Hardy 23:59, 1 February 2006 (UTC)

I've not seen the Math Monthly article, but I first learned about it from a short note by Ian Agol. The proof is fairly simple but ingenious. I don't know if we need two references, but this has the advantage of being freely available through the Internet (instead of say, through JSTOR). --C S (Talk) 01:34, 2 February 2006 (UTC)

I made a variation of Borromean Rings with a tangle toy. Each ring is pringle shaped, though.


[edit] Recent edits

The recent edits have, in my opinion, been rather lacking. Poor wording and even misleading statements have been inserted. --C S (Talk) 20:33, 12 March 2007 (UTC)

I do like the introduction of gallery tags. Anyway, I changed some things. --C S (Talk) 00:04, 14 March 2007 (UTC)

[edit] Diagrams in "Mathematical properties"

Surely the "ellipses of arbitrarily small eccentricity" are actually in the disgram to the left rather than the right as mentioned in text. I'll edit it but feel free to correct me.--Andyk 94 10:07, 3 August 2007 (UTC)

Anyone want to explain why it keeps being changed back?--Andyk 94 02:46, 10 August 2007 (UTC)

The 2D picture (with the yellow circle) doesn't represent a geometric situation with either circles or ellipses. It shows the basic configuration but the circles cannot actually be made that way without bending them. The 3D picture (with the green) shows realizable ellipses. I hope that clears up your confusion. --C S (talk) 13:02, 9 March 2008 (UTC)
I agree. Andyk is wrong. The picture in the center is the one that shows how this link can be realized with ellipses. The one on the left shows a link with circles that is not realizable in Euclidean space. Michael Hardy (talk) 13:45, 10 March 2008 (UTC)

[edit] Images added to French version of article

Overhead view of circular-appearing 3D shapes linked in Borromean rings configuration

Side view of same shapes

-- AnonMoos (talk) 08:44, 19 February 2008 (UTC)

[edit] Pargraph needs editing

"The Borromean rings give examples of several interesting phenomena in mathematics. One is that the cohomology of the complement supports a non-trivial Massey product. Another is that it is a hyperbolic link: the complement of the Borromean rings in the 3-sphere admits a complete hyperbolic metric of finite volume. The canonical (Epstein-Penner) polyhedral decomposition of the complement consists of two ideal octahedra." - what the hell does any of this mean? -- anon —Preceding unsigned comment added by 81.99.106.40 (talk • contribs)

I means something that can be understood by those who know algebraic topology and some advanced geometry. Maybe that should be stated at the beginning of the paragraph. Michael Hardy (talk) 17:04, 22 March 2008 (UTC)