Talk:Borromean rings

Mobius Strip

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

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)

In the figure, those are not geometric circles. Perhaps the lighting from the simple raytracing is misleading. That particular projection sends them to true planar circles, but in 3-space they are pringle shaped as has been mentioned. You can also do it with ellipses, but then they won't project to true circles, as in the figure. Rybu (talk) 19:38, 10 February 2024 (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.

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)

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)

Eccentricity is a measure of how much a ellipse deviates from being circular. An ellipse with arbitarily small eccentricity would have it's foci aribitrarily close to the center, so it would look circular--that's the point, right? This should be mentioned, and the picture reference removed.

I'm not sure what you are asking to have mentioned. The middle picture of ellipses demonstrates a feasible configuration. The one on the left with circles is not feasible, even were you to replace them with ellipses, since the rings would have to bend out of the plane. That's the point of the picture reference which you want removed. Without reference to the middle picture, it would be very difficult to see how to realize the Borromean rings with ellipses of small eccentricty. --C S (talk) 20:52, 26 June 2008 (UTC)

The phrase "arbitrarily small eccentricity" indicates an ellipse that is almost circular, IE the picture on the far left. Or, to put it another way (and I hope this clarifies more than confuses), it displays the attempt at using circles but, as pointed out in te text and above, these circles would have to be bent, or, to keep them as 2d shapes, stretched (and, I believe, rotated slightly) IE turned from circles into ellipses. I shan't re-edit it because we're just going back and forth here. But if I've made it clear to those who were removing my edits, it should be corrected imediately.--Andyk 94 (talk) 16:06, 27 June 2008 (UTC)

No, again, let me repeat. The ellipse picture is NOT the one on the far left. I don't know what you mean by rotated slightly. The obvious interpretation doesn't work. I have no idea why you are latching onto the left picture like you are. The way to have a configuration with ellipses is to use the middle picture, which I think you don't understand. The eccentricities of those ellipses can in fact be made arbitrarily small. There is no correction needed, although I'm starting to wonder from your confusion whether there should be some rewriting to make the points clearer. --C S (talk) 19:04, 27 June 2008 (UTC)

Your misconception stems from being fixated on the leftmost picture with the circles, apparently because the middle picture doesn't look circular enough to you. The middle picture can in fact be made as nearly circular as you wish. Do you see how it uses ellipses to realize the Borromean Rings (topologically the same, but looks arranged differently)? Next, do you see how you can make the eccentricities smaller (but not all the same) realizing that configuration? (Note the picture is drawn in 3D perspective) From that, you can easily tell why ellipses make the configuration possible. Now, you insist that the leftmost picture is realizable by ellipses. Why? How does making the rings ellipses help avoid the "bending" problem? Even if you try rotating each one this way or that, you can't avoid the bending. And it's certainly not clear why using ellipses would help avoid it. According to you, somehow this "rotating" doesn't work when they are circles but works when they are nearly circles. BTW, I wrote the material, and what you are thinking is certainly not what I had in mind, and it's not correct either. --C S (talk) 19:28, 27 June 2008 (UTC)

I'm claiming that the central picture shouldn't be described as containing "ellipses of arbitrarily small eccentricity". By "rotated slightly" I was refering to rotation out of the plane in which we are working, as suggested above by another user. I have latched onto the leftmost picture because that is the one that seems to be best described by the text that currently references the central picture. I'm not certain how the central picture differs from the leftmot picture, other than the spacial arrangement and ecentricity of the rings, the relationship of the rings with each other seems to be identical. In which case, the alteration of eccentricity along with this rotation out of the single plane would surely make the 2 diagrams equivalent?--Andyk 94 (talk) 00:34, 19 July 2008 (UTC)

Andyk, you know how to miss a point. The point is: (1) Borromean links cannot be realized by circles, whether coplanar or not, and (2) They can be realized by ellipses that are not exactly circular, and (3) No matter how close to being circular they are, as long as they're not exact circles, the ellipses, if suitably positioned, can still realize the Borromean links. You can't have a picture of particular "ellipses of arbitrarily small eccentricity". Any particular ellipses that are not perfect circles are NOT of "arbitrarily small eccentricity"; they have whatever particular eccentricity they have and it's not zero. The words "arbitrarily small" merely express the point identified as (3) above, somewhat less long-windedly, but in language so completely standard that everyone (except non-mathematicians) will know that's what it means. Michael Hardy (talk) 11:57, 19 July 2008 (UTC)

OK, just to clarify further: it's not just that you can't have a picture of particular ellipses of "arbitrarily small" eccentricity. It's that no particular ellipse can have "arbitrarily small" eccentricity. Any particular ellipse has whatever particular eccentricity it has. The point of the picture is that ellipses that are not circles can serve in that role. The "arbitrarily small eccentricity" part means they can still serve no matter how close they are to perfect circles as long as they're still not perfect circles. But since the picture's purpose is to illustrate that ellipses that are not exactly circular can serve, it has to make clear from their appearance that they are ellipses that are not perfectly circular. Michael Hardy (talk) 17:30, 19 July 2008 (UTC)

