Talk:Surface

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[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)