Talk:Surface (topology)

Opinion From a Reader
Please rewrite the entire article, and put the existing content into other articles that go into the technical and esoteric details of surface. Here is what I mean.

The current draft of the article is too esoteric and too technical. It should be rewritten for the general audience and have references to other articles that go into more technical and esoteric detail. It does not provide a starting place for learning about surfaces. The general reader is a high school and university freshman who needs to quickly learn about surfaces and learn where they can get more information if they need it.

The article does not clearly and simply tell the reader what is a surface, and what are the different classes or types of surfaces. For example, what is a closed surface and what is an open surface.

An introduction should start from a short historical point of view, and remain conscience of the reader whose purpose is to learn about surfaces, and learn where they can get more information or details about subtopics to surfaces. No esoteric jargon should be used in simple discussions about an introduction to surfaces, and only limited and defined technical jargon should be included.

--Thomas Foxcroft, July 16, 2009 —Preceding unsigned comment added by Thomas Foxcroft (talk • contribs) 01:13, 17 July 2009 (UTC)

Two suggestions
Suggestions:


 * Better graphics under "Some models", and add a higher genus example such as genus 2.


 * A discussion of constant curvature metrics on surfaces, connections with the Gauss-Bonnet theorem and the Uniformization Theorem.

--Mosher 14:41, 21 September 2005 (UTC)

Reverts
I reverted the following sections, for resons below.


 * Open and closed surfaces


 * Surfaces have two directions, called u and v.

This sentence is akward. Surfaces don't have directions, but surfaces are two-dimensional. A more appropriate edit would say something like "the coordinates on a surface are commonly called u and v."
 * Surfaces most definitely do have directions. You can travel in a direction on a surface.  A sphere has compass directions.  Did I just imagine all of this ? StuRat 03:30, 16 October 2005 (UTC)


 * There are an infinite number of compass directions, not just two directions. In slang-jive, one can say a surfaces have directions, but in formal, proper grammatical talk, this is not a correct way to phrase the property of two-dimensionality of surfaces.


 * So your criticism comes down to it being written in simple language while you prefer language too complex for a general audience to understand. StuRat 19:33, 16 October 2005 (UTC)


 * Open surfaces are not closed in either direction. This means moving in any direction along the surface will cause an observer to "fall off" the edge of the surface. The top of a car hood is an example of a surface open in both directions.

This is just plain wrong. The definition of open and closed sets have nothing to do with direction, they have to do with topology, and with the existence of boundaries. A more appropriate statement might be "An open surface has a boundary. By traveling in some direction, one might reach the edge of an open surface."


 * Who is talking about sets here ? The rewrite is acceptable, but should say one would reach the edge. StuRat 03:30, 16 October 2005 (UTC)


 * You seem to be confusing "open" and "closed" with "having a boundary" and "not having a boundary".


 * Unless you are considering a surface with a seam, then having no edges should be the same as being a closed surface. StuRat 19:33, 16 October 2005 (UTC)


 * Surfaces closed in one direction include a cylinder, cone, and hemisphere. Depending on the direction of travel, an observer on the surface may fall off the edge of such a surface or eventually return to the same point.

OK, this can almost pass, except that the last bit is wrong. For a cylinder, there is only one compass direction that gets you back to the same point. Movement in any of the other (infinite number of) compass directions will cause to you spiral around, forever. If the cylinder is truncated, then you will spiral till you fall off.


 * Ok, just change it to "...or continue to travel on the surface forever." StuRat 03:32, 16 October 2005 (UTC)


 * Surfaces closed in both directions include a sphere and a torus. Moving in any direction on such surfaces will eventually bring the observer back to the same location.

True on a sphere, and utterly false on a torus. Pick a point and a compass direction for a torus, and you will, with probability one, spiral around forever and never return to the same point. The only time you get back to the same point is when the compass direction is a rational multiple of 2pi, say p/q, in which case you will spiral around p times one way, and q times the other way, and then come back to the same point. (Such a path is called a periodic orbit). However, the rationals are a set of measure zero compared to the real numbers, and so almost all directions will never return. (But they will get close, this is called Poincare recurrence.) This is a basic result of ergodic theory, and is true not just for tori, but for all surfaces of negative curvature, and more generally for all spaces of negative curvature. (Although a torus has zero curvature).
 * Geez, do we really need to get into all these details ? Just change it to "There is no direction of movement which will result in falling off the edge of the surface." StuRat 03:30, 16 October 2005 (UTC)


 * Flattening a surface


 * Some open surfaces and surfaces closed in one direction may be flattened into a plane without deformation of the surface. For example, a cylinder can be flattened into a rectangular area without distorting the surface distance between surface features. A cone may also be so flattened. Such surfaces are linear in one direction and curved in the other (surfaces linear in both directions were flat to begin with).

Wrong or confusing or I don't know what. You can flatten a cylinder by cutting it. But a cut cylinder is no longer a cylinder, its topology is different. Also, to assert flat vs. non-flat requires some concept of curvature, which hasn't been introduced. It also sounds like you are ironing a shirt, which is not the impression you want to give. I think you are trying to say that "a cylinder, a cone and a torus can carry a coordinate system that is flat". A fancier set of words would be that "a torus and a cylinder carry a metric with zero scalar curvature."


 * Your criticisms here are quite vague and seem to boil down to "I don't like it, not complex enough." Flattening something is a basic concept that requires no detailed discussion on the nature of curvature.  Saying it the way you want would make it completely inaceessible to the general public. StuRat 03:30, 16 October 2005 (UTC)


 * No, I said "I don't like it because its WRONG".


 * No, you said it's "Wrong or confusing or I don't know what." StuRat 19:33, 16 October 2005 (UTC)


 * Other open surfaces and surfaces closed in one direction, and all surfaces closed in both directions, can't be flattened without deformation. A hemisphere or sphere, for example, can't. Such surfaces are curved in both directions. This is why maps of the Earth are distorted. The larger the area the map represents, the greater the distortion.