OK, now I've rephrased it for the benefit of those not fluent in mathematicians' linguistic conventions. Michael Hardy (talk) 17:35, 19 July 2008 (UTC)

Sorry Mike, I'm not sure if you realise your attempts to partonise me are actually re-afirming my point or not but I'd love to go over it again. 1) Borromean links cannot be realized by circles, whether coplanar or not. I know, however, if stretched infinitesimally and rotated infinitesimally (again out of the initial plane) then a set of circles become a set of ellipses with "arbitrarily small" eccentricity. Now, assuming that these defections (eccentricity and planar rotation) from the impossible case (ideal circles in a single plane) the diagram of apparently circular rings fits the description perfectly, so long as the defections are too small to be recognised by the naked eye (and indeed, possibly too small to be displayed in pixels). Assuming the viewer doesn't assume a physically impossible configuration is being displayed, (s)he would have to assume that this is the case.

I'm not attempting to patronize you; I'm attempting to get correct information here and in the article. The point of the second picture is that Borromean links can be realized by ellipses that are NOT circles. The picture should make that clear by making it clear that the curves are ellipses that are NOT circles. The second picture does that. The first picture fails to do that. The words "arbitrarily small eccentricity" mean only that ellipses can still serve no matter how small their eccentricity, as long as it is not exactly zero. It is unreasonable to ask for a picture of ellipses of arbitrarily small eccentricity, since there is no such thing as an ellipse of arbitrarily small eccentricity. Michael Hardy (talk) 20:55, 19 July 2008 (UTC)

Perhaps it would help to include a more realistically rendered image. Here is a true-life picture of essentially the same configuration, but in the "bad" projection they appear as hexagons, rather than circles. https://www.printables.com/model/161860-borromean-rings-hexagons Rybu (talk) 19:42, 10 February 2024 (UTC)

I've found some Wikipedia pages that can help

I've found these two pages:

• Arbitrarily large
• Primes in arithmetic progression

The first addresses the issue directly, explaining how mathematicians use this terminology. The second explicitly says there are "arbitrarily long, but not infinitely long" arithmetic progressions of prime numbers.

To say that there are arbitrarily long arithmetic progressions of prime numbers DOES NOT mean that there is any particular arithmetic progression of prime numbers that is "arbitrarily long". There is no such thing. It means that no matter what length you pick, no matter how big, there are arithmetic progression that are at least that long (but still may be finite, and in this case are always finite).

That's just part of the standard jargon of mathematicians. Michael Hardy

I understand that, and I'm not claiming that the picture does fit the description of "ellipses with arbitrarily small eccentriity". I'm claiming the left most picture comes closest to displaying the notion that any finite eccentricity will suffice ie by using ellipses with incredibly small though not "arbitrarily small" itself(since, as you point out, to claim any particular quantity is arbitrary makes no sense).--Andyk 94 (talk) 02:04, 21 July 2008 (UTC)

The second picture illustrates the idea that it can be done with ellipses. The fact that they can have eccentricity as small as desired is probably not something that could be illustrated by a simple picture. The fact that ellipses differing only infinitesimally from circles cannot be distinguished from circles in an accurate picture presents a problem. Michael Hardy (talk) 03:43, 21 July 2008 (UTC)

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)

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)

The Massey product thing sounds like gibberish, I agree. But the other things are actually very down-to-earth statements. --C S (talk) 21:01, 26 June 2008 (UTC)

It doesn't sound like gibberish; it sounds like something that hasn't been defined here in this article. It has a link. The article it links to is unfortunately pretty terse, though. Michael Hardy (talk) 12:02, 19 July 2008 (UTC)

Mathematics section error

When you look closely at the first picture it is not a Borromean ring at all, the blue ring is not connected to the other two as it is on top of the yellow and under the red, but not interlocked. If rings were placed in such a fashion as that one, the blue could easily be taken apart without breaking it, and the other two would remain locked, clearly not borromean fashion. —Preceding unsigned comment added by Pyrofyr (talk • contribs) 04:25, 7 February 2009 (UTC)

If you mean File:BorromeanRings.svg, no two rings are directly interlinked, and that's actually the point. Blue is "under" yellow, red is "under" blue, and yellow is "under" red. The whole thing has perfect threefold rotational symmetry, so I'm not sure why you're singling out blue... AnonMoos (talk) 08:29, 7 February 2009 (UTC)

Photo of older depiction

Classic medieval depiction of the rings as literal rings... AnonMoos (talk) 18:27, 6 May 2009 (UTC)

"Borromean nucleus"

Apparently a term sometimes used by physicists: http://www.anl.gov/Media_Center/News/2004/PHY041029.html -- AnonMoos (talk) 18:52, 9 May 2009 (UTC)

