Talk:Möbius strip/Archive 1

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Axeeeeeel. Nice work. One day I'll put some pictures with strip and some outstanding Maple code or something like that.FireJamXRasta 3 Wednesday [2002.02.27]

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Wouldn't it be cool to have a small java applet of the moebius strip in 3D?

Yes, (and Maple or Mathematica code too)

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I took out the R from the parametrization for three reasons:

• It was not explained.
• It looked as if it was a parameter, but it was in fact a fixed number representing the radius of the Möbius band.
• Not all values of R work (you can get self intersections if R is too small.

I explained the parametrization a bit better. AxelBoldt

Nice Axel. Yes R seems to be a constant and not a parameter. We should investigate for which R Moebius strip is really defined. I would like to say something more: I didn't mean that those presumptions about connection Universe<->Moebius strip come from SF - they come from science world (physics, cosmology) I guess. We should correct that fact somehow. Uh, Axel I don't want to be your student, ha, ha. I still owe to this page another picture of a strip... XJam [2002.03.25]] 1 Monday (0)

I haven't seen any serious cosmology suggesting a Moebius strip universe, but if you find anything, make sure to put it in the article. AxelBoldt

I assume by "generalized Moebius strip" something like a finite volume universe with orientation reversing paths is meant. My understanding of cosmology (very weak admittedly) is that such universes are possible. But for some reason most physicists suppose the universe to be orientable.--C S 10:09, Sep 7, 2004 (UTC)

I have checked 'very briefly' for R. As it seems strip degenerate near 0 and probably R must be positive or non-positive real number. Very intersting - how small should R be to get self intersections. We can also split R to R₁ and R₂. Does any self intersection appear if R₁ = - R₂. (I guess not - but how can you be shure?) Another output picture is coming...
XJam [2002.03.26]] 2 Tuesday (0)

The Klein bottle isn't a 3D analogue, since it's also a surface -- it's more an extension. -- Tarquin

The Klein bottle is actually two Moebius strips glued together along their edges. --C S 10:09, Sep 7, 2004 (UTC)

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I have a question regarding

Another equation for a Möbius strip is log(r)*sin(θ/2)=z*cos(θ/2).

I assume this is in cylindrical coordinates (r,θz)? This equation describes an unbounded figure though (you can enlarge r and z beyond all bounds), so I don't see how it can describe a Moebius strip. AxelBoldt, Sunday, June 2, 2002

It is in cylindrical coordinates. It describes an unbounded Moebius strip. If you want a bounded strip, you can take the part inside a torus, or restrict r and z. --phma

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moebius in fiction: there's also A Subway Named Moebius, AJ Deutsch.

Wouldn't it be better to have "Boston subway authority" link to MBTA? Counterfit

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Am I the only person bothered by the phrase "Mobius strip is a topological object with only one surface"??? What does this mean??? I know what it is meant to mean, (i.e. that it's not orientable), and that this is the way to put it in as comfortable a language as possible, but the way it's phrased this way it seems too inaccurate (or meaningless) to be worth the benefit. Revolver 10 Nov 2003

I wouldn't say that's meaningless. It's implicit in that statement that the Moebius band under consideration is in 3-dimensional Euclidean space. There the band is one-sided. So I interpret "only one surface" to mean "one-sided". --C S 10:09, Sep 7, 2004 (UTC)

Harmless in the intro, I'd say. Anyone reading on is given a clearer idea. Generally speaking the first para of an article has some license to use looser language, and not to define everything with exactitude. There again, the surface link is probably unhelpful there.

Charles Matthews 18:00, 10 Nov 2003 (UTC)

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I removed this:

A family of 3D solids that closely relate to Möbius strips are the Sphericons. They are like a Möbius band, but without the hole in the middle. If you make a Möbius band out of a n-sided polygon sectioned strip, rotate it k amount and count the number of sides and edges created, more parallels can be found with the Sphericon.

I'm convinced at this point that the sphericon is that closely related; certainly not topologically.

Charles Matthews 13:48, 7 Apr 2004 (UTC)

I apologise, I think I was wrong. I was regarding a mobius band as not just a 2D rectangle cuved round into a circle and twisted, but also as 3D prisms curved round and twisted. These result in shapes with one side and one edge just as a mobius band does, and I believed they all fell under the same name. At simplest you can have a triangular prism, but a prism with many sides approaches a torus. Similarily, the simplest sphericon is based on a sixty degree apexed cone split and twisted, and a more complex sphericon approaches a sphere. What page would you reference it to?

You could add it to List of polygons, polyhedra and polytopes, perhaps as a 'see also'; also to the list of geometry topics under the 3D shapes.

Charles Matthews 20:01, 9 Apr 2004 (UTC)

Thanks, it's done.

Proberts2003 20:50, 9 Apr 2004 (UTC)

Möbius strip with a circular boundary

Don't know how to edit this -- the second to last paragraph

A cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. The term 'cross-cap', however, often implies that the surface has been deformed so that its boundary is an ordinary circle. This cannot be done in three dimensions without the surface intersecting itself.

is wrong. This error is repeated in the cross-cap entry as well (word for word!). It is possible to embed a Mobius Band in $\mathbf {R} ^{3}$ with boundary a perfect circle. Here is the idea: let C be the unit circle in the xy plane in $\mathbf {R} ^{3}$. Now connect opposite points on C, i.e., points at angles $\theta$ and $\theta +\pi$, by an interval which is an arc of a circle. For $\theta$ between $0$ and $\pi /2$ the arc lies above the xy plane, and for other $\theta$ the arc lies below (there are two places where the arc lies in the xy plane).

I don't know what this embedding of the Mobius strip is called. sam Mon Aug 16 17:49:53 CDT 2004

Thanks for pointing this out. I have adapted what you wrote and added it to the page. Charles Matthews 10:20, 7 Sep 2004 (UTC)

Unfortunately, I can't make sense of the given description. I think an improved version is needed. The way I personally visualized this is hard to verbalize, but I found some nice descriptions in George Francis' A Topological Picturebook. They rely extensively on some key pictures however. Because of copyright I can't just scan them in, but maybe I could draw my own versions. At least, for now, we can give Francis' book as a reference (for a doubting Thomas).--C S 11:45, Sep 7, 2004 (UTC)

I'm having some problems with this description as well. It is a little ambigouous. A picture would be worth a thousand words here. If someone can give me a more concrete description, I'd like to plot this in Mathematica. -- Fropuff 17:10, 2004 Oct 20 (UTC)

Perhaps these photos help. I made an actual physical model, with a wire coat-hanger and a tee-shirt. In the first picture, quadrants 1, 3, and 4 are just a single layer. Quadrant 2 inside the circle has three layers; the middle layer connects to the boundary circle, while the top and bottom layers make the corner-shaped pouch in the upper left. The rope demonstrates that there's really a hole. Travelling from the end that goes off the edge (towards the yellow taped end), the rope passes up through the circle twice: it wraps around the circle inside the pouch.

The second picture is just the first one flipped over.

Fropuff, does that help at all? I've been fortunate enough not to have to learn Mathematica, so I can't make a plot of it. dbenbenn | talk 11:49, 29 Jan 2005 (UTC)

Thanks dbenbenn, I appreciate the effort. Though I must admit that I've stared at this picture for quite awhile and I'm afraid I still can't picture it. How can the rope pass twice through the circle? Where exactly is the circle in this picture? -- Fropuff 07:16, 2005 Feb 1 (UTC)

Tonight, I'll cut a "window" in the model and upload a new photograph. Perhaps that'll help. I'll also try highlighting the boundary with the Gimp. dbenbenn | talk 23:43, 3 Feb 2005 (UTC)

Okay, I procrastinated ridiculously, but it's here now. Does that help? dbenbenn | talk 07:24, 19 Feb 2005 (UTC)

— — —

Uh, no. I just read this for the first time, and I'm also baffled by this proposed method of embedding a Mobius Band in $\mathbf {R} ^{3}$ with a perfect circle boundary, without self-intersections. Alas the "shirt & coat hanger model" photos aren't helping me, but perhaps adding some more detail in the paragraph?

• Let C be the unit circle in the xy plane in $\mathbf {R} ^{3}$.
• Now connect opposite points on C, i.e., points at angles $\theta$ and $\theta +\pi$....

Okay, so far so good. I agree if we can connect these opposing points, we'll get an embedding of a Mobius Band. But I don't believe this surface won't intersect itself.

• .. by an interval which is an arc of a circle.

Is the author suggesting connecting the opposing points using arcs which extend into the z direction, so the middle of each arc will lie on the z axis?

• For $\theta$ between $0$ and $\pi /2$ the arc lies above the xy plane, and for other $\theta$ the arc lies below (there are two places where the arc lies in the xy plane).

If I understand this right, the author is suggesting that:

1. For $\theta$ equals 0, the arc lies in the xy plane
2. Then as $\theta$ increases away from 0, the arc starts going higher and higher above the xy plane
3. Until some critical $\theta$ is reached (maybe $\pi /4$), when the arc goes highest above the xy plane.
4. Then as $\theta$ continues increasing, approaching $\pi /2$, the arc starts going lower and lower, but is still above the xy plane.
5. When $\theta$ reaches $\pi /2$, the arc lies in the xy plane again.
6. Then as $\theta$ increases away from $\pi /2$, the arc starts going lower and lower below the xy plane.
7. Until some critical $\theta$ is reached (maybe $3\pi /4$), when the arc goes lowest below the xy plane.
8. Then as $\theta$ continues increasing, approaching $\pi$, the arc starts going higher and higher, but is still below the xy plane.
9. And when $\theta$ reaches $\pi$, the arc joins with the arc created when $\theta$ equalled 0, completing the surface.

If this is what is being suggested, and the middle of each arc lies on the z-axis,

I never said that the center of the arc lies on the z-axis. In fact the arc "topples" over. At angle 0 the arc in in the xy plane and is inside the circle C. At angle 90 degrees the arc is in the xy plane and is outside the circle C. I hope that helps! Sam nead 20:07, 7 September 2005 (UTC)

then it looks like there will have to be self-intersection on the z-axis. During "step 2" in my list above, each arc uses a higher point on the z-axis until an arc with maximum height is reached, so when the arcs try to descend in "step 4", they must use those same z-axis points coming down. Each z-axis point that is used once going up must be used again coming down. That sounds like self-intersection to me.

