Talk:Möbius strip/Archive 2

That Scarf

The scarf in the shape of a Möbius strip in no way provides a "demonstration of the mathematical basis of crochet." If I cut a potato in the shape of Alexander Hamilton, I have not shown that he invented tuberous vegetables. —Preceding unsigned comment added by 68.63.170.169 (talk) 03:11, 25 April 2008 (UTC)

Agreed, it's interesting but it doesn't add to the article. Unlike the link that was recently removed (and restored) to a demonstration of a paper strip music box implemented as a mobius strip. Not only does show the physical properties but demonstrates the direction change.--RadioFan (talk) 15:43, 7 December 2010 (UTC)

Careful not to imply other actors here, it was restored by you after I removed the addition by you. DMacks (talk) 16:21, 7 December 2010 (UTC)

Careful to talk about content and not editors.--RadioFan (talk) 20:56, 7 December 2010 (UTC)

Mobius strip with circular boundary

I found the following reference online. "D. Lerner and D. Asimov. The sudanese mobius band. (video). In SIGGRAPH Video Review, 1984." Does anybody have access to this video? I'd love to see some stills! Sam nead 00:30, 6 September 2006 (UTC)

Hi all, could someone provide a clearer description of the projection that makes the Mobius edge a circle? I mean, how do we embed the mobius in the 3-sphere, and how do we project the C² object to R³ with the right point? Equations or any kind of computer program would help. Thanks

That's discussed at length in Talk:Möbius_strip/Archive_1 76.77.182.66 (talk) 10:56, 12 December 2009 (UTC)

I've added a link to my animation of this surface at the bottom, but I'm also happy to contribute it to wikimedia if thats better. As for the clearer description of the projection and embedding, I might have a go at writing something, but probably not for a while. Also, I think this surface was first described by Blaine Lawson - he should be credited. Would a separate article for this 'Sudanese' band be in order? I just think that its a really nice embedding that deserves to be better known. —Preceding unsigned comment added by Anodised (talk • contribs) 11:52, 3 November 2008 (UTC)

I'll edit to add a reference to Blaine Lawson. Just a sec. Sam nead (talk) 11:23, 27 August 2013 (UTC)

And done. Sam nead (talk) 11:37, 27 August 2013 (UTC)

3D versus 2D

Could someone please answer me this:

If you put a hole in the m�bius strip, where does it go to? It goes through the strip and to the same side it started. How does that work?

"Where does it go to?" To "go" somewhere, you have to go "through" something. That is, you are assuming that the Mobius strip has some thickness. It doesn't.

If you do insist that the strip has some thickness (atoms, etc) then you are treating it like a solid, three-dimensional object, and not like a surface. Again there is no mystery: lots of three-dimensional things have only one side, like an orange or a bagel. Sam nead 16:58, 15 April 2006 (UTC)

The m�bius strip is a surface in three dimentions, and each point on the strip is part of two areas on the strip. Thus, removing the points on an area of the strip would create two breaks in a line drawn around the strip. --Roger Chrisman 22:40, 4 September 2006 (UTC)

Imagine a piece of thick paper, 10 mm wide, 0.5 mm thick, and however long you want. I give the end a half-twist and join the ends together in whatever normal or magical way you like. This is unquestionably a 3D object, being that it has a 0.5 mm thickness. But as far as I can tell, this does not prevent it from having any of the amazing properties of a legitimate mobius strip. If one makes a hole in it, that hole starts on one side, goes through the thickness of the paper, and ends up on the SAME side!!! And you can prove it is the same side, by crawing the line as described in the article! Anyone have a problem with that? That's why I say that a mobius strip is (or can be) a 3D object. --Keeves 00:43, 5 September 2006 (UTC)

Keeves: What you are describing is three-dimensional, but it is not a Mobius strip. Mobius strips are surfaces, and so are two dimensional. Said another way -- a Mobius strip is a mathematical abstraction and not a real object. It is a special kind of surface. Sam nead 03:22, 5 September 2006 (UTC)

Roger: The Mobius strip need not be embedded in three-space. It is a "space" all on its own. Sam nead 03:22, 5 September 2006 (UTC)

I stand corrected. Thanks! --Keeves 16:14, 5 September 2006 (UTC)

I view the mobius is more like 3cm wide, 10cm long and 0cm high. In our dimension te figure does not exist, but theoreticly it does. But in our world everything has thickness, so a truely 2-D object does not exist in our universe. —Preceding unsigned comment added by Jodra13579 (talk • contribs) 01:34, 2 May 2008 (UTC)

We must not give kids the impression that we are talking about a continuous flat surface. Möbius strip is a 3D solid thus all surfaces are curved. Kids are capable of understanding a gimmick demonstrated for fun. If a strip is cut several times and undone it can be always assembled from the cuttings and "fused" together to make a solid ring, which is what it really is ... with ridges, but a solid ring nevertheless. Edit here is to remove the awe and to foster true understanding. Rstafursky (talk) 22:03, 27 December 2011 (UTC)

In response to the initial question, as people have pointed out, it has no thickness and needn't be embedded in R³ -- it may be easier to think of them as being made not of paper but of transparent plastic film. So cutting a hole "through" one doesn't "connect" any points; rather, it just removes a contiguous set of points from the manifold. If you cut a long door out of one and twist the door and glue one edge of it to the cut edge, then you have changed the topology, but it doesn't mean you can "suddenly" get to the other side because each point is already assumed to correspond to both sides. —Ben FrantzDale (talk) 15:17, 28 December 2011 (UTC)

Cutting Concerns

Alternatively, if you cut along a Möbius strip about a third of the way in from the edge, you will get two strips: One is a thinner Möbius strip - it is the center third of the original strip. The other is a long strip with two half-twists in it (not a Möbius strip) - this is a neighborhood of the edge of the original strip.

This contradicts what is said above:

If the strip is cut along the above line, instead of getting two separate strips, it becomes one long strip with two half-twists in it (not a Möbius strip). This happens because the original strip only has one edge which is twice as long as the original strip of paper. By cutting you have created a second independent edge, half of which was on each side of the knife or scissors. If you cut this new, longer strip down the middle, you get two strips wound around each other.