Borromean castle

http://rfcwalters.blogspot.com/2009/06/borromean-rings.html claims the admission tickets to the Borromeo family castle show an incorrect picture of the rings. I haven't traced it carefully enough to spot the error, but if there really is one, it may be worth mentioning in the article. 67.122.209.126 (talk) 08:20, 28 June 2009 (UTC)

That a version of the crest containing another 3 ring configuration is on a ticket is not so interesting, particularly when the crest itself was depicted that way by the Borromeos themselves, including on many carved doors and such. the Borromean family used several versions of their crest, some of which were not Borromean (hm, that sounded a bit funny now that I read it...). This was discovered by some topologist who examined all the carvings, depictions, etc. of the crest in the castle. The actual Borromean ring configuration wasn't used in a substantial portion of all of them. So the ticket can be considered to depict a "legit" crest, unless you want to take the position that many depictions used by the Borromeos is wrong. --C S (talk) 04:27, 29 June 2009 (UTC)

Literary reference

In Heinlein's Space Cadet, the Borromean rings were used as an emblem. AnonMoos (talk) 21:02, 8 February 2010 (UTC) --

“ Above the stage at the far end were the three closed circles of the Federation - Freedom, Peace, and Law, so intertwined that, if any one were removed, the other two would fall apart. ”

Multiple Borromean rings - Discordian mandala

example - the blue is above both the red and green, the three loops are not entangled

How many Borromean rings configurations does the Discordian mandala contain? In my understanding, each configuration consists of three of the five 9-sided 'rings'. There are ten ways of choosing three from a group of five so I suggest the mandala contains not five but ten configurations. Less theoretically: Red-Yellow-Green; Red-Yellow-Blue; Red-Yellow-Magenta; RGB; RGM; RBM; YGB; YGM; YBM; GBM. 221.243.91.229 (talk) 06:02, 6 May 2011 (UTC)

Sorry I didn't see your remark before, but I really don't think so; if you choose a non-adjacent configuration, then they are not Borromeanly-linked... AnonMoos (talk) 01:07, 7 August 2011 (UTC)

Proof?

Is it just me, or is the synopsis of the Lindström & Zetterström proof incomprehensible? It goes like, "if one assumes that circles 1 and 2 touch at their two crossing points" – OK, but what if one does not assume that? Is the touching here assumed to take place in the "link diagram"? Isn't that supposed to be a plane figure? Then how do spheres enter the picture? (I have to confess I don't find the full proof as given in the AMM particularly convincing either: the "lifting" operation is not well defined; for C₂' to meet C₁ in two points, just a translation will not do in general; additionally a rotation is needed, which makes it unclear what the statement means that C₂ is lifted "away" from C₃.)  --Lambiam 11:27, 31 January 2012 (UTC)

Clover loop

This loop/link was drawn from some rearrangment of the Zeppelin knot,

, but also could seen as a figure of eight with a particular extra loop. Would this match any of the knot theory work?SignedJohnsonL623 (talk) 11:58, 27 February 2013 (UTC)

It's not Borromean/Brunnian because it only contains one loop (i.e. is not a "link" in knot theory terms). It appears to be a non-alternating 9-crossing knot. You could run it through the "Knotscape" program to see what it translates to in terms of standard knot inventories (see http://katlas.math.toronto.edu/ etc.). AnonMoos (talk) 14:58, 28 February 2013 (UTC)

Two cirlces, one ellipse of arbitrarily small eccentricity

While it has been mentioned that the Borromean rings can be realized as a link using three ellipses with arbitrarily small eccentricity, it is actually the case Borromean rings can be realized with two circles and one ellipse of arbitrarily small prescribed eccentricity. Here is a rather canonical example. Consider the circles (with 0 < b < a):

A: y²/a² + z²/a² = 1; x = 0 in the yz-plane

B: x²/b² + z²/b² = 1; y = 0 in the xz-plane

and the ellipse

C: x²/a² + y²/b² = 1; z = 0 in the xy-plane

Obviously, A & B do not intersect, for they meet the z-axis at points a units and b units from the origin respectively. A & C meet the y-axis at points a units and b units from the origin, respectively. And, finally, B & C meet the x-axis at points b units and a units from the origin, respectively. Since 0 < b < a, these points are all distinct, and since a & b can be chosen as close to each other as desired, the eccentricity (1 - b²/a²)^1/2 of ellipse C can be as small as desired. Chuck (talk) 05:22, 28 July 2013 (UTC)

Lacan

Some reference, however fleeting, should be here.--Dnavarro (talk) 00:46, 9 November 2015 (UTC)

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Editing needed

The first paragraph is mathematically incorrect at several points.   In the first sentence, the word unconnected should be replaced by unlinked.   The phrase "In other words" in the second sentence is not correct, since the second sentence is not equivalent to the first sentence.   The fact that:  no two of the three rings are linked with each other as a Hopf link, is not particularly relevant and is awkwardly phrased.   A suggestion would be to replace unconnected with unlinked in the first sentence, to delete the second sentence, and to add a final sentence to the paragraph: The Borromean rings are characterized by being the nontrivial 3-component Brunnian link with minimal crossing number.  