In fact, even if one doesn't try to guess what is meant to be happening off the xy plane, the description itself says: "there are two places where the arc lies in the xy plane". As I read it, these two places are ($\theta =0$), when we're trying to connect the east and west points of the circle, and ($\theta =\pi /2$), when we're trying to connect the north and south points of the circle -- yet both arcs must lie in the xy plane. Well, I can't draw a curve connecting the east and west points of a circle, and another curve connecting the north and south points, without those two curves intersecting.

I would love to see better what the author did with his shirt; otherwise, I'm proposing deleting this fragment from the Mobius band page. I think it's fairly well established that embedding a Mobius band in $\mathbf {R} ^{3}$ such that its boundary is a circle must have self-intersection. Anonymous Reader -- UTC 12:24 Sep 6, 2005

— — —

Hmmm.. Thinking about it some more, I think I see how it might be done, if you don't mind that the resulting surface is infinitely large.

In my list of "steps", above, the arc is rising higher and higher in "step 2". But maybe it doesn't reach any maximum, so there is no "step 3" and no subsequent descent in "step 4"; rather, the arc just keeps getting higher and higher, and wider and wider, so that when $\theta$ reaches $\pi /2$, the highest point of the arc reaches infinity.

Technically at this point, the arc is in the xy plane again, because it's so huge that the only thing you can see at any scale, is a line going north from the north-point of the circle, and another line going south from the south point of the circle, and we're supposed to believe that these two lines meet at the point at infinity. So that's how the north and south points of the circle can "connect" in the xy plane, without intersecting the perfectly normal horizontal-line-segment that connects the east and west points.

Then as $\theta$ increases past $\pi /2$, the extreme point of the arc jumps to negative infinity and continues rising. So instead of "steps 6,7,8", the extreme point of the arc just rises smoothly from negative infinity back to zero, as $\theta$ increases from $\pi /2$ to $\pi$.

In this version (as above) the arc at 90 degrees is in the xy plane, is outside of C, but now it is a pair of rays. This is totally fine -- in fact, your version gives a nice Mobius strip in the three-sphere, the one-point compactification of R^3. To turn your version into mine, just apply a Mobius transformation (like a linear map, but more general) which fixes the circle C. Sam nead 20:07, 7 September 2005 (UTC)

Okay. If this infinite-sized-curiousity is what the author intended, I agree that an embedding with no self-intersection is technically possible, I no longer think it should be deleted. Though I still don't understand is how this shirt-photo was supposed to be a "physical model" of a surface with infinite extent!  :)

The paragraph in the article doesn't really explain the resulting surface at all -- a plot would be very helpful. Anonymous Reader -- UTC 13:37 Sep 6, 2005

I did the best I could! Sam nead 20:07, 7 September 2005 (UTC)

— — —

Oh wow, you're totally right, Sam Nead! I can see it now, it's finite, smooth, and .. really pretty. Now that I understand what the thing is supposed to look like, I made up a parameterization and created the following plot of the object: Anonymous Reader -- 25 September, 2005

What beautiful pictures!! I have added them to the article.Sam nead 00:58, 28 September 2005 (UTC)

Nice pictures, thanks. I would be very curious to see the parametrization you used to make these. -- Fropuff 02:20, 28 September 2005 (UTC)

Well, I just basically plodded through the verbal description above, making up functions as needed to make the thing look nice.

Let the first parameter be $\theta$ running from -π/2 to π/2.

For any given $\theta$ we want to connect antipodal points (cos$\theta$,sin$\theta$,0) and (-cos$\theta$,-sin$\theta$,0) with the arc of a circle that lies in a plane which "tips over" as $\theta$ increases away from zero. So I decided that the arc would lie in the (u,v) plane, with $\rho$ indicating the amount of tipping.

${\vec {\mathbf {u} }}=\left\{\cos \theta ,\sin \theta ,0\right\}$

${\vec {\mathbf {v} }}=\left\{\sin \theta \sin \rho ,-\cos \theta \sin \rho ,\cos \rho \cdot \operatorname {sign} \theta \right\}$

And of course $\rho$ had to be 0 when $\theta$ was 0, and π/2 when $\theta$ was ±π/2, so it looked best if I just set:

$\rho =\theta \,$

Now within the (u,v) plane, we want a circular arc to connect (-1,0) with (1,0), with the shape of the arc changing as a function of $\theta$. The shape of the arc can be expressed by $\tau$, defined as half the angle subtended by the arc. So if $\tau$ is π/2, the arc is a semicircle, connecting (-1,0) with (1,0) through (0,1). For lower $\tau$ the arc is shallower and in the limit as $\tau$ approaches 0 the arc becomes a straight line through the origin. For higher $\tau$ the arc is more than half a circle, and u gets to take on values outside the {-1...1} range.

Let the second parameter be t running from -1 to +1 indicating our position along a given arc.

Then the arc in the (u,v) plane is given by:

$u(t)={\frac {\sin(\tau t)}{\sin \tau }}\qquad v(t)={\frac {\cos(\tau t)-cos\tau }{\sin \tau }}$

so the only thing remaining is to select how $\tau$ should vary as a function of $\theta$. When $\theta$ is 0, we select $\tau =0$ to get a horizontal line connecting the east-west points of the circle. When $\theta$ is ±π/2, (and the (u,v) plane is tipped 90 degrees, so (u,v) = (±y,x)), we need $\tau$ to exceed π/2 so that the arc will bulge outward and connect the north-south points of the circle without ever going inside the circle. I arbitrarily selected $\tau =3\pi /4$ when $\theta$ was ±π/2, but a linear relationship between $\tau$ and $\theta$ didn't look so good. So fiddled with it and ultimately selected:

$\tau ={\frac {3\pi }{8}}\left(1-\cos \left({\sqrt {2\pi |\theta |}}\right)\right)\,$

this being an arbitrary function that starts $\tau$ off at 0 for $\theta =0$ and gives it a sort of smooth rapid rise up to its maximum of 3π/4. So I just told my plotting program to compute a grid of values for $\theta$ and t, found the (u,v) basis vectors and the u(t),v(t) values to map each point into (x,y,z) coordinates, and plotted the resulting surface.

Alas the resulting surface looked squashed, so I scaled it by 150% along the z axis.  :)

--Anonymous Reader 8:35, 29 September 2005 (UTC)

A nice description

After sifting through the above discussion and a lot of tinkering around, I have stumbled upon a very nice desription of the Möbius strip with a circular boundary.

The first step is to embed the Möbius strip in the 3-sphere (thought of as a subset of C²) via the maps

$z_{1}=\sin \eta \,e^{i\phi }$

$z_{2}=\cos \eta \,e^{i\phi /2}$

Here η runs from 0 to π and φ runs from 0 to 2π. The boundary of the strip is given by $|z_{2}|=1$, which is clearly a circle on the 3-sphere.

The second step is to map S³ to R³ via a stereographic projection. Stereographic projections map circles to circles and will preserve the circular boundary of the strip. In order to get a finite surface one must project from a point not on embedded Möbius strip in S³. After some trial and error I found that projecting from the point (½,−½,−½,−½) looks quite nice. The resulting surface looks similiar to the one shown above. (I can upload Mathematica graphics if anyone is interested).

Absolutely, I'd love to see it! Your description is a lot cleaner. (I guess it has fewer knobs for sculpting the surface, but that's a good thing. More natural, less hacky. I can't visualize 4D projection in my head (yet!), but the Möbius-Strip-In-4D with which you're starting has an awesome symmetry to it.) -- Anonymous Reader. 8:35, 29 September 2005 (UTC)

Should we replace the description in the article with this one? I think this description is a little cleaner and less ambiguous.

-- Fropuff 05:57, 29 September 2005 (UTC)

Okay, here is a plot of the surface I described above. It was just done with Mathematica so it doesn't look quite as nice as the other one.

-- Fropuff 14:59, 29 September 2005 (UTC)

I implemented your 4D-stereographic-projection idea so I could look at it myself, and find a view of it that looked as good as my first hacky one. Using your equations for the strip in C², I set the projection-point to be that point on the 3-sphere closest to { Re(z₁), Im(z₁), Re(z₂), Im(z₂) } == { X, Y, 1, 0 }, and proceeded to modify (X,Y) to see how this changed the surface.

With (X,Y) == (0, .33), the circle forms the opening of a deep well, that twists like a snail-shell into a slender tube. Striking, but asymmetric and kind of lumpy..

With (X,Y) == (-.33, 0), the film is flat between the circle, and nearly a disc, but with the top and bottom "sides" connected by a tube that is centered and much neater!

I'm really impressed at the variety of different-looking Mobius-Strips-With-Circular-Boundaries that your technique makes possible! Great job! In both cases the "slender tube" bloats up as the magnitudes of X or Y increase, until the "tube" is fatter than the disc. The symmetric one, bloated up at (X,Y) == (-1.5, 0) and with the colors reversed, looks like the closest match to my first manually-made version from above.

I agree that this stereographic-projection-of-a-4D-mobius-strip explanation is a much more precise and powerful description than the arc-that-flops-over paragraph, and it should replace the paragraph in the article. -- Anonymous Reader -- 18 October 2005 (UTC)

The 4D Möbius strip on which this is based is generally thought to have been first discovered by Blaine Lawson in his Stanford Ph.D. thesis (1960's). It has not only tremendous symmetry (I think its symmetry group is O(2)), but it also happens to be *minimal* in S^3. With the help of Doug Lerner, I made an animation of this strip -- stereographically projected into R^3 in the most symmetrical way so as to create a non-compact surface in R^3 (that's technically the Möbius strip minus one point) in 1983; it appears in Siggraph Video Review #17.Daqu 12:50, 17 January 2006 (UTC)

P.S. For historical reasons we called the 4D strip the "Sudanese Möbius band". It can be geometrically described as follows: There is a way to think of the 3-sphere S^3 via an "open book decomposition". That is, choose some great circle, calling it C. There is a circle's worth of great hemispheres all having C as their common boundary, and whose union is all of S^3. Denote these hemispheres by H_t for 0 <= t < 2pi according to the angle H_t makes with H_0. Each hemisphere H_t has a "pole" -- the unique point farthest from its boundary -- which we call P_t. We *also* parametrize C itself by c(t), 0 <= t <= 2pi, where c(t) traverses C at unit speed.