Right? There the same steps. Joerite 04:23, 1 October 2006 (UTC)

You lost some context for the second quote: "the above line" was drawn in the middle (half-way from the edge), not ⅓ from the edge. Try it for yourself and see what happens:) It sounds weird, and perhaps counterintuitive to people not familiar with the Möbius strip, which is why it's mentioned. DMacks 16:09, 1 October 2006 (UTC)

So cut halfway across the width and you get one strip, but cut one third of the way and you get two strips? What happens at 2/5 of the way across? 3/7 of the way? 4/9? Seems fishy to me - how close do you have to get to "halfway" before you end up with one strip? Where did I put those scissors....?143.252.80.100 21:15, 21 November 2006 (UTC)

Interesting thought experiment. I think what actually happens if you cut 1/3 of the way across, is, when you get all the way around once your scissors WON'T intersect your starting point. Instead you'll be at the 2/3 point, and can continue cutting all the way around again, until you do get back to the 1/3 point. But wouldn't that give you THREE strips when you do get back to the start? That doesn't make any sense either, though...when you finally finish the cut you're clearly only separating two strips. Help, my head is going to explode! Middlenamefrank 21:26, 21 November 2006 (UTC)

You're right in that the "first" time around, you wind up 2/3 across. As I visualize it more generally, the issue is that there is only one edge (in the sense of a normal sheet of paper) on the strip. So if you start "some distance x from the edge", when you have traveled once around the loop, you are now that same distance from the edge, but now "the edge" is now across the loop from where it was when you started. You haven't really crossed' the middle so much as wrapped around it. So you peel the edges off the strip as one loop and leave the middle section as the other loop. DMacks 22:08, 21 November 2006 (UTC)

There's nothing particularly special about 1/3. The point is whether you cut along the center circle or not. If you cut along the center circle, you will get an orientable band. If you basically trim off the edge (cutting along some distance less than half the width of the band) when you come once around, you find you need to make another trip around to finish off the cutting. Of course, if you decide not to go around again, but just join the ends of the cut together, that's fine and gives you the same result as cutting along the center circle. In real life, where you can't really cut along the exact center (everything is approximate), that's what you're really doing. That should answer your question about how close you can get. If you get close enough, when cutting you will probably decide to join up the cut after only one trip around the band because the ends of the cut are close enough. --C S (Talk) 08:29, 22 November 2006 (UTC)

How to add the twist mathematically?

I understand how a Mobius strip is a fiber bundle, the result of crossing the unit interval with a circle, but in that formulation how does one describe the twist soas not to just get a cylinder? (I haven't taken a course on topology; let me know if this isn't a sensible question.) �Ben FrantzDale 04:27, 25 January 2007 (UTC)

Your first sentence is incorrect. The Mobius strip is not ...the result of crossing the unit interval with a circle... The Mobius strip is not a direct product. However, you are correct when you say that the Mobius strip is a fiber bundle. best, Sam nead 15:29, 25 January 2007 (UTC)

OK, so then how is it a fiber bundle? �Ben FrantzDale 15:55, 25 January 2007 (UTC)

A parametric plot of a Mobius strip

Take a piece of lined paper (what a great adjective! Much better than ruled) and with a pen draw five closely spaced (and evenly spaced) vertical lines, perpendicular to the lines of the paper. Cut along the outer two lines to get a skinny rectangle of paper. Glue the short ends together, with a half twist, to get a Mobius strip. Notice each of the paper lines meet the middle pen line in a single point. Also, for every point on the middle pen line, there is a paper line going through that point. That is, for every point of the pen line there is one and only one paper line associated to it. These paper lines are bundled together to form the Mobius strip. So the Mobius strip is a bundle over the pen line (a copy of the circle) with fibres being the paper lines (copies of the closed interval).

ps. The second and fourth pen lines don't play any role in this discussion. I just added them to match the picture above. Also, my description above is very similar to the discussion of the Mobius strip in the article on fiber bundles. You might also be interested in the articles on ruled surfaces and ruled paper. best, Sam nead 15:31, 26 January 2007 (UTC)

Ben FrantzDale seems to be confusing a fibre bundle with a trivial fibre bundle. For a fibre bundle you have a base space X, a total space E, a typical fibre F, and a continuous map $\pi :E\to X$. We require that the fibre bundle be locally trivial, i.e. for every open neighbourhood $U\subset X$ there exists a homeomorphism $h:U\times F\to \pi ^{-1}(U),$ i.e. over each neighbourhood of X we can write it locally as a product. A fibre bundle is trivial if we can take U to be the whole of X. For a cylinder we can write it as $\mathbb {S} ^{1}\times [0,1]$, so h is just the identity, U can be taken as $\mathbb {S} ^{1}$, and F = [0,1]. This isn't true for the Mobius strip. A more abstract view is the structure group consisting of homeomorphisms between fibres. For a trivial bundle the group is just the identity. For the Mobius strip it's $\mathbb {Z} /2\mathbb {Z}$. I recommend the wonderful book:

• H. OsbornVector Bundles, Volume 1: Foundations and Stiefel-Whitney Classes. Academic Press 1982.

Dharma6662000 (talk) 21:17, 24 August 2008 (UTC)

Power logo

I want to add the Power Architecture logo (Image:Power-architecture-logo.png) to this page because it's a modern and current example of a logo that uses a M?bius strip. It wasn't chosen because it was pretty but because it represented something to those who use it, in this case "flexibility", "diversity", "infinity", "curiosity" and "discovery".[1] The way they charge logo and use the symbol is relevant to this article. Just like the Recycle-logo is relevant for the same reasons.