False. Any bounded, connected subset of the Euclidean plane is going to be open and flat. There is also the confusion about "direction" and "openness". For example, if I cut a hole in a donut, I get an open surface. Is this surface "closed on both directions"? "open in one direction"? Who knows? That's because its not a precise definition. linas 01:34, 16 October 2005 (UTC)


 * I have no idea what that "bounded, connected subset of the Euclidean plane" stuff means. A surface with a hole in it is not a canonical form, which is what I am discussing here. StuRat 03:30, 16 October 2005 (UTC)


 * If you don't know the topic, perhaps you should not be writing about it. linas 19:23, 16 October 2005 (UTC)


 * If you can't write in simpler terms than that, perhaps you should be writing for mathematical journals, not for publications intended for the general public, like Wikipedia. StuRat 19:33, 16 October 2005 (UTC)


 * Perhaps the article could be split into "mathematical" and "non-mathematical" part, leaving some room for people who like to write about mathematics but do not understand what they are writing about. Tomo 06:30, 18 October 2005 (UTC)


 * While you stated it quite rudely, I will take the suggestion anyway, and make a new article surface (computer). I was essentially writing about the CAD entity called a surface, and since mathematicians have control of this article and refuse to allow any discussion of any other type of surface, I will move my material there, instead. StuRat 15:19, 21 October 2005 (UTC)

Open and closed surfaces
The new section "Open and closed surfaces" is vague and contain mathematical errors. A computer program written with this level of vagueness and error would either not compile or would crash, and a Wikipedia reader trying to understand this section would turn away confused.

The section should be either substantially rewritten to be precise and correct, or should be removed.

Part of the vagueness and incorrectness stems from what I think is a lack in the present article. Namely, the article needs a discussion of local coordinate systems on a surface, along the lines of standard undergraduate textbooks in multivariable calculus, or more advanced textbooks in topology or differential topology. A local coordinate system on a surface is a pair of real valued, continuous functions, which may be denoted u and v, which are defined on a part of the surface, and which make that part look like a part of E2 (Euclidean 2-space). In more advanced language, the domain of u and v should be an open subset W of the surface, and these two functions should define a homeomorphism between W and an open subset of E2.

For example, on the surface of the earth, the meridian and longitude give a local coordinate system defined on all of the earth except at the north and south poles (where longitude is undefined), and at the 180 degree longitude line (where there is an ambiguity between east longitude and west longitude). On the other hand, although the north and south poles are singularities for meridian and longitude coordinates, the Conversion between polar and Euclidean coordinate systems can be used to give a local coordinate system near those points. Also, along the 180 degree longitude line, the ambiguity can be overcome by allowing longitudinal coordinates to take values between 0 and 360.

This example points out that local coordinates rarely apply to the surface as a whole, and that usually several different coordinate systems are needed to describe the surface globally. Moreover, on any surface, there are infinitely many different coordinate systems, because one can carry out infinitely many coordinate transformations starting from any one coordinate system.

The notion of direction that is used in the section "Open and closed surfaces"  is confusing and ambiguous. At the opening of the section, directions are described as follows:


 * Surfaces are said to have two directions, commonly called u and v. That is, any point on a surface can be described by a u and v coordinate pair.

Under this description, a direction seems to be identified with a coordinate function of a local coordinate system. But since there are infinitely many local coordinate systems, there are infinitely many directions. This makes the sentence


 * This means moving in any direction along the surface will cause an observer to hit the edge of the surface

meaningless because, in a local coordinate system, usually the observer will hit the edge of the open set W before every encountering the edge of the surface. The sentences


 * Surfaces closed in one direction include a cylinder, cone, and hemisphere. Depending on the direction of travel, an observer on the surface may hit a boundary on such a surface or travel forever

are meaningless for similar reasons.

I tried to understand these last two sentences using a different notion of direction, namely a continuous path on the surface. But under that interpretation, on a cylinder, cone, and hemisphere there are infinitely many directions in which the observer hits the edge, and infinitely many others in which the observer does not hit the edge. Also, the sentences


 * Open surfaces are not closed in either direction. This means moving in any direction along the surface will cause an observer to hit the edge of the surface.

are either false or meaningless under either of the two possible interpretations of directions.

(My browser is misbehaving and not letting me sign. I am Mosher, who wrote the suggestions at the top of the page)

Reorg/rewrite on 9 Sep 2006
I just did a reorganization and rewriting of this thing, adding some stuff, removing some redundancy (two treatments of the classification), etc. I've tried to preserve all of the examples, although they are now spread throughout the discussion. Sorry if I've stepped on any toes. I'm not sure what to do about this:


 * For the nature of real surfaces see surface tension, surface chemistry, surface energy, roughness. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the surface of a snowflake, which has a great deal of fine structure, is to go beyond the simple mathematical definition.

I'll move some of it to Surface (disambiguation). I don't understand why the snowflake's surface would be illegitimate (except on an atomic level). Joshua Davis 06:43, 9 September 2006 (UTC)


 * I think there is a need for a applications section, where some of these could go. This article does need to be quite a general article covering all aspects. A snowflake surface is a surface formed as an energy minimisation process, so is technically a form of minimal surface. See for example Mathematical Existence of Crystal Growth with Gibbs-Thomson Curvature Effects. --Salix alba (talk) 09:25, 9 September 2006 (UTC)


 * Nice sphere picture. I agree that applications are desirable; that's why I threw in the blurb on aerodynamics (not very good, I know). Perhaps at some point we should move this to Surface (topology)?


 * In retrospect, singularities were coming up a lot, so I moved them up into the Definitions section so we could refer to them freely. I made your treatment of the implicit surfaces with singularities a little more vague, since it did not assume the function was analytic; I've left the specifically algebraic stuff to the algebraic geometry section. Let me know if you disagree. (There's still a slight problem in that zero sets of smooth functions with vanishing gradient can be very nasty, as I recall -- not anything we'd call a surface.) Joshua Davis 14:34, 9 September 2006 (UTC)

Fundamental polygons?
The treatment of fundamental polygons here (and on the page dedicated to the subject) seems meaningless.

For example, there is no definition of the mechanism where attaching sides with labels yields the indicated surface.