Finally, on each H_t we draw the unique great semicircle S_t passing through both P_t and c(2t). The union of all the S_t's forms the Sudanese Möbius band. (Thus S_(t+pi) = S_t for 0 <= t < pi.)

Note: Sam nead's description sounds like exactly what the Sudanese Möbius band looks like in my Siggraph film of the same name . . . after it is stereographically projected S^3 - {pt} --> R^3 (with projection point being one of the poles P_t above).Daqu 13:27, 17 January 2006 (UTC)

Handedness of Möbius strip

Is the photo at the top of Möbius strip a right or left handed strip? I can't tell from reading the definition in the article. Perhaps someone who knows the convention could improve the description. dbenbenn | talk 23:42, 14 Mar 2005 (UTC)

In topology, a Möbius strip (like any surface or topological space) is defined independent of its surroundings. It's thought of as a thing-in-itself. More accurately, it's thought of as not only a rectangle-with-one-pair-of-opposite-sides-identified-by a-flip, but actually any topological space that's topologically equivalent (aka homeomorphic) to that surface.

Also, a Möbius strip *might* be identified with a subset of another topological space, and of course it's usually depicted in 3-space. There's no standard terminology for the "left-" and "right-" handed versions, but in fact there are many more than just these two! To start with, any odd number of twists, clockwise or counterclockwise, in a strip -- before glueing its ends -- creates a topological Möbius strip. (The logo of the wool industry, and the logo for recycling, are each 3-twist Möbius strips). And to make matters worse, its core circle could also be knotted!Daqu 20:09, 5 May 2006 (UTC)

Pronunciation

The IPA for the article says (pronounced /ˈmøbiʊs/), but isn't the ø a long vowel? 惑乱 分からん 18:34, 27 February 2006 (UTC)

I'm not well-versed in the IPA, but the German pronunciation of Möbius (deemed correct by international mathematicians for pronouncing "Möbius strip") has the first vowel pronounced as a hybrid of the (English) u in "but" (ŭ) and the oo in "look" (denoted here by ʊ). More accurately, it's a diphthong composed of the ŭ followed by the ʊ. That is, just as the long i (ī) is the diphthong ah-ee when spoken quickly, the ö is at least very close to ŭ-ʊ (spoken very quickly, with the ŭ extremely short).

The common pronunciation in the U.S. by the *non*-cognoscenti is with long o (ō), but I feel this error should not be encouraged.Daqu 20:42, 5 May 2006 (UTC)

Should we replce Möbius with Mobius?

It would certainly make for easier editing. Klosterdev 20:29, 6 March 2006 (UTC)

If you can't type an "ö" easily, you could always just write "Mobius" and let someone with an "ö" key fix it. The regexp replace user script makes fixing such things particularly easy. —Ilmari Karonen (talk) 21:07, 6 March 2006 (UTC)

There is an entirely standard way to represent a German o-umlaut when you don't have an umlaut symbol: you just put an e after the o. So among mathematicians, at least, an entirely standard alternative to "Möbius" is "Moebius". (But definitely *not* "Mobius".)Daqu 19:27, 7 August 2006 (UTC)

Humanist symbol

The strip is the symbol of the Humanist International and also of many humanist parties (ex: Portuguese Humanist Party [1] ). I think that should be refered in the article. Mário 00:05, 23 March 2006 (UTC)

Good Article, but...

Hi to the contributors - great work. I'd love to nominate this as a Good Article but I fear it would be shot down by reviewers for a lack of references. Anyone think they could add a few? Inline citations would be great too - no harm in asking eh? Cheers SeanMack 16:37, 11 April 2006 (UTC)

Searching the net I came across chemistry papers talking about aromatic compounds with Möbius strip shapes, for example here,

http://www.rsc.org/ej/CC/2000/b002462g.PDF, any chemists know enough about this to do it justice? SeanMack 18:10, 30 May 2006 (UTC)

who can help me to add

Ruban de Möbius animé

diagram missing

in the topology subsection the first paragraph talks about a diagram on the left which is nowhere to be found (at least on mobile version) Alexthunder27 (talk) 16:07, 3 March 2020 (UTC)

I moved the image. It should be visible in that section now. — Anita5192 (talk) 16:53, 3 March 2020 (UTC)

We should not be talking about positions of images at all. They are usually different for mobile viewers than for viewers on large screens. See MOS:SEEIMAGE. —David Eppstein (talk) 18:32, 3 March 2020 (UTC)

GA Review

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

Reviewer: Mover of molehills (talk · contribs) 17:10, 28 March 2022 (UTC)

I'll start working on this review soon! It may take me slightly longer than a week, but I'm excited to look at this one.

I'll order the review by the Good Article Criteria:

Well-written

Following is a list of changes I would recommend making for overall clarity. Please feel free to debate with me if you don't think any of these changes would improve the article - I don't have that much expertise with topology, so I might be wrong in some of these places.

Lede

• Second sentence: for more specifity, could you change "but its appearance in Roman mosaics long predates their work" to "but it has appeared in Roman mosaics as far back as INSERT YEAR"? Mover of molehills (talk) 17:30, 28 March 2022 (UTC)
  • Ok, done. —David Eppstein (talk) 20:52, 28 March 2022 (UTC)

    Thank you! Small nitpick for clarity: could you say "it had already appeared... since" instead of "it already appeared... from"? Mover of molehills (talk) 21:15, 28 March 2022 (UTC)

    • Ok. —David Eppstein (talk) 22:41, 28 March 2022 (UTC)

      Strangely, I still don't see the changes - have you made them yet? Mover of molehills (talk) 22:47, 28 March 2022 (UTC)

      Special:Diff/1079845424. I changed the tense, but I did not change "from" to "since". The mosaics were from the 3rd century. It would be incorrect grammar to say that the mosaics were since the 3rd century. —David Eppstein (talk) 01:38, 29 March 2022 (UTC)

      I see, I think I phrased that poorly. The revision I was suggesting was something along the lines of "but it has been featured in Roman mosaics since the third century AD." I understand that this is a very minor change, but I feel like it reads better on my end. Does this work for you? Mover of molehills (talk) 12:40, 29 March 2022 (UTC)

      But the Roman mosaics that the references discuss were ONLY from the third century. This does not appear to have been a continuing practice from then until now, which your wording would make it seem. —David Eppstein (talk) 16:16, 29 March 2022 (UTC)

• How is the strip's "embedding in three-dimensional space" any different from the three-dimensional strip itself? Specifically, could you cut this clause to tighten the sentence without sacrificing any meaning?
  • No, it would make it incorrect. It's explained in the "properties" section. The strip itself is a two-dimensional topological surface, with multiple embeddings into three-dimensional spaces (for instance, twisting it three times or knotting it makes it different from one time as an embedding, but the same as a topological surface). In three-dimensional Euclidean space it is always one-sided, no matter how it is embedded, but it has other embeddings in other three-dimensional spaces that are two-sided. This sentence of the lead is intended as a summary of the more detailed explanation later. —David Eppstein (talk) 20:52, 28 March 2022 (UTC)

  Got it, thank you for letting me know. Mover of molehills (talk) 21:15, 28 March 2022 (UTC)

• It would be useful to include a brief one-sentence definition of what "non-orientable" means ("it is the simplest non-orientable surface, meaning that...") Mover of molehills (talk) 17:30, 28 March 2022 (UTC)
  • Ok, I added a little more here. —David Eppstein (talk) 20:52, 28 March 2022 (UTC)

    What you have is helpful. However, do you think you could replace/add to this with something a little bit more conceptual (like "meaning that ideas of "clockwise" and "counterclockwise" are not well-defined on it")? What I'm really looking to do is explain what this means to the reader. Mover of molehills (talk) 21:23, 28 March 2022 (UTC)

    • Ok, changed to a lead gloss involving inability to define clockwise, and added more detail in this same interpretation in the later "Properties" section so that the lead would continue to summarize the rest of the article. —David Eppstein (talk) 04:51, 29 March 2022 (UTC)

      This is exactly what I was talking about, thank you. The lede already looks much better. Mover of molehills (talk) 12:38, 29 March 2022 (UTC)

• The second paragraph goes into a little bit too much detail into all of the specific ways to create a Möbius strip. I would simplify this part to "There are many geometrical ways to generate a Möbius strip, and a physical model can be created by twisting a strip of paper and gluing together the opposite ends." (This could replace everything up to "an embedded Möbius strip can be stretched"). Mover of molehills (talk) 17:30, 28 March 2022 (UTC)
  • Again, see MOS:LEAD, one of the core requirements of WP:GACR. The lead is supposed to provide a brief summary of the entire rest of the article. The specific ways of constructing the Möbius strip occupy a section of the article with six subsections, averaging roughly three paragraphs and many illustrations each. It is a large fraction of the total article. It seems reasonable to me that summarizing this material properly is worth a whole paragraph, and that each of the subsections deserves at least a sentence (although in fact there are fewer sentences in this summary paragraph than subsections). —David Eppstein (talk) 20:52, 28 March 2022 (UTC)

    Okay, I think that we can leave the detail as is for now. Mover of molehills (talk) 21:18, 28 March 2022 (UTC)

• The section about all the different variations on the Mobius strip could use a sentence of introduction. Specifically, I would preface the sentence "An embedded Möbius strip..." with the sentence "Mathematicians have explored many different variations on the basic Möbius strip, all of which have slightly different properties." Mover of molehills (talk) 18:53, 28 March 2022 (UTC)
  • What, you don't think the sentence "Many different ways of forming geometric structures with the topology of a Möbius strip are known. " already serves this purpose? —David Eppstein (talk) 20:52, 28 March 2022 (UTC)