Even if I don't have any problems comming up with reasons why the logo should be on this page (I think it's obvious, certainly since there's already a logo on it) I object to the suggestion that I should state my case here and getting it approved before adding it. The way I've used Wikipedia, and seen it used, one certainly should argue when deleting something, not adding something. The last deletion was seemingly without merit or cause, just petty. -- Henriok 17:24, 28 March 2007 (UTC)

The big difference between the recycling logo and the Power logo is that the recycling logo is public domain, whereas the Power logo is fair use. mrholybrain's talk 18:15, 28 March 2007 (UTC)

I think it's a pretty clear fair use of the logo here. It isn't here to brand a product but as a piece of art, clearly on topic and relevant to the sub section (art and technology). -- Henriok 14:56, 29 March 2007 (UTC)

I would argue that another difference between the two logos is that one is essentially everywhere you look, while the other is somewhat obscure. The Mobius strip page should be about Mobius strips, not about Power Architecture. Perhaps it would be most appropriate to edit the Power Architecture page. Currently there is only a short sentence describing the logo. You could add to that page "Notice that the logo, in the shape of a Mobius strip, represents..." Best, Sam nead 00:21, 29 March 2007 (UTC)

Are you really playing the obscurity card here? It should be here even more because it's not well known, it's an encyclopedia after all and people come here to find stuff they didn't know about. People interessted in Möbius strips should get a fair sample of when it is used, and such examples should not be removed when they are readily available. There's already listed several instances where Möbius stips are used, and I think there should be even more examples available on this page. I understand that some are frightened about logos and think "fair use" is more like "not use" so is it OK with you guys if I just write a blurb and just link to it? And seriously.. There's no risk of this page being about Power Arch even if it's logo is shown inline. No one is thinking this page is about recycling stuff now is it? -- Henriok 14:56, 29 March 2007 (UTC)

Look, the fair use label says that the Power arch logo can be used to discuss Power architecture. Nothing else. That is United States copyright law for you. Putting the logo on this page is copyright infringement. mrholybrain's talk 16:23, 29 March 2007 (UTC)

I agree that there are other obscure uses of the Mobius strip on the page. I would be happy to discuss removing some of the less interesting ones. On the other hand, none of those references come as close to being advertising as the Power logo does. Hmmm. Perhaps we could create a "Popular culture" section for the rock band references, and so on. Then the Power logo could go there - advertising counts as popular culture.... I am not really excited about this suggestion, however. Best, Sam nead 12:48, 30 March 2007 (UTC)

Section on technology

The section on technology and science is very cryptic and poorly typeset. Please edit it into something reasonable. Best,Sam nead 23:43, 12 April 2007 (UTC)

This is Wikipedia. Please don't be shy. Why don't you bring it to your own very high standards? Dharma6662000 (talk) 00:36, 25 August 2008 (UTC)

two sided strip

i'm too lazy to read so does it say that a strip op paper in a mobeous strip two sided? —The preceding unsigned comment was added by 69.158.185.61 (talk) 23:54, 8 May 2007 (UTC).

If you're too lazy to read the article, you're probably too lazy to read the answer here either. DMacks 00:47, 9 May 2007 (UTC)

I bet most people will be too lazy to try and figure out what on earth you could possibly be asking. --C S (Talk) 05:36, 9 May 2007 (UTC)

And I bet that this is some idea of a joke and as such we should waste our time replying. So I shouldn't even be writing this, so I'll stop... Dharma6662000 (talk) 00:37, 25 August 2008 (UTC)

Animation

I have created a short animation on Mathematica showing that a moebius strip is non-orientable. It might be worth putting it in the article or creating a link to it. How would I go about this and what would be a good video format to use? John Wallbank 11:16, 26 July 2007 (UTC)

The lack of any response to your post is typical. If I were to know then I would tell you. The majority of people only ever reply to threads to argue some silly little point, and to make themselves feel smart (you know who you are). If anyone does know how to do this then please include some instructions. Dharma6662000 (talk) 00:41, 25 August 2008 (UTC)

Recycling Mobius Loop

Is it just me or does the Mobius loop not look Mobius at all? I can see an inner side and an outer side. Am I viewing it wrong? Gelsamel 03:56, 16 August 2007 (UTC)

The sides switch places on the half-twist. Thus it has only one side, which is both inner and outer. Admiral Norton ^(talk) 00:26, 3 January 2008 (UTC)

As it is demonstrated and marketed, it isn't a mobius. As was originally submitted, it met the definition.http://logoblink.com/wp-content/uploads/2008/03/recycling_symbol_garyanderson.pdf—Preceding unsigned comment added by 76.205.134.31 (talk) 20:10, 12 August 2009 (UTC)

Introduction

The introduction of this article was very hard to read for a non-mathematician. Is it necessary to include all those mathematical details at the very start? This detracts from the article's other interesting points about the properties of the strip that are curious and interesting for the non-technical reader. (I'm talking about things like "boundary component," "non-orientable," "ruled surface," "developable surface" - etc.) Thoughts? Jpp42 10:28, 27 August 2007 (UTC) i thought that the mobius strip had something to do with space and was an object in the space so it was expremly hard to read and the mothmatical terms were hard and made no sense for my project so i did not understand what it was and was even more confused than when i started. —Preceding unsigned comment added by 68.116.165.144 (talk) 19:40, 31 March 2008 (UTC)

I think that Wikipedia askes that the first few lines give a short technical statment of what the article's about. I understand what you're saying, but if it weren't that way then there'd be thousands of people going on and on about how the article doesn't comply with this, that, and the other, and hundreds of admins with their power-hungry delete-button itchy fingers just poised to use their newly found powers. It's better if you just try to read on. Any good article will try and fill it out. N.B. There are many more editiors than 100 so I'm not talking about all of you, but you guys know who you are. For example: one admin had deleted more than 97,000 pages which, by sveral orders of magnitude, was more that he'd editied. Dharma6662000 (talk) 00:49, 25 August 2008 (UTC)

Transmissionsriemen.jpg

Conveyor belt in the shape of a Möbius strip to power an threshing machine by an steam tractor

This image from Commons was just added to the article with the caption: "Conveyor belt in the shape of an Möbius strip to power an threshing machine by an steam tractor". However, if I'm not mistaken, the belt in the picture does not actually seem to be a Möbius strip; if you trace the belt, you'll find that the same side is always the outside. Or am I just looking at it wrong? —Ilmari Karonen (talk) 23:16, 31 August 2007 (UTC)

I just noticed the exact same thing - unless there's some part we're not seeing. It's quite a confusing image for readers though, most of whom will probably notice it too. Seaserpent⁸⁵_Talk 13:33, 6 September 2007 (UTC)

Ok, so I'm not alone. Thanks. I've removed the image from the article for now, at least until someone disagrees, and have added it here instead. —Ilmari Karonen (talk) 15:13, 6 September 2007 (UTC)

True. It has 2 half-twists. Admiral Norton ^(talk) 00:29, 3 January 2008 (UTC)

What is a 3D Möbius strip called?