 * On a non-rigorous level, imagine that the polygons are rubber sheets. Sew or glue the matching sides together. At least in a couple of cases (torus, sphere) it is easy to convince yourself that you get the claimed surface. A rigorous proof would not be useful for Wikipedia, I think.

Is this a purely arbitrary notation? Or is there some reason for its use?


 * The choice of symbols (A, B, etc.) for the sides is arbitrary. All that matters is that we know which sides to match with which (i.e. A with A), and which direction to match them in (i.e. so that the arrows point in the same direction).

(The page dedicated to fundamental polygons seems to indicate that this notation has something to do with group isomorphisms, but without any examples or other basic orientation the discussion seems inaccessible to someone like me without a background in the topic.)

159.54.131.7 14:26, 2 October 2007 (UTC)


 * To understand this part of the article, you need to understand groups, group presentations, the fundamental group, and the Seifert-van Kampen theorem. This article cannot explain all of these points, but it does at least mention all of them. I think the detail you want should be provided by the fundamental polygon article, not this article. Joshua R. Davis 16:16, 2 October 2007 (UTC)

Gauss-Bonnet theorem and curvature
This aspect of the geometry of surfaces and its tie-up with Euler characteristic seems to be entirely missing from the article at the moment. This classical material can be found in standard textbooks, such as those of Barrett O'Neill and Singer & Thorpe. --Mathsci 20:24, 6 November 2007 (UTC)


 * Agreed; this is a serious omission at present. By the way, good work on your recent edits. Joshua R. Davis 23:10, 6 November 2007 (UTC)

Previewing
Over the past three days there have been about 60 edits to this article from just two editors. They're good edits, but very close in time -- sometimes within a minute of each other. Surely this is an opportunity for the preview button? Joshua R. Davis 16:36, 14 November 2007 (UTC)
 * The preview button is being used, but this is not so easy to write. At the moment I am working out how to put the Alexandrov inequality in completely elementary terms. I also had to write all the maths of the article on the Cauchy-Kowalevski theorem. M.H. is the consultant beautician for the added section. Please be patient: major additions are bound to require a lot of edits and reordering/corrections as the content evolves. --Mathsci 17:05, 14 November 2007 (UTC)

Okay. I don't mean to be so bossy. :-) Joshua R. Davis 18:52, 14 November 2007 (UTC)

Proposal to split
An article on differential geometry of surfaces would be one of the top-level articles within the subject of differential geometry, and indeed there are many links to this article. However, at the moment, the emphasis here is on the topological aspects, especially in the first half; geometry makes its first serious appearance in section 6. I think that it makes sense to split the article in two, one dealing with topological surfaces (sections 1–5) and the other devoted to differential geometry (sections 6 and 7). In my opinion, both topics deserve their own separate articles. Such a split would probably have minimal impact on the first part, but would allow for a more leisurely treatment of surfaces in R3. Any thoughts or objections? Arcfrk (talk) 02:53, 31 January 2008 (UTC)


 * I agree. Indeed, the entire differential geometry of surfaces stuff was added pretty recently, mostly by one author. I've been thinking it should be split off for a while. Joshua R. Davis (talk) 04:06, 31 January 2008 (UTC)


 * Agreed. -- Fropuff (talk) 04:14, 31 January 2008 (UTC)


 * Agree, it should be dubbed two-manifold also (at least as a redirect :))--kiddo (talk) 06:01, 31 January 2008 (UTC)

Done. Arcfrk (talk) 21:22, 31 January 2008 (UTC)
 * As the "one author" referred to above, may I say that I wished for a split, but that I think that it was not very cleanly done. The current section on the geometry of surfaces is a bit uninformative: it should contain a short summary of the contributions of Gauss (just a short non-technical paragraph). This material, as before, is completely lacking from the article. Arfrck removed it without replacing it by even a hint as to what it contained. (The lead that he added to the new article contained a lot of vaguely decribed material not reflected in the subject or main body of the article: if he wishes to write a section on applications in partial differential equations specific to real surfaces, he should do so and then afterwards add comments in the lead.) There is now a new CUP undergraduate text by Pelham Wilson on Curved Spaces, concentrating on surfaces from this point of view. I propose to add a a very short summary of the differential geometry of surfaces to this article, adding as references Wilson's text along with the standard undergraduate texts by Singer & Thorpe, do Carmo and O'Neill. Mathsci (talk) 07:03, 2 February 2008 (UTC)


 * Yes, the split-off article needs continued work, but it is already a good start (because the section was good). If we all agree that the split was a good idea, then let's discuss the improvement of that article at its talk page.


 * I'm fine with adding a bit more to the Geometry of surfaces section of this article. In particular, I vote for mention of the Gauss-Bonnet theorem, since it's a surprising result that connects the topology of surfaces (the topic of this article) to the geometry. We could even precede it with a brief exposition of metrics and Gaussian curvature. But we should leave all details and elaboration (including, say, Theorema Egregium) to the split-off article. Joshua R. Davis (talk) 14:39, 2 February 2008 (UTC)


 * The top of the page states that this article is written from the point of view of topology (how is that for a hint? Differential geometric perspective can be mentioned in the section on surfaces in geometry. Gauss–Bonnet theorem can be summarized in a paragraph or two, with a "main" link to the corresponding article. Arcfrk (talk) 22:22, 3 February 2008 (UTC)

Orientability
I think that the first paragraph contains quite an innacurate sentence:

"On the other hand, there are surfaces which cannot be embedded in three-dimensional Euclidean space without introducing singularities or intersecting itself — these are the unorientable surfaces."

I do not agree. What about a Mobius band, considered as an open surface, or possibly as a surface with boundary?. It certainly isn't orientable and certainly can be embedded into three-dimensional Euclidean space, as you should be able to convince yourself by constructing one yourself in what we conceive as 3-dimensional space... —Preceding unsigned comment added by Jjw19 (talk • contribs) 20:21, 25 March 2010 (UTC)


 * [I moved your comment to the end of the page.] You are certainly correct. I'll fix it. In the future, feel free to fix such things yourself. Mgnbar (talk) 18:35, 8 April 2010 (UTC)

Ok, thanks. Just to note, it may be nicer to still give the generalised statement about the Klein bottle and say that all the closed unorietable surfaces cannot be embedded into three-dimensional Euclidean space; this is certainly true.