• It does serve the same purpose, I was just trying to reword it so it could be clearer. Do you agree with the rephrasing? Mover of molehills (talk) 21:18, 28 March 2022 (UTC)
  • To me your rewording is vaguer not clearer. It replaces a sentence that contrasts geometry with topology (the point of the section) with inaccurate "variations on the basic": they are not really variations (they are all Möbius strips) and I don't understand which one is called basic and which ones variations. Also "all of which have slightly different properties" doesn't convey any useful information to me: if they are different from each other, of course their properties are different, and "slightly" appears meaningless here. But more to the point, it suggests that the "variations" came first, and then people investigated how they might differ from each other in the properties they had. Really, what came first are the additional properties that mathematicians hoped nice surfaces might have (ruled, polyhedral, developable, curvature-free, etc), and then they found constructions of Möbius strips with those properties. I rewrote the whole paragraph in an attempt to make this point clearer. —David Eppstein (talk) 01:45, 29 March 2022 (UTC)

• For brevity, I would replace "the same circular boundary shape" with "this same shape." Mover of molehills (talk) 18:53, 28 March 2022 (UTC)
  • In your proposed replacement, does "this same shape" refer to the circular boundary or to the shape of the entire Sudanese Möbius strip? It is unclear, in a way that we should try to avoid. —David Eppstein (talk) 20:52, 28 March 2022 (UTC)

    That's fair. Maybe you could just say "circular boundary" instead of "circular boundary shape"? Mover of molehills (talk) 21:12, 28 March 2022 (UTC)

    Ok. I took the opportunity to rearrange the prose to be a little tighter here. —David Eppstein (talk) 22:43, 28 March 2022 (UTC)

    That's great, thank you. My last comment would be that it's confusing to mention a cross-cap without defining what it is, so I would say "another version, the self-intersecting cross-cap..." Mover of molehills (talk) 22:46, 28 March 2022 (UTC)

• I almost think that it would be worth cutting out the whole chunk of words "has realizations... and its points" (second-to-last sentence of second paragraph). The sentence seems to flow better without it, and I feel once again like this is a level of detail not appropriate for the lede (it would make a lot of sense to move the info into a "variations" section or subsection later if you want to). Mover of molehills (talk) 18:53, 28 March 2022 (UTC)
  • So you want to completely omit the subsection "Surfaces of constant curvature" from being summarized in any way in the lead? —David Eppstein (talk) 20:52, 28 March 2022 (UTC)

Thank you for moving through this first round of edits. I apologize if my cuts sometimes feel too draconian, and I agree with a lot of the counterarguments you have made. My main goal here is to get a lede that can be understood fairly well by the average person, because I think that Möbius strips are a topic which is potentially of interest to people with non-mathematical backgrounds. This doesn't mean that you shouldn't go into the more technical details later, I'm just trying to make the opening of the article easier to read while still summarizing the article's contents. Mover of molehills (talk) 21:25, 28 March 2022 (UTC)

I definitely agree that this part of the article should, as much as possible, be readable by non-mathematicians. There is some unavoidable technicality later, but this is a widely-known enough topic that we cannot reasonably expect all readers to be mathematicians. —David Eppstein (talk) 22:44, 28 March 2022 (UTC)

More feedback:

• I like the sentence about applications of the Möbius strip, but the part about the world map is a little bit confusing. Could you phrase that as "so that points on opposite sides of the earth appear on opposite sides of the strip"? Mover of molehills (talk) 22:50, 28 March 2022 (UTC)
  • But the strip only has one side. What are its opposite sides? Instead I tried rewriting it as "opposite each other", without talking about sides. —David Eppstein (talk) 05:37, 29 March 2022 (UTC)
• "Möbius strips appear in molecules..." What does this mean? Are there actually molecules with a Möbius-strip geometry? Mover of molehills (talk) 22:50, 28 March 2022 (UTC)
  • Yes. Most obviously in graphene, which naturally has a two-dimensional structure that can be formed into twisted ribbons. I tried looking on commons for molecular diagrams that we could use to illustrate this point but didn't find any. —David Eppstein (talk) 05:42, 29 March 2022 (UTC)
• I'm kind of confused by the final sentence of the lede, where it describes stories with "events that repeat with a twist." This feels like conflating a topological twist with the word "twist" as a plot device. Do you have a reference for this idea, or could you rephrase it? Mover of molehills (talk) 22:53, 28 March 2022 (UTC)
  • This is supposed to summarize the paragraph about fictional Möbius strips, which cites a reference for exactly this conflation of ideas. It appears to be a fairly common metaphor in literary analysis. Our article for of the pieces of fiction linked in that paragraph, Lost in the Funhouse, suggests that at least in that case it was deliberately written with the Möbius strip metaphor in mind, and that this idea was incorporated into its typography. —David Eppstein (talk) 02:03, 29 March 2022 (UTC)

History

• For simplicity, would it be possible to replace "lemniscate-shaped" with "figure eight"? Mover of molehills (talk) 14:48, 29 March 2022 (UTC)
  • Ok. —David Eppstein (talk) 18:42, 29 March 2022 (UTC)
• Similar edit: could you change "annuli" to "simple rings"? I know that "annuli" carries a slightly more precise meaning, but I think that the History section is one that should be accessible to the general reader, and the repeated use of longer words like this makes it a much denser read. Mover of molehills (talk) 14:48, 29 March 2022 (UTC)
  • "Simple ring" may be less confusing for non-mathematicians, but it is much much more confusing for mathematicians. The reason is that "simple ring" is the kind of phrasing used for technical jargon in mathematics, compound terms with a specific meaning that you have to know if you want to understand sentences that use them. So a mathematician is either going to want to know "what exactly is a ring? what property of a ring makes it simple? where is this defined?" or is going to already know a meaning of "simple ring" and plug that meaning in. But the only standard meaning of the term "simple ring" is something totally unrelated from algebra. Instead, I used "untwisted ring". —David Eppstein (talk) 18:42, 29 March 2022 (UTC)

    I think "untwisted ring" is a good compromise, thanks. Mover of molehills (talk) 19:21, 29 March 2022 (UTC)

• There are a few parts throughout this section where it feels like a lot of the sentences are run-ons. I would recommend breaking them up differently, while preserving the same content and wording, so that they read better:

• Suggested rephrasing: "In particular, Roman mosaics have contained images of coiled ribbons since the third century AD. These ribbons can be either Möbius strips or simple rings, depending on whether the number of coils is odd or even, and the existence of odd ones may be purely coincidental."
• Suggested rephrasing: "Independently of the mathematical tradition, machinists have long known that mechanical belts wear half as quickly when they form Möbius strips. By using both sides (or rather the single continuous side) of the strip at once, the Möbius shape also evens out any curvature that may develop in the belt."
  • I copyedited the paragraph with some attention to avoiding overlong sentences. I didn't use your specific wording, though. —David Eppstein (talk) 18:42, 29 March 2022 (UTC)

• Grammar nitpick: Change "Much older, an image of a chain pump... depicts" to "Much earlier, an image of a chain pump... depicted." Mover of molehills (talk) 14:48, 29 March 2022 (UTC)
  • Ok. —David Eppstein (talk) 18:42, 29 March 2022 (UTC)
• Change "without clear dates for the origin of this task" to "it is unknown when this practice began." Mover of molehills (talk) 14:48, 29 March 2022 (UTC)
  • That's a stronger assertion and would need a better source. The source for this part of the article does not identify a date. It does not, however, state that the date was unknown. The date is unknown to me, because it is not stated in the source. But I am unsure whether the date is unknown to historians of this practice, or merely unstated in the source I used. —David Eppstein (talk) 18:42, 29 March 2022 (UTC)
• Other than that, this section is overall very clear! Mover of molehills (talk) 14:48, 29 March 2022 (UTC)

Verdict

• Not promoted. I'm sorry to jump to this section so early in the review, but it seems so far that improvements on this article have been pretty difficult. I still think that it has the potential to become a GA in the future, but it will need some work, and your comments make it seem like it will take a good deal of time to make the wording of this article more clear without sacrificing mathematical precision. In the meantime, I encourage you to focus on making the wording simpler in sections that are of interest to a general reader, and also to add a bit more information (and maybe a new section) on the properties of geometry on the surface of the strip (as opposed to hyperbolic and Euclidean geometry). Thank you for submitting to GAN! Mover of molehills (talk) 15:31, 30 March 2022 (UTC)

@Mover of molehills: WTF WTF was that? It was progressing through minor copyedits and then suddenly complete shutdown for no reason? Can you please point to ANYTHING in the article, anything at all, that is actually far from any of the GACR criteria, before I escalate this serious miscarriage of a review to higher levels? If it's merely that you feel incapable of actually completing the review, because it's going to be too much work (it can be a lot of work), then failing the article is the wrong way to back out. It is punishing me and my efforts for a choice to review that was entirely your responsibility. —David Eppstein (talk) 16:05, 30 March 2022 (UTC)

@David Eppstein: I'm sorry if this feels abrupt to you, but I didn't feel like you were cooperating with some very reasonable edits to improve the wording of the piece and make it more accessible to the average reader. Right now, this piece does not meet the GA criteria because many parts of it are not very well-written, too dense for an average reader to understand, and because the article's scope is missing a good deal of information and additional sections about the properties of geometry on a Mobius strip's surface. After progressing through a few rounds of edits, I just decided that I think there is too much work left to go on the article for it to become a GA right now.