I know that a true Möbius strip has no thickness, so do 3D representations have a particular name? Since any way to "create" one in the real world would give it thickness, i.e. the thickness of the sheet of paper used, these are true Möbius strips. —Preceding unsigned comment added by 64.16.40.18 (talk) 07:20, 1 October 2007 (UTC) The name of the figure would be a Diahedron. Di- meaning 2 and hedron meaning 3-D figure. —Preceding unsigned comment added by Jodra13579 (talk • contribs) 01:38, 2 May 2008 (UTC)

If I understand your question then you're asking what an abstract three-dimensional Möbius strip would look like. I don't know the answer, but I'll try and put it into a better frame work. The standard model for a Möbius strip is the unit square $[-1,1]_{x}\times [-1,1]_{y}$ where we identify the points $(-1,y)$ with the points $(1,-y)$. There is an idea of a thee-torus, i.e. a three dimensional analogue that lives in four-space. So what we need to do is take a cube, and identify some faces with certain twists. Quite how this is done, I'm not sure. Dharma6662000 (talk) 00:59, 25 August 2008 (UTC)

A physical representation of a Möbius strip with non-zero thickness would topologically be simply a torus. Its single (orientable) surface separates the solid inside (paper) from the outside(air). —Preceding unsigned comment added by Anodised (talk • contribs) 17:32, 22 October 2008 (UTC)

A Heart on the Right.

I read somewhere that if a Flatlander traveled around a Mobius Strip it would find itself normal but the other Flatlanders reversed (if they were left-hearted, now their hearts are on the right). The others would think that the adventurous Flatlander was reversed. I understand the relativity of this, but I don't understand why either one would be reversed. I tried this by drawing it on a MS already made and one not yet created and the results didn't match up with what I read. COuld someone describe why this is, dare I say it, true? In laymen's terms (id est: no mathematics involved), please. Also, should this be included in the article? Please help! 76.188.26.92 01:41, 27 October 2007 (UTC)

I hope you figured it out! If not and you are still interested, I think the section on orientation by triangulation in orientability should be of great help. --C S (talk) 08:33, 18 May 2008 (UTC)

This doesn't seem right to me. If the flatlander made one lap of the Möbius strip then he would be on the otherside, i.e. he's be walking upside-down on the ceeling whilst everyone else would be standing normally on the roof. If he did another lap then he'd be the right way up. That's why you get the $\mathbb {Z} /2\mathbb {Z}$ as a torsion group in the first homology group $H_{1}(M,\mathbb {Z} )$ (and the structure group of the fibre bundle is $\mathbb {Z} /2\mathbb {Z}$. If you do two laps then you're back to normal. If he was reversed after two laps then he'd need to do four and the torsion group would be $\mathbb {Z} /4\mathbb {Z}$. Dharma6662000 (talk) 20:38, 24 August 2008 (UTC)

Dharma, there is no roof when you live INSIDE the surface... like Flatlanders do :)--83.39.61.55 (talk) 17:35, 12 October 2008 (UTC)

Dharma[2⁴·5³·3331] writes: "That's why you get the $\mathbb {Z} /2\mathbb {Z}$ as a torsion group in the first homology group."

But this is not correct. The first homology group of the Möbius band has nothing to do with its non-orientability. H₁(M; Z) = Z. There is no torsion. (Nor in any other homology group with Z coefficients: H₀(M; Z) = Z, and all the other H_k(M; Z) for k ≠ 0,1 are 0, meaning the trivial group.)

Why "Sudanese"?

How did the Sudanese Möbius acquire that name? Is there some connection to the African country? Ishboyfay 20:37, 15 November 2007 (UTC)

I coined the name Sudanese -- a combination of the names of two researchers -- to designate the minimal Möbius band embedded in the 3-sphere as half of the minimal Klein bottle discovered by Blaine Lawson in his thesis. In the 3-sphere in 4-space, it has a very high degree of symmetry and already has a round circle as its boundary. There is no actual connection with the African country. A computer graphics film of it was made and appears under that name in Siggraph Video Review #17 at, e.g., http://www.cg.tuwien.ac.at/resources/video/svr001-119.html .Daqu (talk) 17:59, 20 January 2010 (UTC)

You could mention the names of the two researchers, for clarity. (I presume they're not Blaine and Lawson.) —Tamfang (talk) 19:06, 23 January 2010 (UTC)

I happen to know that one of the researchers is Sue Goodman and the other one's first name is Dan... As soon as I remember the second's last name I'll add it. A13ean (talk) 20:32, 14 April 2010 (UTC)

The other researcher's name is Dan Asimov(nephew of Isaac). 152.23.99.141 (talk) 00:38, 14 April 2011 (UTC)

I was able to confirm the above. I have added it into the article since the section seems a bit confusing without it, but a good reference for it is still needed. A13ean (talk) 00:25, 11 May 2011 (UTC)

Pop culture section overdone?

The section on pop culture is turning into the tail that wags the dog. I know the Möbius strip is fascinating, but imagine if the article on the circle likewise contained a list of occurrences of the circle in popular culture. —Preceding unsigned comment added by Ishboyfay (talk • contribs) 05:22, 19 February 2008 (UTC)

I agree, by all means cut it down to size or eliminate it. It's used as a common symbol and cataloging every time it pops up in a video game, song lyrics, or a corperate logo makes verifying and maintaining the section practically impossible. Zytsef (talk) 07:21, 19 February 2008 (UTC)

I also agree. There is a lot of spam in that section. Sam nead (talk) 18:49, 19 February 2008 (UTC)

I think WP:TRIVIA applies here. I agree the section needs to be trimmed. -- Fropuff (talk) 03:03, 20 February 2008 (UTC)

I've gone ahead and removed the section entirely. I think this is the best solution to the problem of crap accumulating. It might be worth mentioning in the intro that Mobius strips come up a lot in pop culture, without dedicating a whole section to it, but I'm perfectly happy without it. Zytsef (talk) 03:18, 20 February 2008 (UTC)

I don't think removing it entirely was the best course of action. To quote WP:TRIVIA:

This guideline does not suggest removing trivia sections, or moving them to the talk page. If information is otherwise suitable, it is better that it be poorly presented than not presented at all.