I tried to edit the second sentence and it was immediately undone.   The ensuing discussion convinced me that I had neither the time nor energy nor knowledge of wikipedia for carrying out the edit. — Preceding unsigned comment added by Jfdavis (talk • contribs) 21:47, 21 August 2019 (UTC)

I'm sorry, but most of the cultural prominence of the Borromean rings among the general public comes from the initially-surprising fact that no two rings are directly linked (Hopf-linked) but all three are collectively interlinked. Without this Brunnian property, the Borromean rings would be of little interest except to mathematicians -- like the Whitehead link or the 6-crossing links other than the Borromean rings (NONE of which have separate Wikipedia articles devoted to them). Therefore the Brunnian nature of the Borromean rings MUST be mentioned in the lead section -- and this needs to be explained (we can't automatically assume that the general reader will know what "Brunnian" means)... AnonMoos (talk) 16:42, 23 August 2019 (UTC)

P.S. Here's how they're described in two math/geometry books for non-specialists which I happen to have conveniently at hand: Cundy & Rollett Mathematical Models (2nd edition) -- "The three rings shown in Fig. 52(d) are such that no two are linked together, but all three are; cutting any one frees the other two" (page 59). Alan Holden, Shapes, Space, and Symmetry -- "The rings are inseparable, but no two are linked together; hence cutting any one frees the other two" (page 182). AnonMoos (talk) 14:38, 29 August 2019 (UTC)

Borromean rings "are" the Ballantine logo? Yes and no.

Using Google images to search for photos of the Ballantine logo, it is clear that some of the images — like this one: https://i.etsystatic.com/11180086/r/il/e72dcd/1829839974/il_1588xN.1829839974_jg7g.jpg — are indeed the Borromean rings.

But some either show an over/under arrangement that is definitely not the Borromean rings, or else contain no over/under information.

So it is not accurate to just state that the Ballantine logo "is" the Borromean link.2601:200:C080:630:49F3:543E:6F88:5D6A (talk) 22:02, 10 January 2021 (UTC)

We need to base our content on published reliable sources, not on our own surveys of the results of Google image searches. Do you have a reliable source that notes the variation in logo topology? —David Eppstein (talk) 23:22, 10 January 2021 (UTC)

Borromean Moebius strips

I have made something like this:

File:Borromean Moebius strips.SVG

Allegra Pstrocski (talk) 22:59, 9 February 2021 (UTC)

Unless you have published sources for the significance of this construction, we can't use it in the article; see WP:NOR. --David Eppstein (talk) 23:58, 9 February 2021 (UTC)

Anyway, they're not actually basic (half-twist) Moebius strips... AnonMoos (talk) 20:18, 13 February 2021 (UTC)

GA Review

This review is transcluded from Talk:Borromean rings/GA1. The edit link for this section can be used to add comments to the review.

Reviewer: Tayi Arajakate (talk · contribs) 16:30, 6 March 2021 (UTC)

Assessment

1. Comprehension: No major issues with comprehension.
Pass

Criteria	Notes	Result
(a) (prose)	Some minor issues might exist. See comments below. (Resolved)	Pass
(b) (MoS)	No manual of style issues were found.	Pass

2. Verifiability: The article appears to meet the requirement for verifiability.
Pass

Criteria	Notes	Result
(a) (references)	The article has inline citations and a list of references.	Pass
(b) (citations to reliable sources)	References used in the article are from reliable sources	Pass
(c) (original research)	No original research found.	Pass
(d) (copyvio and plagiarism)	No copyright issues found; earwig's brings up a few false positives.	Pass

3. Comprehensiveness: The article is comprehensive.
Pass

Criteria	Notes	Result
(a) (major aspects)	The article covers all major aspects, one elaboration may be possible. See comments below. (Resolved)	Pass
(b) (focused)	The article is on topic without any unnecessary deviations.	Pass

4. Neutrality: The article is neutral.
Pass

Notes	Result
No issues with neutrality found.	Pass

5. Stability: The article is stable.
Pass

Notes	Result
No major changes or ongoing disputes.	Pass

6. Illustration: The article is well illustrated.
Pass

Criteria	Notes	Result
(a) (images are tagged and non-free images have fair use rationales)	All images are free to use and tagged with their appropriate copyright statuses.	Pass
(b) (appropriate use with suitable captions)	Image use is proper and with suitable captions.	Pass

Comments

• Hello David Eppstein. I will be taking up the review for this nomination and will present it shortly. Hopefully my feedback would be helpful. Tayi Arajakate _Talk 16:30, 6 March 2021 (UTC)

@Tayi Arajakate: Hello? It's been over a week since you promised a review. Any progress? —David Eppstein (talk) 20:00, 14 March 2021 (UTC)

David Eppstein, sorry about that. I'll complete it within the next 24 hours. Tayi Arajakate _Talk 06:55, 15 March 2021 (UTC)

