# Talk:Möbius strip

As for ALT1 below. Aion is NOT holding a Möbius strip. Give me a break and take a better look. It is just a strip and the Romans perished (due to lead poisoning) before even inventing zero.  — Preceding unsigned comment added by 2A02:2168:B03:3F14:0:0:0:4 (talk) 12:42, 23 August 2024 (UTC) 

The link to "Umbilic torus" in the "See also" section is accompanied with a claim that the Umbilic torus can be obtained from a Möbius strip by gluing the latter along its single edge. IMHO, that statement is false, at least in regular 3D space, where the result would have to be a self-intersecting surface, assuming that the gluing is performed in the 3D space, rather than through some topological magic (sorry, I am not an expert in this field and lack the right terminology; my judgements are restricted to common sense intuition). Note that the article about the umbilic torus doesn't establish any relation of it with the Möbius strip (the link in the "See also" section doesn't count as such). I think that the misleading sentence has to be either removed or reformulated in a less confusing way and a more detailed explanation can be provided in the article about umbilic torus. Leon.Manukyan (talk) 08:41, 23 September 2023 (UTC)

You are confusing the standard embedding of the Möbius strip (with one half-twist) with the Möbius strip itself (a topological surface independent of how it is embedded into space). If you perform any odd number of half-twists to a strip before gluing it back to itself, you get a Möbius strip, with a different embedding for each different number of half-twists (see the second paragraph of the lead). The umbilic torus has a boundary that, if you cut along the ridges, is a Möbius strip. It is a Möbius strip embedded with three half-twists rather than one, but that is still a Möbius strip. —David Eppstein (talk) 10:15, 23 September 2023 (UTC)
You provided the same justification in your reversal of my earlier edit. My feeling was that something was wrong with it. A Möbius strip is single-sided whereas the Umbilic torus apparently has inside and outside. Now thinking about it more carefully, I guess that I found the flaw in your reasoning. The fact that the boundary goes three times around the center of the torus doesn't mean that the strip makes three half-twists. Actually it only makes a full twist (two half-twists). Hence it is not topologically equivalent to a Möbius strip. Leon.Manukyan (talk) 10:31, 23 September 2023 (UTC)
Ok, this time I think you may be correct. If it really were a Möbius strip then (regardless of number of twists) traveling  around the boundary (without crossing any ridges) until you return to your start should take you to the inside, but it doesn't. —David Eppstein (talk) 11:53, 23 September 2023 (UTC)

The hyperbolic plane can be represented as one sheet of the hyperboloid of two sheets in pseudo-Euclidean 3-space of signature −++ (the points with squared distance 1 from the origin, and first coordinate positive), or if you like as both sheets with antipodal points identified. From there the conformal disk/half-plane models are two kinds of stereographic projections and the Beltrami–Klein model is a gnomonic projection. It's only occasionally discussed but both hyperbolic planes on opposite sheets can also be considered as one object containing separate points – the opposite sheet projects to the exterior of the conformal disk or the other conformal half plane, while both sheets collapse to the same gnomonic disk.
The space of oriented lines in the paired hyperbolic planes is a (metric, not only projective) dual, and each oriented line can be visualized as the oriented intersection of the two-sheet hyperboloid with a plane through the origin in −++ space, analogous to great circles on the 2-sphere as the intersection with a plane through the origin in +++ space. If we take the oriented lines in −++ space perpendicular to those planes, they can be intersected with a hyperboloid of 1 sheet (the points with squared distance −1 from the origin in −++ space). This is analogous to taking polar duals between points and oriented great circles on the 2-sphere. The space of oriented lines is topologically a punctured plane.
If instead of considering oriented lines we consider the usual un-oriented lines, this is equivalent to identifying antipodal points on the hyperboloid of one sheet. This space is topologically a Möbius band (as mentioned in passing in the article), but it's worth pointing out that metrically, like the whole one-sheet hyperboloid in −++ space, this space is pseudo-Riemannian with signature −+. As the dual of the hyperbolic plane, it has "hyperbolic" angles and "circular" distances between points (the hyperbolic plane has "hyperbolic" distances and "circular" angles). One more point perhaps worth mentioning: when gnomonically projected the one-sheet hyperboloid projects to the exterior of the disk that the two-sheet hyperboloid projects to the interior of, and under projection one circle of the one-sheet hyperboloid projects to the circle at infinity, but the association of antipodal points also requires associating antipodal points on the circle at infinity; this might be harder for readers to visualize though – maybe it's enough to point out that every line in the projection is joined at its opposite ends.
One source mentioning this is: Khesin, Boris; Tabachnikov, Sergey (2009). "Pseudo-Riemannian geodesics and billiards". Advances in Mathematics. 221 (4): 1364–1396 (Ex. 2.4 p. 1367). doi:10.1016/j.aim.2006.10.004.
A source mentioning that the exterior of the gnomonic disk represents the space of lines and is homeomorphic to the open Möbius strip is doi:10.1007/978-3-642-61777-5_4, p. 94: "As in the Euclidean case, the space of all lines in H is a Mobius band because there is a homeomorphism (obtained by taking the pole) between the space of lines and the space of points lying outside [the conic]; the outside of a conic is homeomorphic to a Mobius band".
I wonder if it's worth mentioning or drawing the one-sheet hyperboloid or the exterior of the gnomonic disk / trying to give some kind of visual impression about why these are topologically a Möbius strip. The current brief mention comes across as a bit of trivia but isn't really explained.
(While we're at it, the metric dual of oriented lines of the Euclidean plane can be geometrically represented as a cylinder with its axis through the origin embedded in a 3-space of degenerate signature, and if we only consider un-oriented lines from the Euclidean plane we get a cylinder with "antipodal" points identified. This space has "circular" distances and "flat" angles, dual to the Euclidean plane with "circular" angles and "flat" distances.)
I also wonder if it's worth pointing out that metrically the duals of the Euclidean/hyperbolic plane are not Riemannian. –jacobolus (t) 20:07, 4 March 2025 (UTC)