81.148.30.150 (talk) 21:30, 7 May 2010 (UTC)

Merging closed surface
I'd like to merge the closed surface stub and the section here on the classification theorem. Should there be a separate article for closed surfaces or should closed surfaces link to here? Richard Thomas (talk) 04:28, 19 July 2010 (UTC)


 * I agree with you; that stub should be merged into this article. Mgnbar (talk) 12:43, 19 July 2010 (UTC)

Reader Suggestion about difference between En and Rn
The article seems to switch arbitrarily between En and Rn to describe Euclidean n-space. It would be less confusing to stick with one; I have never seen En used, and suggest sticking with the standard Rn. HawaiianEarring (talk) 14:24, 12 April 2011 (UTC)

Name
Maybe this article should be called "Real surface", and "real dimension two" should be put in the first line? — Preceding unsigned comment added by 78.239.179.184 (talk) 19:40, 8 November 2011 (UTC)


 * Real surfaces are by far the most common kind of mathematical surfaces that readers are going to seek. On the other hand, it could be argued that mathematics doesn't own the concept of surface at all. For example, there's surface science and even surface art. My long-term hope is that Surface will be a disambiguation page. When that happens, maybe "Real surface" will be the ideal title for this article. Mgnbar (talk) 00:17, 9 November 2011 (UTC)

Should this article be renamed to Surface (topology), for the purpose of disambiguation?
I think this article's title may be misleading to some Wikipedia editors, since many links to surface may not be about surfaces in the context of topology. I think this article should be renamed so that Surface will redirect to Surface (disambiguation), since many articles about other types of surfaces would otherwise link to this page misleadingly. Jarble (talk) 15:25, 22 March 2013 (UTC)

Requested move 18 August 2015

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section. 

The result of the move request was: no consensus. Despite the page view stats, there is a strong argument that the topology article remains the primary topic due to long-term significance. When we have a situation like this, where the two primary topic criteria conflict with each, there really needs to be a clear numerical majority for there to be a consensus. Jenks24 (talk) 18:27, 27 August 2015 (UTC)

– See the section above; I was actually just assumed that this page must be the primary topic until recently. But as part of checking view stats in another discussion I was involved in I realized that the view stats between this page and Microsoft Surface show many more views at Microsoft Surface compared to this article. When I continue sampling randomly back in recent time I continue to find much higher view stats for the tablet, example 52957 vs 8797 in July 2013. While I still don't think that makes this the tablet family the primary topic, I'm not sure that this page is either. Surface (geometry) which already redirects here might work as well for a parallel with Edge (geometry). An alternate to moving Surface (disambiguation) would just be to redirect surface there if this page is moved. PaleAqua (talk) 02:08, 18 August 2015 (UTC)
 * Surface → Surface (topology)
 * Surface (disambiguation) → Surface
 * Clarification: Since I am getting misunderstood. I am suggesting the surface article should either be a redirect to the current disambiguation page or a general broad concept article is created at this location, while the this page should be moved to Surface (topology) ( or alternatively Surface (geometry) or the like. ). PaleAqua (talk) 03:26, 21 August 2015 (UTC)
 * Oppose: as with any other IT product, the brand name will likely fade away in a few years. fgnievinski (talk) 05:40, 18 August 2015 (UTC)
 * That would have been my original thought as well, especially given my math background; and actually argued as such in a another related move discussion. But for the last several years this article has appeared to be getting less than 20% of the total hits of the other surface articles. And looking back before the launch of the tablet say 2011 was getting less than 50% when I spot checked a couple of articles. Not saying that the tablet is primary, it might have usage but not necessary long-term significance, while this article has long-term significance but not usage. PaleAqua (talk) 07:51, 18 August 2015 (UTC)


 * Comment fgnievinski, PaleAqua, While admitting to knowing little of the maths involve it seems immediately apparent, at least to me, that the content presented in the article has significant difference to any basic dictionary definition of the subject. The central and I would argue primary definition of "surface" is presented as a simple definition of "The outside part or uppermost layer of something:".  http://www.oxforddictionaries.com/definition/english/surface
 * In contrast to this, the article dives straight into maths. I would suggest that the article should either begin with more of a dictionary compatible introduction or that the article should be moved to a disambiguated name space.  Perhaps editors who understand the subject more intrinsically can work out a viable solution.  GregKaye 09:25, 18 August 2015 (UTC)
 * I agree with your assessment, though I disagree with the solution, as the present article is beyond rescue as a general introduction to the subject; so after the current nomination is closed, I propose the article be renamed to Surface (mathematics), and a stub WP:Broad-concept article be left behind at Surface. fgnievinski (talk) 04:00, 19 August 2015 (UTC)
 * I agree with you both: it is embarrassing that the dictionary definition of surface is not addressed anywhere on Wikipedia (not even in the dab page!); yet I do not see the need to dilute the coherent content here by including very loosely related references to physical surfaces of objects. An eventual move of this article to surface (mathematics) appears to be a good compromise. Arcfrk (talk) 07:29, 20 August 2015 (UTC)


 * Oppose This is pretty much the most standard definition of surface possible. That said the lede of the article needs a major rewrite and needs to better reflect views in chemistry and physics. There is no need to have the lede so crammed with jargon. This is a high level article, and just an example sentence that is especially severe –
 * "The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3, such as a sphere." This does not help explain surfaces to the general audience! -- CFCF  🍌 (email) 10:26, 18 August 2015 (UTC)


 * I disagree pretty much exactly because of the many of the reasons you give. The page at surface should cover the the chemistry and physics views in addition to the general case, while this article should stay focused on the mathematical view. PaleAqua (talk) 14:34, 19 August 2015 (UTC)
 * Oppose, per mathematical definition that this would be the common name, or if it's not I'll eat my hat (which already is in the microwave). And per CFCF for reasoning and additional editing. Randy Kryn 13:10, 18 August 2015 (UTC)