David Eppstein, I've completed the review. The article is pretty much a good article, there are couple or so issues so I'm putting it on hold for them to be addressed or clarified. Tayi Arajakate _Talk 02:04, 16 March 2021 (UTC)

The article appears better now, congratulations on improving another article to the status of a good article. Tayi Arajakate _Talk 07:16, 16 March 2021 (UTC)

• The PDF linked in ref 24 doesn't seem to be reachable. The sentence it is cited for, "[a]nother argument for the impossibility of circular realizations, by Helge Tverberg, uses inversive geometry", could perhaps be elaborated a bit as well?
  • Fixed the link — it needed http not https. I elaborated the sentence. —David Eppstein (talk) 04:42, 16 March 2021 (UTC)
• The sentence, "[i]n knot theory, the ropelength of a knot or link measures the minimum length of a curve realizing the knot that can be thickened to a tube of radius one around the curve without intersecting itself" is a bit awkward to read. It could be split into two sentences.
  • Well, I tried, but it meant making the first of the two sentences significantly more handwavy. —David Eppstein (talk) 05:03, 16 March 2021 (UTC)
• In the line, "making the Borromean rings one of at most 21 links that correspond to uniform honeycombs in this way", doesn't make more sense to mention that it is one of at least 19 links?
  • I think the information that the number is small and finite is more interesting than the information that mathematicians have found 18 others. —David Eppstein (talk) 05:08, 16 March 2021 (UTC)
• Stylistic suggestion; all the images are on the right, some of them could be moved left. The images are also a bit cluttered, some of them can be removed, resized or their captions modified to prevent them from crossing sections. For instance, the diagram of the non-Borromean three triangle link and the one with the Siefert surface, don't particularly seem to add much. The caption "[r]ealization with smallest known ropelength, the logo of the International Mathematical Union" could omit the first part being already mentioned in the section.

Tayi Arajakate _Talk 01:47, 16 March 2021 (UTC)

• I shortened the IMU logo caption. Putting images on the left tends to run into issues with WP:SANDWICH: with images on both left and right, on narrow screens, the

article

text

can be

squeezed into very narrow columns. Keeping everything right works better for a layout that is flexible over varied window sizes (you can try resizing your browser window to see this effect on articles with left images). The non-Borromean three triangle link is included because there are incorrect claims in the literature that it is Borromean and without being able to see what the link is, it is hard to understand whether to believe those claims. The Seifert surface is included to connect to the article text on the Martin Gardner column, which doesn't otherwise explain what a Seifert surface is. If we're going to be removing images, I think the most disposable one is the Discordian logo, so I dropped that one, and simplified some other captions. —David Eppstein (talk) 04:42, 16 March 2021 (UTC)

Did you know nomination

The following is an archived discussion of the DYK nomination of the article below. Please do not modify this page. Subsequent comments should be made on the appropriate discussion page (such as this nomination's talk page, the article's talk page or Wikipedia talk:Did you know), unless there is consensus to re-open the discussion at this page. No further edits should be made to this page.

The result was: promoted by MeegsC (talk) 10:10, 23 March 2021 (UTC)

(

• Comment or view
• Article history

)

Borromean rings

• ... that the common depiction of the Borromean rings as three linked but pairwise-unlinked circles (pictured) is an impossible object, because they cannot actually be circular?
  Sources: Aigner, Martin; Ziegler, Günter M. (2018), "Chapter 15: The Borromean Rings Don't Exist", Proofs from THE BOOK (6th ed.), Springer, pp. 99–106; Freedman, Michael H.; Skora, Richard (1987), "Strange actions of groups on spheres", Journal of Differential Geometry, 25: 75–98, doi:10.4310/jdg/1214440725; Lindström, Bernt; Zetterström, Hans-Olov (1991), "Borromean circles are impossible", American Mathematical Monthly, 98 (4): 340–341; Tverberg, Helge (2010), "On Borromean rings", The Mathematical Scientist, 35 (1): 57–60. The Aigner/Ziegler and Zetterström sources are paywalled but Freedman/Skora and Tverberg are both open, albeit probably unreadable to nonmathematicians.

• Reviewed: Marijn Heule

Improved to Good Article status by David Eppstein (talk). Self-nominated at 07:47, 17 March 2021 (UTC).

General: Article is new enough and long enough
New enough: Long enough:

Policy: Article is sourced, neutral, and free of copyright problems
Adequate sourcing: Neutral: Free of copyright violations, plagiarism, and close paraphrasing:

Hook: Hook has been verified by provided inline citation
Cited: Interesting:

Image: Image is freely licensed, used in the article, and clear at 100px.
Freely licensed: Used in article: Clear at 100px:

QPQ: Done.

Overall: Nice, interesting hook with illustration. Well referenced article. Good to go. EchetusXe 15:47, 18 March 2021 (UTC)

Pronunciation?