There was some useful information in there. I would suggest that it be restored and merged with the following section. -- Fropuff (talk) 20:27, 20 February 2008 (UTC)

Please go ahead if you think there's something useful. However, I believe that none of it is terribly suitable to the subject other than to show that Mobius strips are common symbols. I think the same effect could be had by adding a sentence saying they're common in the intro with maybe a footnote or a couple of examples. Otherwise we get an ever increasing list of junk as people visit the article and say, "hey, I saw a mobius strip the other day. I should add it to this list of trivial occurances in pop culture". But again, if you feel there's something valuable in there, add it back. Zytsef (talk) 21:14, 20 February 2008 (UTC)

I agree with Zytsef that removing it all is the cleanest way of fixing the problem. There are a couple of instances that are maybe sufficiently notable to be worth keeping (I would have wavered over the Escher one) but they're not essential to the article and having them there just attracts more cruft. If someone somewhere wrote and published a scholarly review of instances and symbolism of the Möbius strip in popular culture, I'd be happy to include a sentence or two somewhere in the article saying that it occurs frequently and citing the review, but unless we can find such a thing I'd be leery of even saying that much. —David Eppstein (talk) 21:27, 20 February 2008 (UTC)

Area

How to calculate the area of a Mobius Strip? Anwar (talk) 11:38, 9 July 2008 (UTC)

Could you clarify your question? A particular M. strip could be specified in many ways, e.g., "glue this rectangle" or by a parametrization. If (to take the first possibility) the M. strip is supposed to be obtained by gluing two edges of a rectangle, and it's assumed the gluing preserves the area, then the area of the strip is the area of the rectangle. But that answer is so trivial that I imagine it's not answering the question you had in mind. Ishboyfay (talk) 22:02, 11 July 2008 (UTC)

I think s/he was asking about the one sidedness of the Möbius strip. It's a natural question to ask, but it's quite simple to answer. Make a Möbius strip out of a strip of paper, then cut it open. The area of the Möbius strip is the area of the paper (just as Ishboyfay was saying). Dharma6662000 (talk) 20:29, 24 August 2008 (UTC)

er, surely the area of the M. strip is twice the area of the rectangle, because both top and bottom faces of the paper become part of the single face of the strip. If you make an M. strip from a rectangle of paper, the amount of paint you would need to paint the strip completely is twice the amount of paint you would have needed to paint the top side of the original rectangle completely. Mooncow (talk) 19:32, 13 November 2008 (UTC)

Mooncow this is a popular misunderstanding. A surface is considered to have no depth. It doesn't make sense to talk about painting sides of a rectangle, just as it doesn't make sense to talk about the top side and bottom side of a line! If you use your paint idea then the area of a given rectangle would be exactly twice the area of the given rectangle (!?). Take a square of 1 cm by 1 cm. It's area is 1 cm². But you would need 2 cm² of paint to paint both sides of the square. If you painted a paper model of a Möbius strip and then cut it to open it you would find that you had a rectangle of paper that was painted on both sides (not just the top side as you say in your example). This is the problem with viewing a Möbius strip (a two dimensional object) in terms of pieces of paper (a three dimensional object).  Δεκλαν Δαφισ   (talk)  21:18, 13 November 2008 (UTC)

Ah, sorry, I answered too hastily and probably only added to confusion. Apologies. However, I certainly did NOT suggest that if you painted a paper model of a Möbius strip and then cut it to open it you would find that you had a rectangle of paper painted just the top side --- my point was in fact quite the converse: if you paint a paper model of a Möbius strip and then cut it open, you will find the paper painted on BOTH SIDES. The surface area of a rectangular lamina embedded in three-dimensional space is TWICE the area of the rectangle that defines it, because the surface has top and bottom faces. Similarly, the surface area of a model of a Möbius strip is TWICE the area of the Möbius strip taken as a geometric space in its own right, even though the model has only one face. It comes down to what the original question meant: the area of a Möbius strip or the surface area of a model of a Möbius strip. I read the question as the latter, but on reflection I'm not sure why because actually it's phrased as the former.

Let me see if I can state this unambiguously: if you construct a model of a Möbius strip from a rectangular sheet of paper, the area of the Möbius strip is the area of the rectangle defining the sheet, or equivalently the area of ONE FACE of the rectangular sheet. However, the surface area of the model is twice that, i.e. the combined area of BOTH FACES of the rectangular sheet. Mooncow (talk) 13:31, 16 November 2008 (UTC)

That's more or less correct. Although the surface area would be a little over twice the area of one of the faces of the rectangle since a rectangle of card is really a cuboid and as such has six faces (two opposing faces will be glued when constructing the model and so will not contribute to surface area). The bottom line is this: If a Möbius strip is given, for 0 ≤ y ≤ 1, by identifying the points [0,y,0] with the points [1,-y,0] on the rectangle [0,1] × [0,1] × {0} in three-dimensional xyz-space then the area of the resulting Möbius strip will be the same as the area of the original rectangle, i.e. one. On another note: you didn't want to say converse in your opening paragraph. The converse of the statement "if A then B" is the statement "if B then A". The converse of the statement "If you painted a paper model of a Möbius strip and cut it to open then you would find that you had a rectangle of paper painted on just the top side" would be something along the lines of "If you had a rectangle of paper painted on just the top side and you glued it together to form a Möbius strip then it would be totally painted", and that is clearly false.  Δεκλαν Δαφισ   (talk)  11:03, 19 November 2008 (UTC)

I completely disagree. If you construct a Möbius strip from a rectangle, and trace the path of the Möbius strip until you reach the point of which you started, you go around both sides of the rectangle. The area of a Möbius strip is twice the area of the rectangle it is constructed with. This is because, in my mind, a Möbius strip is essentially 3-D because of the twist, although the rectangle when viewed in two dimensions has an area only consisting of one side, because the other side doesn't even exist. Saying the area of a Möbius strip is almost like giving area a 3-dimensional property, as if the area "soaked through" the Möbius strip to the "other side" (which is actually the same side, of course; in fact, this fact supports my opinion that this side must also be measured) I can see your point, however. Tntarrh (talk) 00:54, 5 April 2013 (UTC)

"It doesn't make sense to talk about painting sides of a rectangle, just as it doesn't make sense to talk about the top side and bottom side of a line!" is complete nonsense. Declan's mathematical point is correct -- the area of a Möbius strip made from a rectangle is the area of only one side of the rectangle -- but as anyone past kindergarten knows, it is entirely possible to paint both (2-dimensional) sides of a rectangular piece of paper.Daqu (talk) 18:27, 20 January 2010 (UTC)

