Talk:Manifold/Archive 4

message: Well, manifold is no longer a featured article candidate. It failed with 7 votes in oppositions and no votes in support. Archive of old debate is Wikipedia:Featured article candidates/Manifold/archive1. 
The objections include
* informal and unencyclopedic tone
* uses first and second persons, diagrams inadequately explained
* a lot of complex mathematical terminology is not even linked to, never mind explained.
* Inline citations
Not all bad though
* The information seems to be excellent, very probably by far the best description available anywhere on the web
so we are making progress, and do have one of the best presentations. Some other critiques, as well from people who have contributed to this Talk page.
I've now created a second archive Manifold/old2 which is an archive of the version of 14:31, 1 February 2006 by Bobblewik which is the last semi-mature version (with dates de wikified) 
i.e. this version .
We now need to reconsider if we wish to work towards another version for featured article status. Its worth reviewing the criteria for featured article:
----
A featured article should have the following attributes:
# It should exemplify our very best work, representing Wikipedia's unique qualities on the Internet.
# It should be well written, comprehensive, factually accurate, neutral, and stable. Read Great writing and The perfect article to see how high the standards are set. In this respect:
#*(a) "well written" means that the prose is compelling, even brilliant;
#*(b) "comprehensive" means that an article covers the topic in its entirety, and does not neglect any major facts or details; 
#*(c) "factually accurate" includes the supporting of facts with specific evidence and external citations (see Wikipedia:Verifiability); these include a "References" section where the references are set out, enhanced by the appropriate use of inline citations (see Wikipedia:Cite sources);
#*(d) "neutral" means that an article is uncontroversial in its neutrality and factual accuracy (see Wikipedia:Neutral point of view); and
#*(e) "stable" means that an article does not change significantly from day to day (apart from improvements in response to reviewers' comments) and is not the subject of ongoing edit wars;
# It should comply with the standards set out in the style manual and relevant WikiProjects. These include having:
#*(a) a concise lead section that summarizes the entire topic and prepares the reader for the higher level of detail in the subsequent sections;
#*(b) a proper system of hierarchical headings; and
#*(c) a substantial, but not overwhelmingly large, table of contents (see Wikipedia:Section).
# It should have images where appropriate, with succinct captions and acceptable copyright status; however, including images is not a prerequisite for a featured article.
# It should be of appropriate length, staying tightly focused on the main topic without going into unnecessary detail; it should use summary style to cover sub-topics that are treated in greater detail in any "daughter" articles. 
----
As a sugestion I think we should look at the other mathematics articles which have featured article status:
Ackermann function, Algorithm, Eigenvalue, eigenvector and eigenspace, Carl Friedrich Gauss, Game theory, Infinite monkey theorem, Margin of error, Monty Hall problem, Blaise Pascal, Prisoner's dilemma, Regular polytope, Marian Rejewski, Trigonometric function.
Personally I quite like the Regular polytope one, good history, good visuals, good applications. --Salix alba (talk) 22:46, 2 February 2006 (UTC)
thread_title:  Failed Featured Article 

message: Thanks for catching the indentations; they were accidental, caused by moving material I had put on the talk page several days ago into the article.
I agree that the section on zeroes of a function needed to be rewritten, but I think the concept too important to eliminate entirely.  We also need more on analytic continuation. Rick Norwood 23:08, 2 February 2006 (UTC)
thread_title:  Comment for Pfafrich 

message: I've added a number of references.
I note that one of the negative votes re: featured article status said that mathematical words were used without references.  I'll check the article for that. Rick Norwood 23:23, 2 February 2006 (UTC)
thread_title:  references 

message: The history section looks good to me.  Any idea what more the person wants, who asked for an exansion?  I've removed the statement of the Poincarė conjecture as too technical, but left the link and the comment.
I do not understand how the last clause of the last sentence in the history section fits with what the rest of that sentence.  The clause in question reads:
:"and developed through differential geometry and Lie group theory."
I don't know enough about Lie groups to fix it.  I hope someone will tackle that job. Rick Norwood 23:37, 2 February 2006 (UTC)
replies: text: What's the problem? "The foundational aspects were clarified … and developed." Perhaps you want "further developed"? -- Jitse Niesen (talk) 00:01, 3 February 2006 (UTC)

text: Well, people did Riemannian geometry without an explicit chart concept; though those guys of course were perfectly adept at changing variables, as a local thing. Lie groups are a more interesting case, because the evidence is that Hadamard didn't really grok the global concept (and he was no slouch at all). This comes up, for example, in thinking about universal covers, where one chart at the origin is not enough. But I think there is a kind of clincher in black hole theory. Where early examples of models of general relativity had singularities, but these were not really noticed initially. Of course, they were noticed, as places at which the functions blew up in some sense. What you need is a concept like lines of longitude at the North Pole, which is one kind of singularity on a perfectly good manifold because the chart (say Mercator projection type) cannot be forced to cover a point; versus real breakdown of manifold structure. So I say around 1920 the chart-style definition was still not formalised enough. Whitney wrote down certain things in order to prove his embedding theorem. Also, around that time, the partition of unity concept was showm to apply to manifolds (paracompact or something). Charles Matthews 10:12, 3 February 2006 (UTC)
thread_title:  History section 