 * Support: This article has always been about the topological-geometric notion of surface, not about surfaces in general science or society. It is reasonable that its name should be changed accordingly. Further, once readers and editors recognize that this article has that focus, then criticisms such as "needs to better reflect views in chemistry and physics" become irrelevant. In other words, changing the title is much easier than doing a massive rewrite to treat a much wider topic. Mgnbar (talk) 17:43, 18 August 2015 (UTC)
 * Oppose per flawed argument used. Per WP:PRIMARYTOPIC, "a topic is primary for a term, with respect to long-term significance, if it has substantially greater enduring notability and educational value than any other topic associated with that term", and article traffic has nothing to do with it. Surface is per common sense the unarguable primary topic. FoCuS contribs ;  talk to me!  20:15, 18 August 2015 (UTC)
 * Oppose. View stats and page hits toward Microsoft's product is heavily skewed due to recentism. A consumer product that is only less than 5 years old (and could be listed under "What's hot and trending since 2010") should not offset an academic topic/definition that has centuries-long term enduring notability and educational value. Zzyzx11 (talk) 09:28, 19 August 2015 (UTC)
 * Oppose. Pure WP:RECENTISM. Clear primary topic. -- Necrothesp (talk) 13:05, 19 August 2015 (UTC)


 * Comment: (Sorry if it's against protocol to vote and later comment.) Many of these votes read as if the proposal is to make Wikipedia's Surface article be about the Microsoft Surface product. That is not the proposal. In fact, the original rationale explicitly rejects that change. Mgnbar (talk) 13:36, 19 August 2015 (UTC)
 * Exactly. I'm not arguing that the tablet be the primary topic by any stretch. Just the the general concept of a surface is more likely to be the primary topic than the mathematical definition. Thus a disambiguation page or an general overview that covers the term more generally should probably be at the title surface. PaleAqua (talk) 14:19, 19 August 2015 (UTC)
 * It's also why I spot checked 2011 page view stats ( the tablet was launched in 2012 ). Yes there was a drop after the tablet launched, but even before hand didn't appear to cross the 50% threshold. I stopped counting page views of other terms when I got to an equal number of views, so don't take the 50% as exact and it was only a spot check but was enough to show my original previous assumptions ( that the topological term was clearly the primary ) were wrong to me. PaleAqua (talk) 14:31, 19 August 2015 (UTC)
 * Gee, that was a poorly worded proposal; I suggest the current one is closed and a new one is reopened. fgnievinski (talk) 02:33, 21 August 2015 (UTC)
 * Comment Concerning this debate in the context of the page's relationship to Microsoft Surface, I strongly oppose per fgnievinski. Page view statistics do not necessarily correspond to the "primariness" of a topic. For example, "country music" (which is often called simply "country") has higher page view statistics than "country", but obviously it is not the primary topic for the word "country". But in the context of the mathematically-specific, jargon-heavy definition of surface as juxtaposed with the broader but related meaning in common usage, I support . For the present, I think the issue is not solved by moving the page, but can only be solved by someone writing a new article on surface more generally or else expanding the mathematically centered current article to include differing but related understandings of surface. — the Man in Question (in question)  00:42, 25 August 2015 (UTC)
 * Oppose. A disambiguation link in the hatnote is sufficient.--Froglich (talk) 19:05, 26 August 2015 (UTC)


 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Redundant intro sentence
I think this example confuses rather than explains the notion of local compactness.

"For example, the surface of the Earth is (ideally) a two-dimensional sphere, and latitude and longitude provide two-dimensional coordinates on it (except at the poles and along the 180th meridian)."


 * It's not about local compactness. It's about how the surface of the Earth is a two-dimensional manifold, rather than a three-dimensional one, as many non-mathematical readers might guess.
 * The sentence would read more smoothly without the parenthetical disclaimers. But they are important to keeping the sentence correct. You are welcome to propose a re-wording here on the talk page. Mgnbar (talk) 14:45, 15 October 2015 (UTC)

New intro
I think that the new introduction is a step backwards. This article, as it is currently written, is a decent introduction to surfaces in topology. Trying to cover other notions of surfaces will cause a lot of incoherence in the treatment.

For example, the introduction gives an intuitive explanation of what "two-dimensional" means. But this intuition makes sense only for manifolds --- not surfaces with crossings. So the recent addition of singular surfaces to the introduction now requires alteration or removal of this intuitive explanation.

The long-term solution, I think, is to move this article to Surface (topology), and write a new Surface article that treats all aspects of surfaces, in a matter befitting a general encyclopedia, as has often been suggested on this talk page. Mgnbar (talk) 13:37, 5 April 2016 (UTC)


 * Long term solution is not acceptable without a short term clarification: The article has many readers who are not topologists (and even not mathematicians), and is linked from many non-mathematical articles (including Humanities). It is thus important to make the lead as understandable as possible for the layman, and to cover, at least in the lead, the general notion of surface. Moreover the previous lead was wrong, because, for everybody, including most topologists, a conical surface is a surface, although it is not a manifold (if I am wrong, references must be provided for the assertion "a conical surface is not a surface"). Thus, although the new lead may certainly be improved, it is certainly more "decent" than the previous one.
 * You assert that "two-dimensional property" make sense only for manifolds. This is wrong, because I have written that this property is true only around almost every point. D.Lazard (talk) 14:35, 5 April 2016 (UTC)


 * Your points are reasonable. How much work would it be, to write a basic (better than stub-level) article about general mathematical surfaces? If we could do that quickly, then I would love to move this article to Surface (topology) or Two-dimensional manifold. Because Wikipedia needs such an article, and it's already written, but editors and readers keep complaining (justifiably) about the name, which is currently Surface.
 * About the manifold requirement: I have never seen conical surfaces discussed in topology, unless you count wedge sums of disks. Regardless of which of us is right, it would be good to substantiate Wikipedia's use of the term surface. Here are some references to corroborate the manifold requirement, although I admit that three of them are imperfect.
 * Bredon, Topology and Geometry, p. 163: "[Spheres, real projective planes, and Klein bottles with n handles attached] form a complete list without repetitions of compact surfaces up to homeomorphism."
 * Massey, A Basic Course in Algebraic Topology, p. 5: "To save words, from now on we shall refer to a connected 2-manifold as a surface."
 * Do Carmo, Differential Geometry of Curves and Surfaces, p. 425: "An abstract surface (differentiable manifold of dimension 2) is..."
 * Guillemin and Pollack, Differential Topology, p. 195: "We apply these generalities to study the geometry of hypersurfaces, k-dimensional submanifolds of Rk + 1."
 * Furthermore, the text that you wrote in the intro supports this view: "Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not." Mgnbar (talk) 18:12, 5 April 2016 (UTC)