How the heck is this pronounced? Plenty of other articles for uncommon terms include pronunciation guides, such as Santorini or Székelys or Cappadocia or CERN. Would someone please add a pronunciation guide to the Borromean rings article for how to pronounce "Borromean"? -- Eiríkr Útlendi |^{Tala við mig} 09:23, 20 November 2021 (UTC)

Huh? It was just added to the first footnote on the first sentence last week, after a big edit war by a multiple-times-blocked edit-warrior over making it more prominent. The short answer (from the source provided) is like the English words "borrow ME an". The source is from the 1920s, is badly formatted as a source, uses a pre-IPA convention for formatting pronunciations, is about the islands rather than the rings, and none of the modern dictionaries I checked have this, but maybe that's close enough. --David Eppstein (talk) 19:51, 20 November 2021 (UTC)

Sorry, didn't see it in the footnote. The other articles linked above include the phonemic guide inline, so it never occurred to me to even look for it in a footnote. Setting aside any discussion of the source (about which I know nothing), just from a usability perspective, a footnote may not be the best approach. -- Eiríkr Útlendi |^{Tala við mig} 06:14, 22 November 2021 (UTC)

You think the most central, most important aspect of the subject, so important that you must learn about it first before anything else, in the first phrase of the first sentence, before you get to anything else, making the sentence run on longer without getting to the point, so that maybe you go away in boredom before even finishing it, is the correct way to pronounce it? See WP:MOSPRON, "Pronunciation should be indicated sparingly, as parenthetical information disturbs the normal flow of the text and introduces clutter. In the article text, it should be indicated only where it is directly relevant to the subject matter, such as describing a word's etymology or explaining a pun." There is no pun and there is no etymologically-explanatory value in this instance, so it should not be in the text. --David Eppstein (talk) 06:57, 22 November 2021 (UTC)

Yes, I do view pronunciation as important.

Part of why I look things up here is to be able to talk about them. As in, verbally and audibly.

If a Wikipedia article doesn't provide at least the superficial level of information needed to be able to do that, it is deficient.

If a parenthetical IPA pronunciation guide is "making the sentence run on longer without getting to the point, so that maybe you go away in boredom before even finishing it", I think that indicates a defect in the reader more than the article.

I refer you again to other articles that include pronunciation guides directly inline, right at the start of the article, immediately after the initial mention of the headword terms: Santorini, Székelys, Cappadocia, CERN.

I don't understand the vehemence or aggression in your response. I don't know if you're aware, but you come across as very angry and borderline insulting / abusive. Is that your intent? -- Eiríkr Útlendi |^{Tala við mig} 17:54, 22 November 2021 (UTC)

You are picking up an edit war recently begun by a four-times-blocked-for-edit-warring edit-warrior. Is that your intent? Re your listing of other articles that violate the guidance I quoted in WP:MOSPRON, see WP:WAX.

As for why I am testy about this: Wikipedia articles, over time, tend to become crufty and bad through the accumulation of small well-meaning edits by people who don't see or don't care about the big picture of how it affects the readability, flow, and overall content of the article, such as exactly the kind of edit you are proposing. I recently spent considerable effort removing the cruft from this article and bringing it up to Good Article status. So it is irritating for it to so quickly become under pressure from you and others to make it crufty again. --David Eppstein (talk) 19:48, 22 November 2021 (UTC)

@David Eppstein: Hello again, sorry for the hiatus. I spend more of my time over at Wiktionary.

I know nothing about an edit war, so I must apologize if I'm bumping up into any bruises in my ignorance. I am definitely a word nerd, which is why I spend more time at Wiktionary. Perhaps due to that proclivity, when I run across an article here at Wikipedia that has a title unknown to me, I'm interested in some of the more word-ish aspects, such as "how the heck do I pronounce this thing?"

Thank you for the links to WP:MOSPRON and WP:WAX. After perusing those pages, I don't understand your mention of "violating" WP:MOSPRON, since the second paragraph seems to be in favor of including pronunciation information on first instance. Indeed, the #Appropriate use section there further lays out how and why to include pronunciation information -- all of which seems germane and appropriate for this Borromean rings article. The Wikipedia:Manual_of_Style/Lead_section#Pronunciation section linked from #Appropriate use further reinforces that we actually should include pronunciation information in Borromean rings: "If the name of the article has a pronunciation that is not apparent from its spelling, include its pronunciation in parentheses after the first occurrence of the name." For me, at least, the pronunciation of "Borromean" is definitely not apparent from its spelling: I can parse that as three different likely possibilities. Meanwhile, WP:WAX looks like a non sequitur, since that covers arguments about whether to keep or delete pages, which is irrelevant to this discussion.