I certainly can't stop you from reporting me to "higher levels," but I think that this is a grossly unfair response to the situation. Given the kind of language that you have used in response to my constructive criticism of the article, I suspect that you may not like the outcome of an admin decision. Mover of molehills (talk) 16:09, 30 March 2022 (UTC)

Ok, ecalating. This failure is an atrocity and I cannot accept your behavior in it. You cannot fail an article on a technical topic merely because it has technical parts. —David Eppstein (talk) 16:11, 30 March 2022 (UTC)

I hope this GA review will go ahead, and succeed. I have a couple of suggestions:

1. From the first paragraph of the lead: "Its embedding in three-dimensional Euclidean space has only one side". Seems to me, it intrinsically has only one side, regardless of embedding. And this correction would avoid the term "embedding" with its implications not known by many readers.
   • As I already stated somewhere above, this intuition is incorrect. This is a very common misconception, so it is important to counter it. This sentence of the lead is intended as a summary of the first paragraph of the "properties" section, which goes into some detail about how embeddings of the Möbius strip in other 3d spaces than the familiar Euclidean space may actually be two-sided, in exactly the same way that a sheet of paper in Euclidean space is two-sided. (According to one of the sources for this material, JSTOR 3026946, it's also possible to find spaces in which a torus can be one-sided, something I find even more counterintuitive, but that's one reason we prove things in mathematics: to distinguish correct implications from intuitions that seem natural but turn out to be false.) —David Eppstein (talk) 21:30, 30 March 2022 (UTC)

Thank you for convincing me that my intuition is was broken. Yes, I can embed a closed curve in real 2-space so as to be two-sided, and in projective 2-space so as to be one-sided. Maproom (talk) 07:20, 31 March 2022 (UTC)

1. Same paragraph: "one on which it is impossible to consistently distinguish clockwise from counterclockwise". I prefer "one in which", though it'll sound odd to many readers (we've disagreed on this before). You can draw little ↻ symbols consistently all over its surface, if you use a pencil. If you use a pen whose ink soaks into the fabric, of course you can't. Maproom (talk) 21:06, 30 March 2022 (UTC)
   • Ok, that reasoning makes sense to me. Done. I hadn't realized there was a unicode for ↻! Maybe it would be helpful to use that symbol in the text. —David Eppstein (talk) 21:30, 30 March 2022 (UTC)

GA Review

This review is transcluded from Talk:Möbius strip/GA2. The edit link for this section can be used to add comments to the review.

Reviewer: Ovinus (talk · contribs) 03:48, 14 April 2022 (UTC)

I'll take this one. Ovinus (talk) 03:48, 14 April 2022 (UTC)

GA review (see here for what the criteria are, and here for what they are not)

1. It is reasonably well written.

   a (prose, spelling, and grammar): b (MoS for lead, layout, word choice, fiction, and lists):

2. It is factually accurate and verifiable.

   a (reference section): b (citations to reliable sources): c (OR): d (copyvio and plagiarism):

3. It is broad in its coverage.

   a (major aspects): b (focused):

4. It follows the neutral point of view policy.

   Fair representation without bias:

5. It is stable.

   No edit wars, etc.:

6. It is illustrated by images and other media, where possible and appropriate.

   a (images are tagged and non-free content have non-free use rationales): b (appropriate use with suitable captions):

7. Overall:

   Pass/Fail:

Initial comments + lead

• To be clear, smoothness in this context is just $C^{2}$?
  • It's explained in footnotes e and f. Is your question whether those footnotes apply to other places where the article uses the word "smooth"? —David Eppstein (talk) 06:20, 14 April 2022 (UTC)
• Quite a bit in "popular culture" for a math article... I think the Bach explanation may be unduly long
  • Well, most of an entire book (Pickover's) is "in popular culture" for the Möbius strip. And there's quite a bit of mathematics for something that most people have heard of and might want to know more about; the popcult parts fill out the article with some less-demanding reading for that audience. Re Bach, really the point I wanted to make is "too much has been made of the Bach–Möbius connection but for this one canon it does work". But then specifically identifying "this one canon" takes some verbiage. Would it help to relegate that identifying text to the Notes section? —David Eppstein (talk) 06:20, 14 April 2022 (UTC)
    • (edit conflict) Gotcha. Imo, "Möbius strips have also been used to analyze ... such as a cylinder could have been used equally well" should be put in a footnote. The explanation that one canon has this property doesn't imply that all his canons do. I guess there's the misconception that the crab canon popularized in Gödel, Escher, Bach (?) is somehow a Mobius strip, but I really don't think it's that important... anyone earnestly seeking to confirm or deny it would probably notice the footnote. Ovinus (talk) 07:10, 14 April 2022 (UTC)
      • Ok, moved to a footnote. —David Eppstein (talk) 17:49, 14 April 2022 (UTC)
• "Put another way, if an embedded Möbius strip is thickened slightly into a three-dimensional object, the surface of the thickened object forms a single connected set." Does "thicken" have a meaning I don't know about? If a paper annulus were thickened into something that looks like the difference of two concentric cylinders, its topological boundary (in $R^{3}$) would be connected too
  • Yes, but connected across edges rather than across flat surfaces. It's the same sense in which a solid cube has six sides and not one. My feeling is that this is the sort of quibble that might be made only by someone who already understands the intended point, and that an attempt at defining this in a rigorous way as some sort of double cover (topology) would make this much more WP:TECHNICAL and lose the people who don't already understand it. Also a paper annulus is already a three-dimensional object with thickness and volume (a very small thickness and volume, but nonzero.) But if you have a suggestion for an alternative way of explaining this that could be simultaneously as understandable to a general audience and more precise mathematically, I'd be interested to hear it. —David Eppstein (talk) 06:20, 14 April 2022 (UTC)
    • I think quite a few others would think of "surface" and "connected" in the same way I do. I don't know who will actually learn anything from the statement because they will need a subtler notion of "thickening" than either of the intuitive ones: "fattening" the surface (e.g., taking the set of all points within $\epsilon$ units of some point on the strip); or "projecting" the surface perpendicularly into a 3D object with edges, in which case there are two "sides"—the infinitesimal edge (now side) of the paper and the writing side of the paper. In view of that, I'd recommend just removing it. Ovinus (talk) 07:10, 14 April 2022 (UTC)
      • Ok, removed. —David Eppstein (talk) 17:49, 14 April 2022 (UTC)
• I think it's worth to be very explicit about the difference between the "Mobius strip" and its embedding in R^3, especially in the lead, and even if it's somewhat imprecise. A lot of non-mathematicians will be reading this article. E.g., "The Mobius strip is an abstract object with many interpretations, but is most familiar in its embedding in three dimensions..." that's crude, but something like that
  • Ok, I'll have to think about that. I don't think the article gives "many interpretations" for the strip as an abstract object, though? It describes many geometric realizations (not all of which are embeddings or even immersions) but doesn't distinguish carefully between different topological interpretations, although one could do so (is it a metric space, a point-set topology, a differentiable manifold, some other type of manifold...). —David Eppstein (talk) 06:20, 14 April 2022 (UTC)
    • For sure. I just think it's important to combat the idea that the 3D embedding is the canonical Möbius strip, which is probably the intuition of many people (read: me a few years ago), whose introduction to topology has been "a topologist can't tell the difference between a coffee cup and a donut," and that topology is simply the study of manifolds in Euclidean space. Ovinus (talk) 07:31, 14 April 2022 (UTC)
      • I tried separating off a new paragraph in the lead distinguishing it as a topological space from its embeddings; please let me know what you think. —David Eppstein (talk) 17:49, 14 April 2022 (UTC)
        • Big fan. Quibbles: "or with a knotted centerline" link Mathematical knot, and explicitly say somewhere that the embeddings are in fact topologically equivalent.
          • Ok, added a wikilink and one more sentence about topological equivalence (wikilinked without explanation to the correct but somewhat technical notion of equivalence, Ambient isotopy). —David Eppstein (talk) 07:14, 16 April 2022 (UTC)
• "as has the space of two-note chords in music theory" Probably not worth mentioning, since it requires a lot of context
  • In the lead, you mean, or at all? It's one of the rare "in popular culture" items that actually has any mathematical depth. —David Eppstein (talk) 06:26, 14 April 2022 (UTC)
    • In the lead. Ovinus (talk) 07:28, 14 April 2022 (UTC)
      • Ok, removed. —David Eppstein (talk) 17:52, 14 April 2022 (UTC)
• "have a circular boundary" In a topological or geometric sense?
  • Geometric. (For the cross-cap, sources differ on this point as well as on whether it has a boundary at all or is just another name for a projective plane. But at least one of the sources I'm using for that point explicitly says that it's a geometric circle. For the Sudanese, it is unambiguously a geometric circle.) —David Eppstein (talk) 06:20, 14 April 2022 (UTC)
    • Cool.
• "and world maps printed ... " A little unclear on the meaning of opposite. Maybe specify "two-sided"
  • But there's only one side? That's why it says "opposite" without saying that it's an "opposite point" (it's the same point). One source said something about how you could find the antipode by pushing a pin through the map but I couldn't find a way to word it as concisely using that idea. —David Eppstein (talk) 06:20, 14 April 2022 (UTC)
    • Got me there... LOL. The new phrasing is sensible, but remove the editorializing "convenient" (I doubt Mobius-strip maps are convenient in any application) and make "antipodes" singular. Ovinus (talk) 07:10, 14 April 2022 (UTC)
      • But our article antipodes is one of those rare articles whose title is plural, because antipodes come in pairs: there is never just one of them. I changed "the antipodes" to "antipodes", though. —David Eppstein (talk) 17:52, 14 April 2022 (UTC)
        • I mean in the Applications section. "and that the antipodes of any point on the map can be found on the other printed side of the surface at the same point of the Möbius strip"—no, the singular antipode of any point may be found on the other side. There are not three antipodes corresponding to each point on a sphere. Ovinus (talk) 19:12, 14 April 2022 (UTC)
          • Oh, that makes more sense. I thought you meant its mention in the lead. In that context, singular is correct. Done. —David Eppstein (talk) 20:00, 14 April 2022 (UTC)
• "The Möbius strip is a non-orientable surface, one in which it is impossible ..." replace "one in which" with "meaning"; otherwise it sounds like it's a special type of non-orientable surface that has the latter property
  • Ok, that makes sense, will do. —David Eppstein (talk) 06:20, 14 April 2022 (UTC)
• Ah, I totally forgot. It'd definitely be a good idea to mention the chirality/# of half-twists question in the lead, since that'll be another common misconception of readers (and perhaps one that will better challenge their notion of topology). Something along the lines of, "Three-dimensional embeddings of Mobius strips are chiral—they have both a "right-handed" and "left-handed" version—and may be generated by any odd number of half-twists. These constructions, however, are all topologically indistinguishable when considered as two-dimensional surfaces." Ovinus (talk) 07:38, 14 April 2022 (UTC)
  • Done as part of the new paragraph on abstract space vs embeddings. —David Eppstein (talk) 17:53, 14 April 2022 (UTC)