Lacking broad-concept article
Thinking again about the discussion of the previous section, it appears that the title of the article must correspond to a Broad-concept article, and that the body is restricted to Topology of surfaces (similarly we have Differential geometry of surfaces). We have thus to rename the article, and to write the broad-concept article. This article, renamed Surface (topology), will be sub-articles of the broad-concept one, as well as Surface (differential geometry) and Surface (algebraic geometry) (these are redirects that I have just created for harmonizing titles and making research easier).

From this point of view, the new lead is not adapted to the body. It is also not convenient for the broad concept article. A witness of this is that the most popular example of surface is the graph of a bivariate continuous function, and that this example is not cited in the article, although it is fundamentally the basis of the concept of a two-dimensional manifold.

Therefore, I'll move Surface to Surface (topology) (through a move request), and start to write the broad-concept article, which will be a stub at the beginning. D.Lazard (talk) 09:41, 6 April 2016 (UTC)

Requested move 06 April 2016

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section. 

The result of the move request was: Moving as requested, and moving disambig page to . Having a broad concept article is probably a good idea, but there isn't one written yet, so the disambig page is the best thing we have. And there's consensus that neither geometry nor topology is the primary topic in its own right. &mdash; Amakuru (talk) 21:12, 25 April 2016 (UTC)

Surface → Surface (topology) – Making place for a broad-concept article, which is obviously lacking – D.Lazard (talk) 09:51, 6 April 2016 (UTC)
 * This is a contested technical request (permalink). Anthony Appleyard (talk) 14:48, 6 April 2016 (UTC)


 * Contesting. Discussion is needed to determine if this is not the WP:PRIMARYTOPIC. SST flyer 14:05, 6 April 2016 (UTC)
 * Support. We have had many objections on this talk page over the years, that the article specifically treats topology of surfaces, rather than surfaces in general. Although a requested move last August was defeated, it seems that many of the editors misunderstood the proposal (thinking that it proposed Microsoft Surface as primary). Regarding the primaryness, surface is a common word probably employed by all speakers of English, while this article is currently written about surfaces in topology, a subject of which most speakers of English have probably never heard. For what it's worth, Google reports 898,000,000 hits for surface, but only 24,000,000 hits for surface topology. Even within pure mathematics, surface is used heavily with other meanings: Riemann surface, complex surface, etc. Mgnbar (talk) 20:11, 6 April 2016 (UTC)
 * Support Per my arguments in the previous RM and arguments above by Mgnbar. In addition to the Google report above, my look at page views prior to the launch of the Surface tablet also showed that the general meaning was primary to the topological term. Also consider the following ngrams . Which shows the total usage of the surface compared to usages that do not mention topology, mathematics, manifold, or math. This clearly shows that topology or mathematical usage of the term is not the primary meaning. PaleAqua (talk) 02:18, 7 April 2016 (UTC)
 * BTW the guideline Broad-concept articles as well the section of the disambiguation guideline are the main reason I believe the surface article should be a broad-concept article. Having the disambiguation page located at or redirected from surface would be another option but a broad-concept article is preferable. In addition to the n-grams, page view stats, search counts, even looking at dictionary definitions do not have the topological meaning as primary. For example . PaleAqua (talk) 08:27, 8 April 2016 (UTC)
 * Comment As the author of the move request, I am in favor of the move. As a consensus seems to exist, I want to warn the administrator who will do the move to not leave a redirect, and, instead, to move to Surface (If the move is done without leaving a redirect, I can do this second move myself). In fact,  was a redirect here, that I use for starting writing the broad-concept article. For the moment it is a two lines stub, that contains also a hatnote aimed for not breaking redirects. D.Lazard (talk) 08:48, 7 April 2016 (UTC)
 * Comment it is much too soon to see a consensus. Only two opinions lodged, and less than 7-days of discussion. What's wrong with you just overwriting the redirect? A redirect should exist until such a time as a new article is created, if this move goes through, or you can just convert this into a multimove request, since that is what you're implying, since you have a stub. -- 70.51.45.100 (talk) 05:48, 8 April 2016 (UTC)
 * I don't think that Surface (geometry) is the primary topic either. Surface should be a broad concept article that covers the general meaning of the term not just the mathematical definitions. If anything the disambiguation page would fit better here in the short term. PaleAqua (talk) 02:38, 9 April 2016 (UTC)
 * NOTE This seems to be an improperly formatted multimove, per the extra notes that the nominator made. -- 70.51.45.100 (talk) 05:48, 8 April 2016 (UTC)
 * Oppose until two things are resolved: 1. Surface (geometry) becomes more than a stub and 2. there is consensus about which is the primary topic. Otherwise this move is premature IMHO. --Bermicourt (talk) 07:37, 8 April 2016 (UTC)
 * Comment. The original proposal, for which I have voted, does not assert that Surface (geometry) should be primary. That topic is roughly as narrow as Surface (topology). Wikipedia is not a math encyclopedia. Surface should probably encompass non-math topics such as Surface science. Mgnbar (talk) 19:18, 8 April 2016 (UTC)


 *  Oppose, Support. There is a lot more to surfaces than geometry and topology. Xxanthippe (talk) 04:18, 9 April 2016 (UTC).
 * I don't understand why you are opposing as the reason you give is exactly why the topology article should be moved and surface should become a general broad concept article that covers more than just geometry and topology. PaleAqua (talk) 04:45, 9 April 2016 (UTC)
 * Understood. Xxanthippe (talk) 05:44, 9 April 2016 (UTC).