To fully paint any surface at all in 3-space, one would need to paint both sides. But when mathematically figuring its area, whether a surface lies in 3-space is irrelevant. If the geometrical surface is creating by identifying the edges of a rectangle, the area of the resulting surface is the same as the area of the rectangle.Daqu (talk) 06:46, 4 July 2013 (UTC)

A Mobius strip is a 2-dimentional object wrapped around through 3-dimensional space. If you had a 2-D rectangle with a width of x and a length of y, then it's area is xy. Thus, if you don't destroy or create area folding it, then it's area remains the same. The rectangle has two sides, but we don't say it's area is 2xy even though we could theoretically paint both sides. The same applies for the Mobius strip. (Samwisedumby (talk) 00:01, 14 November 2013 (UTC))

Just to throw some randomness in to the mix. Since the area of a 2d surface is defined by its edges, you could argue that since the strip has only 1 edge, its area is simply the length of that edge. The problem is that that since the edge has no start or end, the edge is infinitely long, thus the area of the strip is infinite. Oddly enough, you may also argue that since only 1 edge is present, there is no area to the strip as any length multiplied by the 2nd (non existent or 0 length) edge would give an area of zero. With that in mind, does that make the strip a line (a one dimensional object) rather than a strip anyway?

In case I needed to point it out - No, I don't know the intimacies of the strip, but logically (to my untrained mind anyway), everything I say above seems to hold some water. MrZoolook (talk) 01:03, 22 December 2011 (UTC)

Fibre Bundles

Maybe it would be nice to included a section about the Möbius strip as a fibre bundle. It's the simplest non-trivial fibre bundle, and a nice little example. It shows some algebra being used on the Möbius strip too (i.e. the structure group of the fibre bundle). Dharma6662000 (talk) 20:29, 24 August 2008 (UTC)

If I dare say it: "Don't be shy!" :) best, Sam nead (talk) 02:54, 15 November 2008 (UTC)

It's a fine example of a fibre bundle, but is it is not the simplest possible nontrivial example. That would be exemplified by the unit circle C, in the complex plane, lying above the base space of C itself via the double-covering map p(z) = z².Daqu (talk) 06:34, 4 July 2013 (UTC)

pronunciation

The opening section suggests two "English" pronunciations, namely /ˈmeɪbiəs/ and /ˈmoʊbiəs/, plus a "German" pronunciation, IPA: [ˈmøːbiʊs]. Does anyone actually pronounce this name /ˈmeɪbiəs/?? This looks like a mistake to me. I'm thinking the two common "English" pronunciations should be /ˈmiːbiəs/ and /ˈmoʊbiəs/. Comments? Mooncow (talk) 19:25, 13 November 2008 (UTC)

100% correct. I'll change this at once. (In case anyone is wondering how such an aberration as /ˈmiːbiəs/ should come about, it's because the German letter ö can be written oe, and in English the digraph oe is most familiar from Latin/Greek words such as phoenix, where it's always pronounced /iː/.) 91.107.159.244 (talk) 23:48, 7 January 2009 (UTC)

No, not 100% correct. Virtually no one pronounces it '/ˈmiːbiəs/ in the U.S. (not counting errors). On the other hand many U.S. mathematicians try to pronounce the names of mathematicians as they are pronounced in their native lands -- so indeed it is quite common to hear at least a close approximation to IPA: [ˈmøːbiʊs].Daqu (talk) 03:53, 12 February 2010 (UTC)

Confusing run-on sentence

"If a strip with a given number of half-twists is cut in half lengthwise, it will result in a longer strip, with the same number of loops as there are half-twists in the original, if the original strip has an odd number of half-twists, or two conjoined strips, each with the same number of twists as the original, if the original strip has an even number of half-twists."

This should be split up into either two or three sentences, or separated with semicolons. But I don't want to do either lest I create even more confusion or misinformation. Anyone want to take a crack at it? 66.234.222.23 (talk) 00:53, 11 March 2009 (UTC)

Ruled surface?

The article states "It is also a ruled surface" - I can see that this applies to the "regular" Möbius strip, but surely it is possible to create a Möbius strip that does not have this property? The Möbius strip is a topological concept, but the ruled surface is a geometric concept. 81.142.107.230 (talk) 14:47, 28 October 2009 (UTC)

I suppose what is meant here is that it can be realized as a ruled surfaces. Unlike, say, the sphere. best, Sam nead (talk) 22:35, 29 October 2009 (UTC)

I'd be happier with "can be" language. Any objections to such a change? —Tamfang (talk) 05:49, 14 December 2009 (UTC)

Go for it. Sam nead (talk) 13:00, 14 December 2009 (UTC)

What is minimum length of a strip of given width that allows creation of Moebius list?

--MathFacts (talk) 10:53, 27 December 2009 (UTC)

For a twice continuously differentiable (C²) isometric embedding of the Moebius band in 3-space, it is known that this is possible for any rectangle whose ratio of longer side to shorter exceeds √3 (approaching, in the limit, three equilateral triangles folded on top of each other). It is believed that this is impossible otherwise.

If a merely C¹ isometric embedding is desired, then the Nash-Kuiper theorem shows that this can be done with any rectangle whatsoever.Daqu (talk) 04:22, 12 February 2010 (UTC)

Why √3, not √2?--MathFacts (talk) 09:15, 17 February 2010 (UTC)

Consider the union of three congruent equilateral triangles in the plane: ABC, BCD, CDE. If now the directed edge AB is identified with the directed edge DE, the result is an abstract geometrical Moebius band which could equally be obtained in the usual way by starting from a rectangle whose proportions are √3 to 1. Any rectangle with proportions P to 1, where P > √3, can be used to create a Moebius band that will embed C²-isometrically in 3-space. For such values of P close to √3, such an embedding will be close to three equilateral triangles folded over one another.Daqu (talk) 07:00, 11 March 2010 (UTC)

There's nothing non-smooth about a stereographic projection of the Sudanese Möbius band

In the article it was claimed that the stereographic projection of the Möbius band in the 3-sphere (in 4-space) -- whose parametrization is given by (sin(η) exp(i φ), cos(η) exp(iφ), for 0 ≤ η ≤ π, 0 ≤ φ ≤ 2π -- is only "quasi-smooth" with some "singularities".