Looks really good overall—excited to read the rest. Ovinus (talk) 03:48, 14 April 2022 (UTC)

History

• "However, it had been known long before, both as a physical object and in artistic depictions. In particular, it can be seen in several Roman mosaics from the third century AD." Prose quibbling here. How can a nonphysical object be artistically depicted in (non-abstract) art? I think "it had been known long before as a physical object" is perfectly fine, esp. given we've never found Mobius strips at archaelogical sites Ovinus (talk) 07:28, 14 April 2022 (UTC)
  • Are you arguing that a depiction of the god Aion holding the zodiac in the form of a ribbon is a depiction of a physical object? —David Eppstein (talk) 18:33, 14 April 2022 (UTC)
    • Well, yes, in the same way that an artist who draws a fictional mathematician holding a Möbius strip without reference to a real-world person is drawing a physical object. They are not drawing something disconnected with physical reality, like a piece of paper that passes through itself. The distinction here is between the mathematical conception and the physical object, right? Ovinus (talk) 19:10, 14 April 2022 (UTC)
      • The intended distinction was between things that exist as physical objects in the real world (presumably, the bucket chain depicted by al-Jazeri) and objects that exist only in the imaginary space of the artwork. —David Eppstein (talk) 20:02, 14 April 2022 (UTC)
        • Okay, I'm convinced
• "In at least one case, a ribbon with different colors on different sides was drawn with an odd number of coils, forcing its artist to make a clumsy fix at the point where the colors did not match up." Now we're getting rude! More seriously, this section has a thoroughly dismissive tone. Is that representative of the current consensus? Ovinus (talk) 07:28, 14 April 2022 (UTC)
  • Yes. The tone here is quite close to that of both sources. From the source on the "clumsy fix": "This mosaic is also an uninspired example ... The artist unwittingly drew his ribbon with an odd number of half-twists ... deduced from the obvious interruption of the coil ... the troublesome half-twist was removed, thus converting a Möbius band into a more pedestrian design". And from the source on the Aion mosaic, we also get a similar dismissive tone, referring to the boundary ribbon examples: "Our example is less dubitable than those [Larison] discussed, principally one from Arles showing a band with five half-twists (Fig. 3) provoked some doubts as to whether it was in fact representing a Möbius strip". —David Eppstein (talk) 20:08, 14 April 2022 (UTC)
    • Great.
• In any case, "but whether they were intended to ... is unclear" is unnecessary. "Alleged" casts enough doubt. Ovinus (talk) 07:28, 14 April 2022 (UTC)
  • It's intended to be a more in-depth explanation for what factor causes their status as Möbius strips to be disputed: because if they depict strips, then the strips they depict are arguably Möbius strips, but it's not clear that what they depict are strips at all. —David Eppstein (talk) 20:11, 14 April 2022 (UTC)
• "using both sides (or rather the same single side) of their material" is rather confusing. There is only one side. Also, it needs to be said what "half as quickly" is relative to. E.g., "Machinists have long known that Möbius-strip mechanical belts, which contact the rollers with twice the surface area as do their two-sided counterparts, wear out half as quickly, and also have another potential advantage in evening out any curvature that might otherwise develop in the belt." Ovinus (talk) 07:28, 14 April 2022 (UTC)
  • This is in contrast to an untwisted belt, which would only wear or curl on a single side. "Half as quickly" is also meant as the same comparison, to an untwisted belt. Locally, the belt has two sides, and this method wears both evenly. Your proposed rewording has a different confusing ambiguity: the area of the patch of instantaneous contact between the belt and the rollers is unchanged by the twist; how could it be twice as much? —David Eppstein (talk) 20:14, 14 April 2022 (UTC)
    • Well, the "untwisted belt" isn't that implicit. In any case the whole "using both sides (or rather the same single side)" is very confusing. To remove the ambiguity you pointed out, you could just remove the "twice the surface area" part: "Machinists have long known that Möbius-strip mechanical belts wear out half as quickly as do their two-sided counterparts", something like that. Ovinus (talk) 20:38, 14 April 2022 (UTC)
      • Copyedited to contrast more explicitly to an untwisted belt. —David Eppstein (talk) 07:19, 16 April 2022 (UTC)
• "develop in the belt" remove "in the belt," not much else can be curved here Ovinus (talk) 07:28, 14 April 2022 (UTC)
  • Is that really true? An axle that started curving would be pretty problematic. —David Eppstein (talk) 20:14, 14 April 2022 (UTC)
    • Hahaha. Fair enough
      • Anyway this wording was redone in the above copyedit. —David Eppstein (talk) 07:19, 16 April 2022 (UTC)
• "but this is after the first mathematical publications regarding the Möbius strip" why is this important? Ovinus (talk) 07:28, 14 April 2022 (UTC)
  • Because it means that this is not strong evidence for prior non-mathematical knowledge of this shape. —David Eppstein (talk) 20:15, 14 April 2022 (UTC)
• "what can only be" Can remove, esp. since there's an image Ovinus (talk) 07:28, 14 April 2022 (UTC)
  • Ok. —David Eppstein (talk) 19:01, 14 April 2022 (UTC)

Properties

• "a small circle with an arrow pointing clockwise around it" How about just "a curved arrow pointing clockwise"?
  • That's much tighter, and clear enough; good. Done. —David Eppstein (talk) 07:24, 16 April 2022 (UTC)
• "embed an uncountable set of disjoint copies" -> "embed uncountably many disjoint copies", "only a countable number of Möbius strips" -> "only countably many ..."
  • I don't feel very strongly about "set" vs "many" here. But there are two wiki-reasons for the current wording. First, the wikilink goes to uncountable set. Second, and maybe more convincing, the "set" wording allows the links on uncountable set and disjoint set to be separated by the word "of", avoiding MOS:BLUESEA issues. —David Eppstein (talk) 07:27, 16 April 2022 (UTC)
    • Fair enough.
• Is there a reason you're using {{nowrap}} for each reference? Never seen it done before and it clogs up the wikitext... I'm surprised <ref> doesn't already insert a word joiner immediately preceding it. Ovinus (talk) 08:09, 14 April 2022 (UTC)
  • See discussion below. To avoid bad line breaks that look like

    Some wikitext.

    ^[1][2]

  —David Eppstein (talk) 07:29, 16 April 2022 (UTC)

• "A path along the edge of a Möbius strip, traced until it returns to its starting point on the edge, has double the length of the original strip." I think it needs to be specified whether this is talking about the embedding. (Is it true more generally? I assume it's metrizable... I'm really only comfortable with basic point-set topology from real analysis, so I'm lost.) Ovinus (talk) 08:09, 14 April 2022 (UTC)
  • I think this is only literally true for the Möbius strip that you get by twisting and gluing a rectangle. For other Möbius strips in roughly the same shape, you go twice around but maybe not exactly twice the length. I rephrased that to be more careful. —David Eppstein (talk) 06:35, 15 April 2022 (UTC)
• "More precisely, two Möbius strips are equivalently embedded in three-dimensional space when their centerlines determine the same knot (or unknot) and when they have the same odd number of twists as each other" Am quite lost here; isn't this just saying embedding is equivalent up to isotopy?
  • It's describing how to tell when there is an ambient isotopy from one embedding to another: you just have to determine the knot type and count twists. It's obvious that two embeddings with the same knot type and number of twists are ambient isotopic. It's less obvious that nothing else can be ambient isotopic. —David Eppstein (talk) 06:35, 15 April 2022 (UTC)
    • (Oops, forgot ambient isotopic and isotopic were totally different things.) I don't think "odd" is needed and is slightly confusing. Also, I don't think "or unknot" is really needed; if someone knows enough knot theory concepts to understand the equivalence of knots, they probably know that an unknot is a valid, if somewhat degenerate, type of knot. Unless I'm wrong; I've never picked up a book on it. Ovinus (talk) 19:55, 16 April 2022 (UTC)
      • Ok, tightened. —David Eppstein (talk) 21:43, 16 April 2022 (UTC)
• "Cutting this cylinder again along its center line" We're really blurring the lines between physical and abstract here... we cut a paper strip, then cut a topological "cylinder" (I know it's still a "cylinder" in 3D space, but most people won't understand that). This description could def. be made more accessible. Ovinus (talk) 08:09, 14 April 2022 (UTC)
  • Changed "cylinder" to "double-twisted strip". Also there was a mistake in here, I think: it said that this strip has "two full twists" when really I think it's two half-twists or one full twist. —David Eppstein (talk) 07:33, 16 April 2022 (UTC)
    • Gotcha.
• "Ringel–Youngs theorem, which states how many colors each topological surface needs" why is this in nowrap?
  • The nowrap was too long, probably because of earlier edits. The reason for the nowrap here is to avoid line breaks between punctuation and footnote markers, which otherwise tend to happen, but it only needs to be on the word "needs", its punctuation, and the following footnote, not the longer phrase. —David Eppstein (talk) 18:15, 14 April 2022 (UTC)
• Say explicitly that Tietze's graph is nonplanar
  • Ok, done. —David Eppstein (talk) 18:13, 14 April 2022 (UTC)
• "on a transparent Möbius strip" Why "transparent"?
  • Because if it's opaque then you're really drawing on the double cover of the Möbius strip, which doesn't have a solution. Look at the way the thick yellow and thin blue lines in the figure are visible on both local sides of the depicted strip. —David Eppstein (talk) 16:00, 14 April 2022 (UTC)
    • Makes sense. "transparent" seems to come out of left field; maybe just "two-dimensional"? (Although that might be slightly imprecise) Ovinus (talk) 20:24, 14 April 2022 (UTC)
      • But that doesn't distinguish transparent from opaque? By two-dimensional you mean as an abstract topological space rather than as an embedded object in 3d? I don't think that way of making the distinction is easy to understand. —David Eppstein (talk) 07:31, 16 April 2022 (UTC)
        • Alright, I'm convinced
• Any reason to use R instead of the more familiar F in the Euler characteristic formula?
  • The reason for that was because I was calling the things it counts regions, not faces, and didn't want to introduce another word only for that purpose. But I can change it to F if you're ok with using F as a variable for the number of regions. —David Eppstein (talk) 16:00, 14 April 2022 (UTC)
    • Changed to F. —David Eppstein (talk) 18:13, 14 April 2022 (UTC)
      • Cool. Ovinus (talk) 20:24, 14 April 2022 (UTC)
• Maybe include somewhere that the strip is homotopy-equivalent to the circle and therefore $\pi _{1}(M)=\mathbb {Z}$
  • That seems like a pretty fundamental thing to include (sorry). I guess this section would be the place for it. It's pretty obvious that it's a deformation retraction (just continuously narrow the width of the strip) but both this fact and the conclusion from that about its fundamental group probably need a source. —David Eppstein (talk) 07:44, 16 April 2022 (UTC)