"As an abstract topological space, the Möbius strip can be embedded into three-dimensional Euclidean space in many different ways: a clockwise half-twist is different from a counterclockwise half-twist ..."
But if M represents a Möbius band with a clockwise half-twist, and M' one with a counterclockwise half-twist, then there is a homeomorphism taking the pair (R3, M) to the pair (R3, M').
So without further explanation, this is simply wrong. 
The further explanation would need to mention orientation-preserving homeomorphisms of 3-space. Which are probably too advanced for this early in the article.
So this unnecessary claim would be best removed, or at minimum placed much closer to the end of the article.

Actually, the intended sense is the one of continuous deformation: You cannot continuously deform the clockwise band into the counterclockwise one. So it's not oriented homeomorphism but rather ambient isotopy. But technical words aside it's a familiar concept in topology: it's the same sense in which you cannot unknot a knot. —David Eppstein (talk) 23:56, 24 April 2025 (UTC)

I am trying to insert again an image of the Möbius band which has been on Wiki for ages. Last year I saw it was removed and I reinserted it because cited on a book on differential geometry by Springer Verlag. AYesterday I discovered that the image was removed again. Now, other than being of higher quality (and that's why was printed in color in a book), it is a citation for Wikipedia. And before removing something cited there must be a real valid reason.
This is the book that cite that picture
https://link.springer.com/book/10.1007/978-3-319-39799-3 JoshDif (talk) 17:00, 16 October 2025 (UTC)

Edit: we apparently created two sections about this in parallel; I'm consolidating them here. –jacobolus (t)
Recently there's been some conflict about the lead image(s), with @JoshDif adding their image and various other editors reverting the change. Without stating any preference about the best end state, we should leave the stable article pending discussion, instead of having a revert war. To simplify a summary of the edit history:

In 2005, @Dbenbenn contributed a photograph of a white paper Möbius strip and later a replacement photograph of a green paper Möbius strip,    which has for most of the time since then been the top lead image on this page. In 2005 and 2006 the color was boosted by commons:User:RadRafe~commonswiki and commons:User:Ttog~commonswiki.
In February 2012 JoshDif replaced the image on the left (liney gradient) with the image on the right (candy-apple gloss):   →   in the "Geometry and topology" section, which was reverted for making it harder to follow the parametrization, so in a follow-up edit included both images in the page
As far as I can tell from a brief skim the candy-apple image stayed there (though some point the liney image was removed) until @Gentlecollapse6, in March 2018 exchanged the candy-apple image with the lead image, which throughout most of that time had been the green paper image
In September 2018 @Anita5192 restored the green paper image to the top and moved other images to a gallery but was reverted by @David Eppstein
In May 2019 @Henstepl returned the green paper image to the lead, below to the candy-apple image.
In March 2021, @Hazhk moved the paper image to be above the candy-apple image.
In March 2022, @David Eppstein removed the candy-apple image and another image from the lead as being "unnecessary". David did extensive work on the article in March 2022.
Recently, in September 2025, @JoshDif returned to restore their candy-apple image to the lead, above the green paper image. Since then there has been a revert war about it.
Personally I think one image of a Möbius strip in the lead gets the idea across okay, but I wouldn't mind including a second one somewhere further down the page. –jacobolus (t) 17:25, 16 October 2025 (UTC)

