Talk:Möbius strip/GA2

GA Review
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):
 * 1) It is factually accurate and verifiable.
 * a (reference section): b (citations to reliable sources):  c (OR):  d (copyvio and plagiarism):
 * 1) It is broad in its coverage.
 * a (major aspects): b (focused):
 * 1) It follows the neutral point of view policy.
 * Fair representation without bias:
 * 1) It is stable.
 * No edit wars, etc.:
 * 1) 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):
 * 1) Overall:
 * Pass/Fail:
 * 1) 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):
 * 1) Overall:
 * Pass/Fail:
 * 1) Overall:
 * Pass/Fail:

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

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 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&#39;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. (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 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. ?
 * 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)
 * 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
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)
 * "(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)
 * 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
Will pass after we resolve the remaining concerns. Ovinus (talk) 22:01, 16 April 2022 (UTC)
 * "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)
 * 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)