  The "Algebraic Topology" chapter of the Princeton Companion to Mathematics (by Burt Totaro) mentions the homotopy equivalence on p. 387 and notes that $\pi _{1}(S^{1})=\mathbb {Z}$ on p. 386 and in the table on p. 390. XOR'easter (talk) 17:20, 16 April 2022 (UTC)

  • Thanks! My copy of that is in my office and I'm home for the weekend so instead I ended up using Massey's A Basic Course in Algebraic Topology. —David Eppstein (talk) 20:03, 16 April 2022 (UTC)

• Is there a standard symbol used in the literature for a Mobius strip? (${\mathcal {M}}$, $M$, etc.)
  • Not that I know of. Confusingly, the Kuiper reference uses $M$ for a generic topological space when you know he is going to have that space be a Möbius strip later in the paper. doi:10.1007/s00233-014-9658-0 (which we're not currently using as a reference) uses $\mathbb {M} ^{2}$ but I don't think I've seen that notation elsewhere. See also a related recent discussion at Wikipedia talk:Manual of Style/Mathematics#Notational conventions for spaces where participants are split on whether spheres should be $S^{n}$ or $\mathbb {S} ^{n}$. —David Eppstein (talk) 07:42, 16 April 2022 (UTC)
    • Horrifying. Never seen $\mathbb {S}$. A unified style across articles would really be nice but seems like somewhat of a pipe dream.
• "while the other has two twists" full or half?
  • Half, I think. Added. —David Eppstein (talk) 07:42, 16 April 2022 (UTC)

Constructions

• "There are many different ways of defining geometric surfaces with the topology of the Möbius strip, depending on the additional geometric properties that are desired for this surface." A slightly strange construction, implying that different constructions of the Mobius strip are made for the express purpose of inducing different properties. I'm sure some are, but how about just "each yielding (or inducing, etc.) additional properties..."
  • Ok, tightened. —David Eppstein (talk) 20:33, 16 April 2022 (UTC)
• "One way to represent the Möbius strip embedded in three-dimensional Euclidean space is to sweep it out by a rotating line segment in a rotating plane" Why not just a segment "following a circle"? The wording is slightly confusing (although the image probably dispels some confusion). When I hear "rotating plane" I'd like to know that it's rotating about an axis not orthogonal to the plane. Ovinus (talk) 20:24, 14 April 2022 (UTC)
  • Because spinning in a rotating plane is a clear way to describe how the segment rotates as it follows the circle. Alternatively I suppose one could say that the axis of rotation of the segment remains tangent to the circle but that seems a big more technical to me. Also, the axis is orthogonal to the plane; why shouldn't it be? —David Eppstein (talk) 20:26, 16 April 2022 (UTC)
    • I don't get it. [2] ?
      • The rotating plane is perpendicular to the circle, and the axis of rotation is a line through the center of the circle perpendicular to the plane of the circle. Think of a wall clock whose hands spin like the line segment, mounted on one of the panes of a revolving door. —David Eppstein (talk) 21:45, 16 April 2022 (UTC)
        • So the line segment is not rotating in the plane, but about an axis in the plane. Hm. Is "attached to" too informal? Ovinus (talk) 22:31, 16 April 2022 (UTC)
          • No! It is rotating in the plane. But the plane is moving. The hand of a clock rotates in the plane of the clock face, regardless of where that plane is attached. The "axis in the plane" is what the whole plane rotates around, at the same time. Another familiar model: consider a tethered model propeller airplane. The propeller blade spins in a plane. The plane of the spinning propellor blade rotates around the point to which the airplane is tethered. —David Eppstein (talk) 23:10, 16 April 2022 (UTC)

          Look at https://scooj.org/2021/02/11/thursday-doors-11-february-2021/ and scroll down to the blue door with the clock on it. The door lies in a plane. If you open the door, that plane swings around an axis through the door hinges. Meanwhile, the hands of the clock are spinning within the plane of the door. —David Eppstein (talk) 23:22, 16 April 2022 (UTC)

          This is reminding me of that old joke about how to frustrate an Italian (first told to me by an Italian friend): ask them to explain a spiral staircase while keeping their hands in their pockets. —David Eppstein (talk) 23:25, 16 April 2022 (UTC)

          • • So the axis about which the plane is rotating is in the plane and not perpendicular to it. Hence my comment "When I hear 'rotating plane' I'd like to know that it's rotating about an axis not orthogonal to the plane"; a plane can rotate in many ways, unlike a door. Perhaps you thought I was referring to the segment's axis of rotation. Ovinus (talk) 23:51, 16 April 2022 (UTC)
              • But it already said, explicitly, "the plane rotates around one of its lines". —David Eppstein (talk) 00:02, 17 April 2022 (UTC)
                • I made a suggested rearrangement to those two sentences. Ovinus (talk) 00:08, 17 April 2022 (UTC)
                  • What, the part about "following a circle"? I don't know what it means for a segment to follow a circle. —David Eppstein (talk) 00:41, 17 April 2022 (UTC)
                    • No, to the "rotating plane" sentences in the article. Ovinus (talk) 01:11, 17 April 2022 (UTC)
                      • Where in this review is your "suggested rearrangement to those two sentences"? Because I seem to have lost track of where it might be and can't find it. That's why I thought maybe your "following a circle" suggestion was the suggestion you meant. —David Eppstein (talk) 07:18, 17 April 2022 (UTC)
                        • [3] Ovinus (talk) 18:21, 17 April 2022 (UTC)
                          • Oh, ok. I thought you meant in the review somewhere. Copyedit looks ok; I reworded it a little more. —David Eppstein (talk) 19:04, 17 April 2022 (UTC)
• "The equations of these coordinates are" replace with something like "An example parametrization is", since there are many parametrizations that satisfy the conditions you set above
  • That's what the "can be" in the previous sentence was supposed to imply. I had it in that order because I wanted to describe what the parameters were before using them. But since you found that confusing, I reordered that material into a single longer sentence with the equations first and the description of what its parameters mean second. —David Eppstein (talk) 20:33, 16 April 2022 (UTC)
• "by rotating the segment more quickly in its plane, relative to the rate of rotation of the plane" Can remove "relative ..."
  • I guess. If you rotate both the segment and the plane more quickly, nothing changes, though. —David Eppstein (talk) 20:34, 16 April 2022 (UTC)
• "Can a 12\times 7 paper rectangle be glued end-to-end to form a smooth Möbius strip embedded in 3d space?" Citation? Also, should be "3D"
  • Do I really need a citation for the fact that 12/7 is between 1.695 and 1.73? I thought this was the kind of "routine calculation" described by WP:CALC. I put it that way in the open problem box to keep the one-sentence statement there as simple as possible. I added an explanatory footnote about why 12/7 (it is the simplest rational number in the range of unknown aspect ratios). Re 3D, ok (apparently this is to avoid confusion with "3d" as an abbreviation for "third"), but I think maybe it's better just to say "space" and avoid the issue. —David Eppstein (talk) 20:41, 16 April 2022 (UTC)
    • Ohhhh I see. For some reason I thought it was the size of A4 paper and thus was probably conjectured explicitly somewhere. Falls under CALC, indeed. Ovinus (talk) 21:14, 16 April 2022 (UTC)
• "The open Möbius strip is the open set formed from the interior of the standard Möbius strip, in any of its other constructions, omitting the points on its boundary edge" Why is "the interior of" necessary, or alternatively "open set"? (Considering the open Möbius strip is clopen relative to itself)
  • Rewrote to use relative interior and otherwise be simpler. —David Eppstein (talk) 21:00, 16 April 2022 (UTC)
• "The Sudanese Möbius strip extends on all sides of its boundary circle, unavoidably if the surface is to avoid crossing itself." Wdym by "extends"?
  • If you sweep a small circle around the boundary circle, all points of the circle will be hit by the strip. Do you have some suggestion for how to say that more clearly? —David Eppstein (talk) 21:02, 16 April 2022 (UTC)
• "to the real projective plane by adding one more line, the line at infinity." why not just "adding the line at infinity"
  • To emphasize that only a single line is being added. You might think that adding lots of points to the plane also creates lots of new lines, not just the line to which all the added points belong, but it doesn't. —David Eppstein (talk) 21:06, 16 April 2022 (UTC)
• The second paragraph in Spaces of lines is far beyond me, so I'll take your word for it.
  • Now I'm sad. I had been hoping, at least, that the second part of this paragraph on group models had brought this material down from the level of "I don't know what any of these words mean" (my reaction on starting to work through the sources to this material) to "this is trivial to anyone who knows what stabilizers and cosets are". I was expecting this paragraph to be the most technical one of the entire article, though, so in that respect I'm not surprised. —David Eppstein (talk) 21:06, 16 April 2022 (UTC)
    • Tragic. Hopefully in a few years. Ovinus (talk) 21:24, 16 April 2022 (UTC)
      • I took another pass at making this paragraph less technical but it's still not easy material. —David Eppstein (talk) 19:05, 17 April 2022 (UTC)
        • Hm. Except for the homogeneous space/solvmanifolds part, I think I mostly get it. The last process of quotienting out the symmetries has clicked. For context, I haven't learned about Lie groups yet. So this same construction works with Möbius transformations and hyperbolic space? Ovinus (talk) 20:09, 17 April 2022 (UTC)
          • Yes, but you have to pick out some particular line to play the role of the x-axis. —David Eppstein (talk) 20:26, 17 April 2022 (UTC)