Ok, thanks for the constructive comment. My point is that the image I am trying to reinsert is officially cited in a published book (https://link.springer.com/book/10.1007/978-3-319-39799-3). And, as a scientist, I find strange that something officially cited is not present on Wikipedia. Moreover, the rendering is more mathematical and the highlighted boundary makes more clear the shape. JoshDif (talk) 17:31, 16 October 2025 (UTC)
Wikipedia should decide based on local editor consensus which images to use, rather than relying on random external book authors/editors. If authors citing Wikipedia want to ensure that the cited version is stable, they can use a permalink rather than a link to a living document. –jacobolus (t) 17:39, 16 October 2025 (UTC)
Why you have to talk about a "random book or author" in referring to a professor. I am a university professor as well, and I would never use those words for a colleague. Please, try to be polite. JoshDif (talk) 17:42, 16 October 2025 (UTC)
I don't mean any disrespect.
This book author drew most of their own figures or hired someone to draw them, so kudos to them, and I'll leave aside critiques of their figures. For the rest it mainly appears that they went to the Wikipedia page about a topic and pasted an image or two from it into their text (alongside one image copied from Make Magazine, one from Flickr, and one from Google Plus of someone's kid, which presumably had a free license). Among those, the author included both the green paper image and the candy-apple image in their book to introduce the concept of a Möbius strip. That's fine and dandy: allowing anyone to use images here however they see fit is why we use free licenses.
But the use of a Wikipedia image in some other context doesn't really prove anything about the image or its relation to the article, and in my opinion isn't a meaningful point in this discussion. We should focus on aspects such as: which image best represents each part of the text, which image is clearest, most attractive, least distracting, and so on.
(Aside: it's somewhat gratifying when authors use one of my images in their books, though I wish they would more often drop me an email about it.) –jacobolus (t) 18:04, 16 October 2025 (UTC)
Do you think removing it from this article (but not from the repository of images) lessens or insults the Professor's book?  Would that argument carry less weight if the book's author were not a Professor, or if the image were cited unofficially? —Tamfang (talk) 21:57, 16 October 2025 (UTC)
I never said that. It would never affect the value of the book. What I was pointing out is that there is objective evidence showing the figure was not unnecessary, contrary to what the person who removed it claimed. Josh (talk) 09:48, 17 October 2025 (UTC)
To make the discussion more concrete, the image can be seen in context on p. 156:
 The most infamous example of a nonorientable surface is the Möbius strip illustrated in Fig.3.24. A model can be built with a thin strip of paper by gluing the ends together with twist. For it to be a surface, its “edge” (shown yellow in the right figure) must be excluded, but before you erase it, trace around it to confirm the interesting fact that there’s only one yellow edge, not two.   If you try painting one side of a paper model, you will discover why it is nonorientable—it has only one side. Conveyor belts in engines, factories and supermarkets are sometimes twisted into the shape of a Möobius strip. Compared to the traditional two-sided belt, this design has a single side that is twice as long, which eﬀectively doubles its lifespan.
In the image credits section (p. 360), there's no link directly to this page, but we have:

 
–jacobolus (t) 13:56, 17 October 2025 (UTC)
As a physical photo I think the green one is a clearer representation of the topic: it doesn't require as much interpretation to see it as a 3d object. And as a lead image the red one is a poor choice because it does not match the lead text "can be formed by attaching the ends of a strip of paper together with a half-twist": the surface in the red image does not appear to be a developable surface as the paper strip would be. We have multiple images that depict the strip in more or less the same 3d shape as the red one: the image of the crab walking on the strip, the double image of the median-sliced strip, the subdivision by Tietze's graph, etc. The topic that most closely fits the red image is Möbius strip#Sweeping a line segment describing the construction that I believe was used to create the red image, sweeping a rotating line segment around a circle, but for that we also have a better image, a gif showing the sweep construction dynamically. So I don't see what value the red image could add to the article. We shouldn't add an image merely because it exists. —David Eppstein (talk) 17:34, 16 October 2025 (UTC)
The point is that you should have not removed the image just because you think it is unnecessary. Especially something published. Sorry, this was a sloppy change. It has been chosen by a book on differential geometry. This has an objective value. For the author of that didactic book, the rendering was useful. And I think the same.
But to me it is funny, that you started to talk like a lawyer about a war and now we discovered that you removed that image by just commenting "unnecessary" (for you). JoshDif (talk) 17:40, 16 October 2025 (UTC)
Re " you should have not removed the image just because you think it is unnecessary": Yes, of course, I should have done that. Editing articles to make them better is what we all do here. Your "because I say so" is totally invalid as a reason for you to assert that you should get your way regardless of opinions of other editors. And your argument that we are somehow required to follow the exact choices made by some book author is not much better: we should include images because they are somehow informative, not as a game of follow-the-leader. That way lies copyvio and plagiarism. —David Eppstein (talk) 17:43, 16 October 2025 (UTC)
You go on giving no reasons about the changes you made. We discovered that you were the cause of what you called a "war". That image was on Wikipedia from 2012 to 2022, until you removed it because, and apparently just you, thought it was unnecessary. JoshDif (talk) 17:54, 16 October 2025 (UTC)
In fact, the first reply I made in this thread entirely consisted of reasons. Go read it again. Also see MOS:PERTINENCE: "Each image in an article should have a clear and unique illustrative purpose and serve as an important illustrative aid to understanding the subject." You have not articulated any actual purpose for the red image. Copying the choice of another source is not a purpose in this sense. —David Eppstein (talk) 17:57, 16 October 2025 (UTC)
For 10 years noone was thinking of it as unnecessary. You should explain why after 10 years there, you woke up and decided it was unnecessary.
As a mathematician (a Professor like you) I already told that I find the image clearer, more mathematical (generated by a formula) and more elaborated given the presence of a boundary. And it is also cited. Think if you cite a paper that noone can then find anymore. Do I really have to explain you this? You were sloppy. It's okey. Now let pass some time and then revert the image. Thanks. JoshDif (talk) 18:00, 16 October 2025 (UTC)
Anything added to a Wikipedia article sits there unchanged until someone else decides to change it. Then the new version similarly persists. Etc. Sometimes that "until" can be a day, or sometimes it can be 10 years. There's nothing inherently wrong with changing something after 10 years, if the change seems like an improvement.
Sometimes there's a disagreement, and if neither party gives up there's a discussion about it, ideally drawing in more editors to weigh in, in search of consensus. –jacobolus (t) 18:11, 16 October 2025 (UTC)