This is wrong: The embedding of the Möbius band M in 4-space given by that parametrization is as smooth as can be: It is real analytic (which implies infinitely differentiable). And the image of that embedding by a stereographic projection (from a point in the 3-sphere off the Möbius band M, just as shown in the pictures below that parametrization) is also real analytic. There are no singularities in any sense of the word. And so I have adjusted the wording to reflect true statements.Daqu (talk) 18:45, 20 January 2010 (UTC)

Need to add word "distinct"?

Correct me if I'm mistaken, but if you remove a single point from the real projective plane, what's left is topologically equivalent to a Moebius strip, right?

Now look where the article currently says "The Möbius strip is the configuration space of two unordered points on a circle." I think that will give the projective plane if you allow the two points to sometimes be the same, so you need to add the word "distinct" to make it clear, to avoid that extra point from the configuration space, right?

198.144.192.45 (talk) 16:42, 29 March 2010 (UTC) Twitter.Com/CalRobert (Robert Maas)

The points of the configuration space coming from equal pairs of points in the circle do not form a single point in the configuration space, they form a circle. And a neighborhood of one of these points has the topology of a 2-manifold boundary, not a 2-manifold interior point — one way to form a combinatorial view of a neighborhood of (x,x) is as the set of points (y,z) for x − ε ≤ y ≤ z ≤ x + ε, and within this square the neighborhood looks like an isosceles triangle with (x,x) in the middle of its edge. So the configuration space has a circle of boundary points in it. I think the description in the article, in which the interior of the Möbius strip corresponds to the unordered pairs of distinct points and the boundary circle corresponds to the pairs where the points are not distinct, is correct. —David Eppstein (talk) 17:15, 29 March 2010 (UTC)

Ah, thanks for the correction+explanation. I guess I was mistaken. Never mind - Emily Littelle (Gilda Radner) 198.144.192.45 (talk) 08:17, 18 April 2010 (UTC) Twitter.Com/CalRobert (Robert Maas)

The use of the word "two" already implies that the points are distinct. There is no way that two points can be equal, for then they would be one point.Daqu (talk) 05:36, 2 July 2010 (UTC)

P.S. I can confirm that what David Eppstein says above is precisely correct:

Let C be a circle, and define an equivalence relation on CxC via: (x,y) ~ (y,x). Then the quotient space is indeed a closed Moebius band with its boundary corresponding exactly to the diagonal {(x,x) in CxC}. This is a bit tricky to see, but here's one way: CxC is a torus, i.e., a square with each pair of opposite edges identified by translation (but we have not done this yet). If we now identify each (x,y) in the square with (y,x), then every point in the lower isosceles triangle is identified with one in the upper one -- so we can throw away the lower one. This leaves a triangle with vertices at say (0,0), (0,1), (1,1), where the (0,0)-(1,1) diagonal corresponds to {(x,x)}. But the torus identifications combined with (x,y) ~(y,x) require us to now identify the directed edge A from (0,0) to (0,1) with the directed edge B from (0,1) to (1,1). This can't be done "physically" without "tearing" the triangle, so let's stretch out the vertex (0,1) to a closed interval -- call it J. Now we can easily imagine identifying A to B, and this clearly gives a closed Moebius band. The closed interval J is now part of the band's circular boundary. To keep us honest we now have to restore J to a single point, and it's easy to verify that the result of doing this is still a closed Moebius band.Daqu (talk) 06:00, 2 July 2010 (UTC)

Ouroboros and ancient times

There is nothing on the Ouroboros page that relates it to the Moebius strip, and the cited document doesn't seem to offer a lot of support for the relation: http://www.scribd.com/doc/18346561/A-Cloud-Thats-Dragonish . Should this perhaps be moved from the second paragraph to some point later in the article? A13ean (talk) 14:59, 19 August 2010 (UTC)

Circumference of Mobius Strip

The circumference constant for a mobius strip is 2 pi, 6.28, a.k.a. tau This should be noted in the article. — Preceding unsigned comment added by RobotHeroLove (talk • contribs) 03:59, 16 April 2011 (UTC)

pronunciation of Mobius.

This is really a phonetic alphabet issue. I personally DO NOT KNOW how to pronounce Mobius in any language, but can figure out the US- English version using the p.alphabet supplied by the link. When I try to do the same for the English version, I am looking through it for a back-to-front epsilon, followed by a symbol I don't know the name of (see the page). I cannot find this combination of the two symbols in the p.alphabet page, and so I am asking is there an error here or there? 92.23.152.173 (talk) 22:41, 12 May 2012 (UTC)

More on pronunciation of Möbius

Currently the lede gives the British pronunciation as UK /ˈmɜrbiəs/. I interpret this as having a rhotic "r" before the consonant "b", which I doubt because standard British English is non-rhotic. Maybe the "r" should be either upraised and minimized, or deleted? Duoduoduo (talk) 21:14, 1 June 2012 (UTC)

Geometry

What is geometry like on a Möbius strip‎? Is it Euclidean? E.g., do the angles of a triangle sum to 180 degrees? Or can we get different geometries by either assuming or rejecting Euclid's fifth postulate? Duoduoduo (talk) 21:18, 1 June 2012 (UTC)

If a geometric figure is local (doesn't wrap all the way around the strip), then it would seem that basic 2D Euclidean geometry would be the default... AnonMoos (talk) 10:38, 11 June 2012 (UTC)

Paradromic rings

This article, the Johann Benedict Listing article and the Aromaticity article refer to paradromic rings and topologies but there is no article for such concepts. I suggest that someone who knows what these are create an article for them. — Anita5192 (talk) 05:03, 11 June 2012 (UTC)

Intend to remove reference to so-called 3D Möbius strip

Under Related objects there is a reference [10] to something said to be "discovered" in 1968 called "three dimensional bodies with Möbian characteristics". More information can be found at this URL: [2].

This in fact has only the remotest connection to a Möbius strip, if any.

(These are just 3-dimensional polyominoes that are homeomorphic to a solid torus S¹×D², such that the boundary torus T contains a closed curve on it -- the edges of the square faces of the individual cubes, as one traverses the hole in S¹×D². This closed curve cuts the boundary torus T into a long annulus that goes around S¹×D² several times. Nothing about this shape is a Möbius strip.)

Such curves on the torus have been known for over 100 years, and are orientable. Nonorientabity is by far the most prominent characteristic of the Möbius band. So objects like these, which have almost no relation whatsoever to the Möbius band, should not be included in the article.