 * Support but ONLY after the new article can be checked out. Torquemama007 (talk) 15:34, 11 April 2016 (UTC)
 * This should be moved to Surface (mathematics) or to Surface (geometry), not to Surface (topology), but only after there is a reasonable draft to replace it at Surface. Michael Hardy (talk) 21:58, 12 April 2016 (UTC)
 * Why not move to Surface (topology), if the article is primarily about the topological properties of surfaces (rather than the geometric ones)? There is plenty of material for separate articles Surface (topology), Surface (geometry), Complex surface, etc. Surface (mathematics) is too big a topic to cover everything in a single article. Mgnbar (talk) 22:18, 12 April 2016 (UTC)


 * Strong Support; this should be disambiguated, and the dab page moved to "Surface".  InsertCleverPhraseHere  08:45, 21 April 2016 (UTC)


 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Broad-concept article
Above requested move is intended for making place for a broad-concept article. As said in a comment in the requested move discussion, I have started writing the broad-concept article in Surface (geometry). The rationale for this is that any surface is a geometrical object and thus "surface (geometry)" is a pleonasm, and therefore Surface (geometry) should be redirected to the broad-concept article, whichever is its final name.

Normally the discussion on the broad-concept article should be in its talk page. However, Surface (geometry) was, until recently, a redirect, and thus has few watchers. Also the content of the broad-concept article is strongly related with above move request discussion. Therefore, I starts this discussion here, even if at some time the discussion should move there.

In Surface (geometry), I have written a lead for the broad-concept article. For being as elementary as possible, only surfaces in $R^{3}$ are considered in this lead. Surfaces in higher dimensional spaces and abstract surfaces are just mentioned. The reason is that their treatment is quite different for implicitw surfaces (i.e. algebraic surfaces) and for locally parametric surfaces (i.e. manifolds). Therefore these questions should not be detailed in the lead, only in the relevant sections. For the same reason, surfaces over other fields are not mentioned (complex manifolds and algebraic surfaces over any field). By the way, the target of is another witness of the need of this broad-context article.

The difficulty in writing the lead is that the different classes of surfaces have a wide intersection. For example the hyperbolic paraboloid $z = xy$ is simultaneously the graph of a function, an implicit surface, a parametric surface, an algebraic surface, a manifold, and a ruled surface. I have tried to clarify, for the layman, this variety of related concepts.

D.Lazard (talk) 11:31, 10 April 2016 (UTC)


 * And what do you think of the concern, voiced by multiple editors in the move request discussion, that Surface should encompass non-mathematical topics? The argument that "any surface is a geometrical object" does not convince me. Real surfaces might not be well-modeled by the mathematical concept, for example if their microscopic structure is non-smooth, and this non-smoothness is important (Supercapacitor, the boundary between phases at the critical point of a phase transition, etc.). Painters, physicists, and engineers might have different priorities for Surface than do mathematicians. Mgnbar (talk) 14:27, 10 April 2016 (UTC)

The proposed article titled Surface (geometry) misses the point with an awesome degree of completeness. The present article titled Surface could appropriately be renamed Surface (geometry). If an article on chemistry mentions that certain molecules cling to surfaces, that's not about surfaces in geometry. There is no pleonasm in "Surface (geometry)"; there are other senses of the word than that in geometry. Michael Hardy (talk) 22:12, 12 April 2016 (UTC)


 * I have now written the skeleton (that is the headings of the main sections) of the broad-concept article (presently at Surface (geometry)). For having more than a stub, there remains a lot of work, mainly:
 * Filling the empty sections
 * Providing sources
 * Providing images for the examples
 * Some of the empty sections are, for me, difficult to write. These are the sections that I have tagged expand section.
 * I hope that, with this skeleton, it would make clearer the kind of broad-concept article that I claimed to be lacking. D.Lazard (talk) 10:18, 19 April 2016 (UTC)


 * Suggestions: fgnievinski (talk) 16:48, 19 April 2016 (UTC)
 * # rename Surface to Surface (topology)
 * # rename Surface (geometry) to Surface (mathematics)
 * # redirect Surface (physics) to Interface (physics)
 * # redirect Surface to Surface (disambiguation), then say that it may refer to either Surface (mathematics) or Surface (physics).
 * Completely agree with the above suggested moves and redirects. PaleAqua (talk) 17:57, 19 April 2016 (UTC)


 * I completely agree with the two first items. These are moves that can only made by an administrator. The first one is the object of the above move request.
 * Third item seems problematic, as there are surfaces in physics that are not interfaces, for example wavefronts.
 * Fourth item deserves further discussion, because there are two possibilities:
 * Redirecting Surface to Surface (disambiguation). It implies that there is no WP:primary topic for "Surface". This redirect would imply to edit a large number of articles that are presently linked to Surface, and for which the natural link would be Surface (mathematics).
 * Redirecting Surface to Surface (mathematics). This implies that the mathematical concept is the primary topic. This is suggested by a quick look to the links to Surface. With this redirect, only those articles, which are already mislinked, would need to be edited.
 * It should be noted that a similar problem could occur with "curve", and that Curve is not a dab page, but an article about the mathematical concept. D.Lazard (talk) 09:44, 20 April 2016 (UTC)
 * I've requested the first two, uncontroversial, moves. I withdraw the third suggestion and have instead inserted a hatnote in Surface indicating the redirect retargeted the redirect so as to point to Surface (mathematics) instead; it'd be useful to split Surface (physics) in a DAB or BCP of its own if we find more related concepts than the two indicated. As for the fourth suggestion, I'm okay with it redirecting to Surface (mathematics) instead of Surface (topology). fgnievinski (talk) 14:28, 25 April 2016 (UTC)