I don't agree that pronunciation information is cruft, so from that perspective, I suppose you and I just don't see things the same way. Considering that pronunciation information, where added, should only be added once, and should be kept brief, even if it were cruft, it isn't much. I continue to find the stridency of your opposition disproportional to the issue at hand. I also note that one possible reason "for it to so quickly become under pressure" to re-add such information might well be that other people (such as me) actually find it relevant and useful. -- Eiríkr Útlendi |^{Tala við mig} 02:03, 8 December 2021 (UTC)

Let me repeat the quote I earlier made from WP:MOSPRON, since you state that you don't understand my mention of this guideline: "Pronunciation should be indicated sparingly, as parenthetical information disturbs the normal flow of the text and introduces clutter. In the article text, it should be indicated only where it is directly relevant to the subject matter, such as describing a word's etymology or explaining a pun." --David Eppstein (talk) 02:25, 8 December 2021 (UTC)

@David Eppstein: The phrasing "such as" is not exhaustive nor restrictive. How to talk about a subject is, to me, "directly relevant to the subject matter", and thus deserving of inclusion in the text of the article. Again, from Wikipedia:Manual_of_Style/Lead_section#Pronunciation,

If the name of the article has a pronunciation that is not apparent from its spelling, include its pronunciation in parentheses after the first occurrence of the name. Most such terms are foreign words or phrases (mate, coup d'état), proper nouns (Ralph Fiennes, Tuolumne River, Tao Te Ching), or very unusual English words (synecdoche, atlatl).

‑‑ Eiríkr Útlendi │^{Tala við mig} 06:24, 18 December 2021 (UTC)

I bet that a lot of mathematicians (both amateur and professional) have seen "Borromean" in writing a lot more than they've ever heard it spoken... AnonMoos (talk) 20:24, 8 December 2021 (UTC)

Your point is? The pronunciation is there. At the start of the article. It is just not the most prominent thing you see before anything else in the article. As it should be not. —David Eppstein (talk) 22:14, 8 December 2021 (UTC)

My point is that that may mean pronunciation isn't actually vitally important for this article... AnonMoos (talk) 22:21, 8 December 2021 (UTC)

Ah, ok, I misunderstood. —David Eppstein (talk) 23:06, 8 December 2021 (UTC)

Borromean rings are not a regular tessellation link

@David Eppstein I just removed the wrong claim that the Borromean rings are one of the up to 21 regular tessellation links defined in https://arxiv.org/pdf/1406.2827v3.pdf

This is obviously false: Table 1 of the paper doesn't contain any link with just 3 components.

In more detail: the complement of the Borromean ring does consist of two regular ideal octahedra. However, the condition in the paper is stronger: the decomposition of the manifold into octahedra needs to have enough symmetries to take any flag to any other flag of the same orientation. That is: each octahedron can be taken to any octahedron by a symmetry. And each orientation-preserving symmetry of an octahedron must be realized by a symmetry of the manifold fixing the octahedron. - Matthias Goerner — Preceding unsigned comment added by 23.93.72.121 (talk) 03:23, 18 January 2023 (UTC)

Ok, that's clear enough. I removed the unused reference left by this change. —David Eppstein (talk) 07:27, 18 January 2023 (UTC)

Ballantine rings

User:Cerberus0 has been trying to remove or downplay the "Ballantine rings" name and history from this article, stating that in edit summaries that it is an "advertisement". On the contrary, we have reliable sources that are independent from Ballantine/Pabst stating that this is (or maybe more accurately was) one of the standard names for this topic, and explaining why. My position is that this topic is relevant, the alternative name should remain in the lead, and the history should remain in the section about history and symbolism rather than relegated to a new cruft-magnet WP:TRIVIA section. But I would welcome reasoned and source-based discussion of that here. Emotional appeals to anti-commercialism are not relevant and not welcome, much as I might agree in spirit with that philosophy. —David Eppstein (talk) 20:14, 5 September 2023 (UTC)

i. I did not try to remove "Ballantine rings". I stated I would not object to it.

ii. To understand why, look at the Google ngram viewer for the two phrases. Cerberus (talk) 21:26, 5 September 2023 (UTC)

You did remove it. You removed it from the lead and from the "History and symbolism" section, instead relegating it to the end of the article into a new worthless catch-all "popular culture" section.

As for ngrams, in Wikipedia we go by what reliable sources tell us, not so much by original research based on web searching. In this case, Cromwell et al 1998 say "In North America, the design is known as the Ballantine rings", and Glick adds that "the rings were widely known in the 1950s when Ballantine beer, "'crisp' as a line-drive double," was a principal broadcasting sponsor for New York Yankees baseball games." Cromwell et al also note the existence of scholarly work in knot theory using this term [2].

I think today that name has fallen into disuse, but that isn't a reason for removing it. —David Eppstein (talk) 22:43, 5 September 2023 (UTC)

You apparently misunderstand what Glick actually said, in your actual quote. The name has not only fallen into disuse, it NEVER had wide use. It should either be removed or demoted. Highlighting a practically nonexistent usage in this way is inappropriate and misleading. You have the evidence. Why aren't you honoring it? Cerberus (talk) 17:57, 6 September 2023 (UTC)

Also, Cromwell cites a single source, Formal Knot Theory, that is NOT about Borromean rings but mentions them in passing and notes (parenthetically) a secondary term.