Especially since there are so many other important things that can be said about the Möbius strip itself.

Opinions welcomed.Daqu (talk) 05:15, 28 December 2012 (UTC)

Infinite types of strips?

In the beginning of this article, it states that there are two types of Möbius strips, clockwise and counterclockwise. But can't you make a completely different Möbius strip by making three half turns before connecting the two ends? Or five? Or seven? Tntarrh (talk) 00:32, 5 April 2013 (UTC)

As a topological space, up to homeomorphism, there is only one Möbius strip. What you get if you twist different (odd) numbers of times, or if you knot it before gluing, are different embeddings of that space into R³. —David Eppstein (talk) 06:53, 4 July 2013 (UTC)

Yes, but in particular the clockwise and counterclockwise versions are also homeomorphic. If you're going to distinguish them, it's because they differ in the way they are embedded in Euclidean 3-space. And then there are infinitely many other different ways of embedding in 3-space: one can use any odd number of half-twists. The same remark applies to the cylinder, which can be obtained by no twisting before gluing, or by any even number of half-twists. Perhaps the remark in the introduction should be edited. Ishboyfay (talk) 17:27, 22 September 2014 (UTC)

The Möbius band is not an orbifold

I changed the sentence:

"A less used presentation of the Möbius strip is as the orbifold quotient of a torus.[5]"

to

"A less used presentation of the Möbius strip is as the topological quotient of a torus.[5]"

This is an example of "A little knowledge can be a dangerous thing." The Möbius band is not an orbifold; it is a topological space -- which is more specifically a surface with boundary. It can also be -- like almost every surface, or perhaps every surface -- the underlying topological space of an orbifold. But the article is not about orbifolds, and the Möbius band is not an orbifold, even though it is in fact the underlying topological space of one.

The relevant fact in the sentence is that the Möbius band is the quotient space of a certain simple identification of the torus T = R^2/Z^2, namely (x,y) ~ (y,x). The waters should not be muddied by throwing in complicated concepts like orbifold, unless someone wants to introduce an entire section of the article discussing in detail how a Möbius band can be the underlying topological space of an orbifold.Daqu (talk) 06:20, 4 July 2013 (UTC)

What detail is needed? The only extra information needed to get a geometry out of the construction you describe instead of just a topology is that we should use the Euclidean metric on R². I agree, though, that "the Möbius strip" properly refers to a topological space that can have multiple unrelated geometries, and so saying that the Möbius strip "is" an orbifold is erroneous. Probably the additional complication of saying it correctly isn't worth it. —David Eppstein (talk) 06:48, 4 July 2013 (UTC)

Popular culture

I just removed the "popular culture" section, which consisted entirely of unsourced and undetailed name-dropping of popular culture artifacts that maybe mentioned Möbius strips somewhere, or that maybe have a name resembling the name for Möbius strips. I'm sure it's possible to say something encyclopedic about the uses of Möbius strips in popular culture, but that means (1) going into nontrivial detail about the significance of Möbius strips (and not just their names) as an important element of some popular culture artifacts, and (2) backing it all up with reliable secondary sources such as published scholarly articles that themselves go into detail about the use and significance of Möbius strips in literature and the arts. As it is, what we had there more closely resembles a disambiguation page than part of an article. —David Eppstein (talk) 18:10, 17 September 2014 (UTC)

A few examples of what look like usable sources for this subject: [3] (The Dwarf (Cho Se-hui novel); in Korean), [4] (Madame Bovary), [5] (Escher), [6] (Friedrich Hölderlin), [7] (architecture). If these examples are more highbrow than the ones I removed, it's not because of any preference for that kind of culture on my part but merely because that's what I could find. —David Eppstein (talk) 18:29, 17 September 2014 (UTC)

David, Perhaps the two films from Argentia and China were worth keeping - I know nothing about them, but maybe an endless loop is somehow integral to their plots. I think Escher would definitely be very good to include - at least in the "see also" section. Xenxax (talk) 18:55, 17 September 2014 (UTC)

I'm sure multiple entries from the removed section are worth re-including, but only with both text in our article describing the significance of Möbius strips to the entry and reliable sources. As the films were described in the removed section, the only part of their description that connected to the subject of Möbius strips was their titles, and neither had reliable secondary sources listed. Additionally, if you go to the actual articles here on those films, neither of them mention Möbius strips. —David Eppstein (talk) 19:14, 17 September 2014 (UTC)

I agree with you completely re. the two films, and didn't mean to say that they should be reinstated. Should have added: "if" good sources were available. As for Escher, there would plenty of sources. The endless loop appeared in so much of his work that it could even warrant its own section, but maybe a link in "See also" would be plenty. Xenxax (talk) 19:52, 17 September 2014 (UTC)

Round circle?

In the Möbius band with round boundary section there is the statement "the boundary is not a round circle". Circles are round by definition - am I missing something here or is this a tautology? Richerman (talk) 14:37, 14 January 2015 (UTC)

It's a way of stating that one is talking about a geometric circle (the round things), not a topological circle (any simple closed curve). —David Eppstein (talk) 16:23, 14 January 2015 (UTC)

Ah, thanks - I wondered if it was some strange topological distinction. Richerman (talk) 17:13, 14 January 2015 (UTC)

Please provide more informations on Paradromic ring

• Please elaborately discuss more about Splitting (Cutting- along with scissors) the strips, and give diagrams to improve this article.

• Also, is there any term for a ring that contains all full-turns (No half-turns)?

• Also, please discuss the impact of splitting (Cutting- along with scissors) such-rings (no half turns).

• btw there is biological significance of it, such as, circularly closed DNA double helix act as Mobius-type ring with all full-turns (to match 3' - 5' ligation). So in theta-type circular-DNA replication) (which is comparable with cutting-along the band with sceissor) ; after completion of replication, 2 interlocked daughter circular- DNA is found Circular_bacterial_chromosome#Termination. From interlocked condition, the 2 daughter-molecules are separated by topoisomerase II.

RIT RAJARSHI (talk) 10:13, 19 July 2016 (UTC)

Update: Article has been requested on Paradromic ring at https://en.wikipedia.org/wiki/Wikipedia:Requested_articles/Mathematics#Geometric_topology. RIT RAJARSHI (talk) 11:31, 19 July 2016 (UTC)