Getting back to the relative merits of various images, let me try to summarize what I've seen of the discussion thus far. (Excuse the bullet lists.) Feel free to add additional points I am missing.
  The main arguments against the candy-apple red rendered image seem to be:
The shape and lighting makes it somewhat difficult for some viewers to visually interpret as a three-dimensional image
The light and dark red shapes inside are somewhat confusing; it's not really clear what they represent or whether they relate to the concept of a Möbius strip
The image doesn't match the description in the lead section of a strip of paper being twisted and taped together, partly because geometrically the shape here couldn't be formed by that method, so an image which does fit that description may be a clearer illustration of what the text means
Some of the other images on the page are geometrically redundant with this one
I'd also add:
The red shading is too uniform and perhaps too colorful overall, which makes it harder than ideal to tell which parts of the strip are supposed to be facing which direction
The yellow border doesn't have enough lightness contrast with the background and blends in, which somewhat defeats the purpose of its inclusion (though the yellow border is quite visible as a line between the two overlapping red parts, which is helpful since they otherwise have indistinguishable shading)
For a computer render, the image has relatively low resolution, so is mainly suitable at thumbnail size
There are some weird shaded ripples along the top, and it's not obvious why they're there (maybe an artifact of a polygonal geometry used for the render?).
 
Arguments I have seen (and some I can think up) against the green paper photograph include:
The green image is a photograph, which doesn't properly convey the abstract nature of the geometrical object (one edit summary had "theoretical image more relevant to theoretical concept")
The green paper photograph has relatively low resolution, and is a bit blurry, with some image artifacts
When the color was boosted on the green paper photograph, the background color got a bit more colorful than ideal, with amplified colorful image compression artifacts in the background
Having the green paper go through an additional two areas of (opposite) overlaps from the camera perspective may make it harder for some viewers to immediately identify the half twist
The lighting could do a yet better job of distinguishing parts of the strip facing different directions
The best solution might be to try to take a new, more carefully arranged and lit photograph of a paper Möbius strip for the lead, or maybe take several alternative such pictures for further discussion. For instance, perhaps the paper strip could include a border. I'm not sure if it would make the image clearer or less clear to use a piece of paper with different colors on each side. –jacobolus (t) 19:29, 16 October 2025 (UTC)
I have created a rendering [1], perhaps it is of interest to this page. I can provide a different resolution/angle if required. Pekkog (talk) 10:46, 4 November 2025 (UTC)
There are a few hundred images in C:Category:Moebius surfaces and its many subcategories. Obviously, we cannot and should not try to include them all. What information in our article is inadequately illustrated already and would be improved by the addition of this image? Alternatively, which image would you suggest removing to make way for it? —David Eppstein (talk) 18:23, 4 November 2025 (UTC)
It doesn't belong in the lead of this article, but

is pretty cute. –jacobolus (t) 20:31, 4 November 2025 (UTC)
Pretty. I would be very happy to have something like that decorating my office. Unfortunately it doesn't appear to be something we can mention under notable artworks. —David Eppstein (talk) 22:27, 4 November 2025 (UTC)