At the very least, referring twice (!) in the article to "Ballantine rings" is misleading the reader by suggesting an almost non-existant usage is common. Cerberus (talk) 18:07, 6 September 2023 (UTC)

On the contrary, I am following sources. You are following your outrage. One of those two is encyclopedic and neutral. The other is unwelcome here. And how is it even possible for a direct quote, stated without annotations or interpretation, to "misunderstand" the quote that it quotes? —David Eppstein (talk) 18:23, 6 September 2023 (UTC)

Because Glick does *not* say that they were know *as* Ballantine rings. Get it?

It is appropriate to treat Google ngrams as published results (by Google) and therefore very reliable rather than as "original research". Cerberus (talk) 15:27, 21 September 2023 (UTC)

More sources for "Ballantine rings" (not needed for the article, but maybe helpful in suggesting that this phrase was widely known):

• "Advertising symbols glorified", Life magazine, 1940: "men and women susceptible to the graphic manipulations of American advertising will be startled by the sight of Coty powder puffs, Ballantine rings, Maxwell House coffee cups, ..."
• Brian Thomson, Judith Bruckner, Andrew Bruckner, Mathematical Discovery, 2011 (self-published?), p. 182: "Many of our readers might prefer to call these Ballantine rings"
• Luciano Boi, Geometries of Nature, World Scientific, 2005, p. 139: "Another example of this sort is the Borromean (or Ballantine) rings"
• Claudi Alsina , Roger B. Nelsen, Icons of Mathematics, Amer. Math. Soc., 2020, p. 157: "In the United States they are sometimes called the Ballantine rings"
• Jon C. Loren, Synthesis of Topological Isomers from Manisyl-substituted Polypyridine Ligands, dissertation, UCSD, 2004, [3]: "The Borromean link is also referred to as the Ballantine rings or the brewer's rings"
• Encyclopedia of Physical Science and Technology, Academic Press, 2002, [4]: "Another example of this sort is the Borromean (or Ballantine) rings"
• Charles Euchner, Extraordinary Politics: How Protest And Dissent Are Changing American Democracy, Routledge, 2018, p. 164: "The parts of an organization overlap like ballantine rings" (lowercased)
• Karen Swenson, A Daughter's Latitude: New & Selected Poems, Copper Canyon Press, 1999, p. 92: "the morning after it's as though all the Ballantine rings dissolved to one and the men just leap through like fancy horses at the circus"
• George J. Lankevich, "Postcards from Times Square", Square One, 2001, p. 72: "Bubbling Bromo, penguins smoking Kools, and a clown tossing Ballantine rings were among Leigh's contributions to the fabulous clutter of the area"
• Don Asher, "Notes from a Battered Grand", Harcourt Brace Jovanovich, 1992, [5]: "Careful examination of the casing reveals the czar's personal crest, an ingenious design of interlocking circles predating this century's famed Ballantine rings"

—David Eppstein (talk) 20:00, 6 September 2023 (UTC)

You ability to dredge a few examples from the internet does not indicate the usage is or even was widespread. This is unreliable original research on the topic of frequency of use. A reliable source on frequency of use is Google ngram. Use it. Cerberus (talk) 15:33, 21 September 2023 (UTC)

No. We go by published sources, not by web searches. Those were published sources, not "examples from the internet". Learn the difference. Also we are not making and should not make claims that the usage was widespread, only that it was used "sometimes". —David Eppstein (talk) 17:18, 21 September 2023 (UTC)

Note, the Don Asher one seems to be originally from a 1913 story in Harper's, and the Euchner book was originally published in 1996. Here's another example from Faye George's poem "Glory Hole": "... in clam shell and chicken bone, / in the rust lace of old tin, a fragment of rosary, / the Ballantine rings of a family's shame. / A civilization is in there, ..." (link goes to the book Back Roads, 2003). And one more example, from a John Roche column in the Washington Court House Record Herald, 1974: "This was not a 'separation of powers,' but a system of checks and balances. Each branch was interlocked with the two others like the Ballantine rings." –jacobolus (t) 19:05, 21 September 2023 (UTC)

---

The logo was well-known in the U.S. during the mid-20th-century, but it deserves only a brief mention on this article. AnonMoos (talk) 20:50, 21 September 2023 (UTC)

Wrong pronunciation

The correct pronunciation of "Borromean" is approximately bore-oh-MAY-in (and not boh-ROME-ee-in as the article claims). — Preceding unsigned comment added by 2601:200:c082:2ea0:444e:7fbc:5a62:4fb6 (talk • contribs) 18:41, 10 February 2024 (UTC)

The article's claimed pronunciation in English is boh-ro-MEE-in. The colon in the IPA indicates the long vowel. It is referenced to a published source. I am sure that other languages would pronounce the penultimate vowel differently. Do you have a published source for an English pronunciation that uses /eɪ/ in place of /iː/? If so we could at least include both pronunciations. —David Eppstein (talk) 19:25, 10 February 2024 (UTC)