 * I have moved Surface (geometry) to Surface (mathematics), but the rest of this matter seems to need more discussion. Page Surface is somewhat of a disambig page because it starts with a description of several meanings of the word "surface". Anthony Appleyard (talk) 14:23, 25 April 2016 (UTC)
 * Thanks. I've moved the non-topological aspects from the lead into the section Surface. If necessary, that portion could be further moved into the article Surface (mathematics), after Surface is renamed to Surface (topology). fgnievinski (talk) 14:33, 25 April 2016 (UTC)
 * Hi all, I've moved the page Surface to Surface (topology), and I've suggested that the disambig page be moved to Surface, unless you guys agree that the other article should be moved there. Let me know if so. I haven't done any move yet into the Surface location, because the incoming links need to be sorted first. It still redirects to Surface (topology) for now. Ping me once that's sorted out, and a decision is made what to move into the primary space. &mdash; Amakuru (talk) 21:21, 25 April 2016 (UTC)


 * I have moved Surface (disambiguation) to Surface. Anthony Appleyard (talk) 21:48, 25 April 2016 (UTC)
 * I've created Draft:Surface and Draft:Surface (disambiguation) which I offer as replacements for their respective mainspace articles. If accepted, sections Surface (topology), Surface (mathematics), and Surface (mathematics) would be deleted. 04:07, 26 April 2016 (UTC)fgnievinski (talk)
 * We can probably work on fixing up the disambiguation page in main space, it can always be moved aside once Draft:Surface is ready. We also probably need to check the pages that link to surface and change the links so that actually point to the right concept. See Talk:Surface. At the Draft:Surface page I don't think we need to cover the tablet, but probably do need to cover surface in terms of computer graphics. PaleAqua (talk) 04:33, 26 April 2016 (UTC)

After moves done by and, the drafts created by , and recent edits, we have two versions for a dab page for Surface, and two versions for a broad-concept article (Surface (mathematics) and Draft:Surface). IMO, the remaining questions are:
 * Should Surface be a broad-concept article or a dab page? From the last posts, it seems that we are near to a consensus for a broad-concept article. (Older posts are difficult to take into account, as the situation has changed).
 * What should be the broad-concept article? I understand the creation of Draft:Surface as the idea of keeping two different broad-concept articles, a general one, and a mathematical one. My opinion is that it is not a good idea, because of the strong relationship between the mathematical concept of surface and the use of this concept in other scientific area. For physical sciences, this is clear from the present version of . For computer science, although the corresponding sections are still empty, it is clear that the main article is about the computer representation of the mathematical concept. For this reason, I oppose having two different broad-concept articles, and I oppose deleting Surface (mathematics), and Surface (mathematics), and I support moving Surface (mathematics) to Surface.
 * What should be the content of the broad-concept article? IMO, because of the various concepts of surfaces, and the very strong relationships between them, the main objective of the broad-concept article should be to clarify these relationships. This would be necessary for non-expert mathematician as well for non-mathematicians (most surfaces encountered in physical science are mathematical surfaces), and this is a further reason for having a single broad-concept article. This what I have tried to do in the (too long) lead of Surface (mathematics). This is also the reason for emphasizing, in the already written sections, on relationships between concepts, rather than summarizing main articles. D.Lazard (talk) 16:16, 26 April 2016 (UTC)
 * Surfaces in nature precede their mathematical formalization; so, no, surface as a mathematical object is not the main concept; the concrete object's property is. Attempting to squeeze Surface (mathematics) into a BCA on Surfaces will either not do justice to the richness in the math concepts or give undue weight to the math jargon. That's why two BCA are justified in this case. There's just so much to say about the topological, algebraic, differential, and fractal aspects. The previous main article, now residing at Surface (topology), lost touch with reality so much that it is only illustrated with abstract cartoons, there are no pictures or photos at all. fgnievinski (talk) 20:25, 26 April 2016 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified one external link on Surface (topology). Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20100612090500/http://new.math.uiuc.edu/zipproof/zipproof.pdf to http://new.math.uiuc.edu/zipproof/zipproof.pdf

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Cheers.— InternetArchiveBot  (Report bug) 05:53, 2 December 2017 (UTC)

complexification of non-metrizable manifoldd
In the article it is stated that 'the existence of the Prüfer surface shows that there exist two-dimensional complex manifolds (which are necessarily 4-dimensional real manifolds) with no countable base. (This is because any n-real-dimensional real-analytic manifold Q can be extended to an n-complex-dimensional complex manifold W that contains Q as a real-analytic submanifold.)'

The statement in parentheses is false. The Long Line carries a real analytic structure but it is not a submanifold of a complex-1-dimensional manifold because that would have to be a nonmetrizable Riemann surface which does not exist by Rado's theorem.

The reason that there is complex-2-dimensional analog of the Prüfer manifold is that it is clear from the construction that you can construct the Prüfer manifold and replace all real numbers by complec numbers.

Am I right or am I missing something?

--2A00:1398:200:202:6146:6249:BF2B:816 (talk) 09:29, 10 September 2018 (UTC)

Closed Surfaces and Surfaces with Boundary
I see a section about surfaces with boundary (6.3) within a section about closed surfaces (6). This suggests the sections need rearranging. Karl (talk) 12:42, 26 May 2021 (UTC)
 * Good point. I did a bunch of reorganizing. There is still a lot of polishing to do. Mgnbar (talk) 17:39, 26 May 2021 (UTC)

Why does hypersurface not belong on either this page, or the other "surface" page?
I have reverted in this article and in Surface (mathematics) a sentence mentioning hypersurfaces as examples of manifolds, together with curves and surfaces. This is wrong as many hypersurfaces are not manifolds. Moreover, even if it were mathematically correct, the sentence would be confusing, as the reason of adding the sentence (linking to Hypersurface) is hidden in a sequence of examples of a topic that is not the subject of the article.

I have added a link to Hypersurface in D.Lazard (talk) 07:57, 28 September 2021 (UTC)

Closed surfaces enclosing volume
Fgnievinski, I understand the point of your edit now. The wording is more confusing than it needs to be. How about: "A surface embedded in three-dimensional space is closed if and only if it is the boundary of a solid?" Mgnbar (talk) 02:02, 10 November 2021 (UTC)

"2-space" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=2-space&redirect=no 2-space] has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached. fgnievinski (talk) 04:51, 7 December 2023 (UTC)