Applications

• Are there any striking topological (or otherwise) theorems resulting from the Mobius strip's non-orientability or failure to embed in $\mathbb {R} ^{2}$? It'd be nice to show at least one application within mathematics that isn't a statement about the strip itself. I vaguely recall using the non-injectivity of a difficult-to-understand mapping $f:M\to \mathbb {R} ^{2}$ to show that a value was achieved at least twice. (Edit: didn't see the social choice theory example. Perhaps that's good enough.) Ovinus (talk) 19:47, 16 April 2022 (UTC)
  • In some sense we already have two more of these, scattered elsewhere in the article: the existence of the Möbius strip demonstrates that not every solvmanifold is a nilmanifold (if you know what those things are), and it provides examples of triangulated surfaces that have no polyhedral embedding (Brehm). There are some papers on scientific computation using the existence of Möbius strips as a counterexample to certain code optimizations that would work if meshes always described orientable surfaces, but I think they would be too technical to explain. Anyway, nothing springs to mind or out of some rudimentary searches, but this is a difficult thing to search for. Part of my preparation for this nomination was to look through articles linking to this one to see whether they had any material that should be mentioned, so I think if there were anything like that in another article we'd probably already be including it. —David Eppstein (talk) 23:45, 17 April 2022 (UTC)

Popular culture

• "(memorialized in a poem by Charles Olson)" Relevant?
  • The significance of the Cagli painting is that it's the earliest artwork inspired by the mathematical study of the Möbius strip that I have a reference to. The poem describes it. You can find more about the poem at this link (not usable as a source because it's a blog post). I added another published reference on the Olson poem that goes into more detail. —David Eppstein (talk) 00:13, 18 April 2022 (UTC)
• "Although mathematically the Möbius strip and the fourth dimension are both purely spatial concepts" How is this relevant? It seems like an unnecessary "debunking" of an artistic liberty (vs the other "debunkings" of overly enthusiastic academics).
  • In this case I think the debunking is completely fair. If the works in question actually depicted people as trapped in a reflecting loop, rather than just in a loop, the connection to the Möbius strip would be more present, but really they use it only as "here's some math we don't understand, therefore maybe something else we don't understand could happen". The more serious debunking would be that, in an actual Möbius strip, all points on all sides of the strip are connected to each other, so if it could be used to describe a situation that characters could get into, they could easily get out of it by reversing their steps and wouldn't be trapped. But that would be original research. —David Eppstein (talk) 00:13, 18 April 2022 (UTC)
• "The Möbius-strip principle" What is that
  • Left over text from long ago that I never edited. I copyedited that paragraph. —David Eppstein (talk) 00:18, 18 April 2022 (UTC)

That's all for now. I'll take a second pass through of the article over the weekend. Ovinus (talk) 20:24, 14 April 2022 (UTC)

Thanks for all the suggestions! I may not have time to get to all of them until the weekend. —David Eppstein (talk) 20:32, 14 April 2022 (UTC)

For sure; take your time. I wrote quite a bit.... Ovinus (talk) 20:39, 14 April 2022 (UTC)

Nowrap

I can't find any evidence that references are pushed to new lines, even with punctuation. In User:Ovinus/sandbox I put a bunch of refs at the end of a sentence and they seem to wrap just fine.... Ovinus (talk) 23:30, 14 April 2022 (UTC)

I can definitely get it to happen repeatably when there is more than one footnote at the end of the sentence, in the viewing combination that I use by default (monobook, OS X, Firefox). In your sandbox, when I narrow my window down to the point where the paragraphs have four lines, and then slightly more than that so that the last line doesn't fit, it breaks between the first two footnote markers at the end of the paragraph and the second two. I can make a screenshot demonstrating this if that would be of any use. When there is only one footnote at the end of the sentence, it seems not to happen as frequently, or maybe not at all, I'm not sure. It is also possible to get line breaks between mathematics formulas and the punctuation immediately following them, which is even more annoying. —David Eppstein (talk) 00:14, 15 April 2022 (UTC)

Can reproduce on Firefox, not on Chrome. Really weird! Ovinus (talk) 01:43, 15 April 2022 (UTC)

Second (quick) pass

• "Both the Sudanese Möbius strip and the cross-cap (another self-intersecting Möbius strip) have a circular boundary." Perhaps just "and another self-intersecting Mobius strip, the cross-cap, have a circular boundary". I'd expect few people to have heard of the cross cap, so it makes sense to have its description be relatively emphasized
  • Ok; I think regardless of what people have heard of this wording is better. —David Eppstein (talk) 00:41, 18 April 2022 (UTC)
• "have been used to prove impossibility results in social choice theory" As I alluded to above, surely there are quite a few interesting results of the strip's topological properties? Why give this one particular weight
  • In the lead? Because I wanted to give an example of a non-physical application beyond pure mathematics. —David Eppstein (talk) 00:41, 18 April 2022 (UTC)
• "plot structure based on the Möbius strip, of events that repeat with a twist, is common in fiction more generally" Important enough to go in the lead? (I'm not familiar with how "legitimate" these analyses are)
  • In general, my goal was to briefly mention in the lead, for half a sentence or a sentence, everything covered at the length of a full paragraph in the rest of the article; see MOS:INTRO "The lead section should briefly summarize the most important points covered in an article in such a way that it can stand on its own as a concise version of the article" and WP:GACR #1b "it complies with the manual of style guidelines for lead sections". So this was the part of the lead where I briefly summarized the paragraph on literary uses of the Möbius strip. Some of these analyses may be speculative; others are obviously deliberate on the part of the author (Lost in the Funhouse was printed in a way that encouraged its readers to cut and rejoin its framing story into a Möbius strip; Delany's author notes to Dhalgren talk about its structure as a Möbius strip) or are in wide circulation (many reviews of Donnie Darko discuss its structure as being like a Möbius strip). —David Eppstein (talk) 00:41, 18 April 2022 (UTC)
• "but without clear dates for the origin of this task" Hm. Wdym vs. "but the origin of this task is unknown"?
  • This clause is here because, without saying something about when this tradition happened, the earlier part of the sentence would legitimately be the target of a {{when}} cleanup tag. But the dates are not in the source. It's not confusing how it started or where it started (in Paris, as a way for older seamstresses to initiate younger ones); it's when it happened. But your copyedit removes any mention of the dates again, making this clause superfluous. I tried rewriting the whole sentence, in part making this part more brief, but keeping its focus on the dates. —David Eppstein (talk) 00:41, 18 April 2022 (UTC)
• "formed by lengthwise slices of Möbius strips with varying widths" Is this addition necessary?
  • It's intended as a clarification that "These interlinked shapes" refers to all of the shapes described in this paragraph – the double-length double-twisted ring formed by a central cut, the two linked double-twisted rings formed by cutting again, or the linked double-twisted ring and Möbius strip formed by a 1/3-2/3 cut – rather than just the last one mentioned. In particular, "with varying widths" means here "not just with the single width of the last example". —David Eppstein (talk) 00:51, 18 April 2022 (UTC)
• "and – reversing that process – a Klein bottle" Maybe just "conversely"?
  • I've gotten in trouble in the past with Wikipedia-editors who care about precision of words for using "conversely" to mean something other than reversing the antecedent and consequent of a logical implication. —David Eppstein (talk) 00:51, 18 April 2022 (UTC)
• "stretch their interpretation of the Möbius strip beyond its recognizability as a mathematical form or a functional part of the architecture" Necessary "debunking"?
  • In this case it merely reflects the tone of the two sources used for this material (one for bridges and one for buildings). Some quotes: "failing to achieve the visual continuity of the Mobius as a whole" ... "the mathematical model of the Möbius is not literally transferred to the building" ... "when one searches the world wide web for pictures of Möbius bridges ... most of them do not satisfy the requirements for an actual Möbius topology". But the intended tone is not really debunking, so much as justifying the selection of the examples in the rest of the paragraph: they were chosen because they were real, unlike many of the other things that you will find if you try to search for a Möbius building or Möbius bridge. (If I were trying to debunk, I would have gone into more detail about why Lucky Knot Bridge has little to do with Möbius strips despite a lot of hype that it does.) —David Eppstein (talk) 01:12, 18 April 2022 (UTC)
    • That's hilarious... I wish there were a compilation of stuff like this. Well, I'm glad you were able to find at least a few meaningful examples. Ovinus (talk) 01:21, 18 April 2022 (UTC)
• "used as clever inventions" probably should be "used in clever inventions", but I don't know exactly what an "invention" is here. I assume a part of some fictional world?
  • In one of the Upson's stories, a fictional WWII army officer distracts another uncooperative soldier by getting him to paint only one side of a pump house drive belt, which he has secretly rejoined into a Möbius strip. In another, miners using a mile-long shaped conveyor belt need to lengthen it, and do so by cutting it down its centerline after one of the miners realizes that this can be done because it is in the shape of a Möbius-strip. So in both cases it is the Möbius strip itself that is used, as a way to highlight the cleverness of a fictional character, but not as part of some more complicated device that includes a Möbius strip. —David Eppstein (talk) 01:04, 18 April 2022 (UTC)
• "Mobius-strip principle" confusion I mentioned above
  • See answer above. —David Eppstein (talk) 01:04, 18 April 2022 (UTC)

Will pass after we resolve the remaining concerns. Ovinus (talk) 22:01, 16 April 2022 (UTC)

• All looks good (if I didn't respond to something I read it and agreed). Thanks for your responsiveness and patience! Ovinus (talk) 01:14, 18 April 2022 (UTC)
  • Thanks, and thanks for yours! —David Eppstein (talk) 01:15, 18 April 2022 (UTC)