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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)
Suggestions:
-- Mosher 14:41, 21 September 2005 (UTC)
I reverted the following sections, for resons below.
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."
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."
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.
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).
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."
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)
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:
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
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
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
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)
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:
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)
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.
Is this a purely arbitrary notation? Or is there some reason for its use?
(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)
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)
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)
Okay. I don't mean to be so bossy. :-) Joshua R. Davis 18:52, 14 November 2007 (UTC)
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)
Done. Arcfrk ( talk) 21:22, 31 January 2008 (UTC)
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)
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)
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)
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)
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)
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)
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 [1] and Microsoft Surface [2] 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)
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)."
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)
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)
The result of the move request was: Moving as requested, and moving disambig page to Surface. 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. — 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)
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 R3 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 complex surface 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)
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)
After moves done by Anthony Appleyard and Amakuru, the drafts created by Fgnievinski, 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:
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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)
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)
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 Surface (mathematics) § See also D.Lazard ( talk) 07:57, 28 September 2021 (UTC)
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)
The redirect
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
Wikipedia:Redirects for discussion/Log/2023 December 7 § 2-space until a consensus is reached.
fgnievinski (
talk) 04:51, 7 December 2023 (UTC)
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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)
Suggestions:
-- Mosher 14:41, 21 September 2005 (UTC)
I reverted the following sections, for resons below.
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."
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."
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.
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).
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."
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)
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:
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
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
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
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)
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:
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)
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.
Is this a purely arbitrary notation? Or is there some reason for its use?
(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)
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)
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)
Okay. I don't mean to be so bossy. :-) Joshua R. Davis 18:52, 14 November 2007 (UTC)
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)
Done. Arcfrk ( talk) 21:22, 31 January 2008 (UTC)
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)
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)
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)
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)
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)
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)
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 [1] and Microsoft Surface [2] 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)
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)."
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)
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)
The result of the move request was: Moving as requested, and moving disambig page to Surface. 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. — 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)
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 R3 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 complex surface 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)
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)
After moves done by Anthony Appleyard and Amakuru, the drafts created by Fgnievinski, 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:
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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)
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)
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 Surface (mathematics) § See also D.Lazard ( talk) 07:57, 28 September 2021 (UTC)
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)
The redirect
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
Wikipedia:Redirects for discussion/Log/2023 December 7 § 2-space until a consensus is reached.
fgnievinski (
talk) 04:51, 7 December 2023 (UTC)