message: You have made your views known.  They were not supported.  A majority of those who commented on the application for Featured Article status complained that the article was incomprehensible.  The introduction you want is even more incomprehensible.  You suggest that your intro can be read by someone with an American high school education or the equivalent (an odd choice of prerequisites).  I am quite sure your introduction cannot be read by anyone who does not already know what a manifold is. Rick Norwood 14:06, 3 February 2006 (UTC)
replies: text: I'm quite aware that my introduction is badly worded, but yours is more so. We need something better than either of us has written, and such has not yet been forthcoming. I do not believe the article lead should be a list of examples of different dimensional manifolds, there is much scope for that in the examples section. As for my example being incomprehensible to high school students, I can't see how that can be as I myself understand it quite well without passing high school. Loom91 12:00, 4 February 2006 (UTC)
replies: text:  That may be strongly worded. But I strongly object to any box of prerequisites slapped on the top of the article or anywhere else. Appropriate wikilinks to related topics as the discussion moves along are more than enough to get the reader an idea of the background necessary for this article. Oleg Alexandrov (talk) 17:31, 4 February 2006 (UTC)

text: Then your objection should be made at the talk page of Wikipedia:Make technical articles accessible and not here. Perhaps people will be happier if the box was called recommended prerequisites? I should also make it clear that the box refers to the article in its currnt state rather than some future article understandable without a knowledge of precalculus, if such a thing is possible. Loom91 12:00, 4 February 2006 (UTC)
replies: text:  I raised the issue at Wikipedia talk:Make technical articles accessible. Oleg Alexandrov (talk) 17:39, 4 February 2006 (UTC)

text:  About the intro being incomprihensible to high school students. Every effort should be taken to make things more acessible. But please do not push that too far. Manifolds are studied seriously only in grad school, and there are good reasons for that. Oleg Alexandrov (talk) 19:18, 3 February 2006 (UTC)
thread_title:  Loom91 

message: I find Pfafrich's edit an acceptable compromise, modulo a few very minor changes.  I hope Loom91 will accept this. Rick Norwood 15:05, 4 February 2006 (UTC)
replies: text: It seems acceptable to me. I cleaned up the wording a bit to make the lead clearer (I hope), and fixed the weird spacing anomolies caused by the inline comments. In my opinion, making an article "accessible" does not imply a need to display explicit prerequisites. Pfafrich's edit is good in this regard. - Gauge 00:21, 5 February 2006 (UTC)
thread_title:  Pfafrich's edit 

message: Once again I must point out that the word "space" is a technical term for mathematicians, leading to misinterpretation by everyone else. But it is clear that all the careful debate and writing of the past is now to be discarded. And so the traditional parents' curse will befall the present editors: may you have children like yourselves. Or in this case, you can watch as the next wave of editors "improves" your version. This is not the way to improve Wikipedia. --KSmrqT 01:22, 5 February 2006 (UTC)
thread_title:  Intro use of "space" 

message: * JA: The number of hidden comment lines in the main article is getting ridiculous.  Could editors please try to carry out their inner polylogues on the discussion page?  Gratia in futuro, Jon Awbrey 04:06, 5 February 2006 (UTC)
replies: text: Good point.  I'll fix it. Rick Norwood 15:49, 5 February 2006 (UTC)
thread_title:  Excessive subtext 

message: Of course Precalculus is a part of Calculus, my point was that you don't need to know proper calculus to understand the beginning motivational example section, but you DO need Precalculus. I doubt many of those who studied elementary real analysis in high school retained memories of the wild logical structures after they passed their exams. What I mean to say that if you don't know calculus, don't worry you will get an elementary idea of what we are dealing with. But if you don't know Precalculus (functions, sets etc) then sorry, but you don't stand a chance of getting past the introduction. What're your objections? Loom91 08:29, 6 February 2006 (UTC)
replies: text: Precalc doesn't comprise a body of knowledge, but rather a level.  You say (functions, sets), but functions are studied in analysis, and sets are studied in set theory.  Anyone who studied enough of analysis and set theory does not need to study precalc which is not a field of mathematics (I say again).  Just because the US educational system spends a year dabbling and names that year "precalc" does not make it a universal requirement.  -lethe talk [ +] 09:41, 6 February 2006 (UTC)
replies: text: As far as I know, the word "precalculus" is American educational jargon, and has nothing to do with mathematics.  It is a way of covering up the failure of the American educational system.  Rick Norwood 13:39, 6 February 2006 (UTC)
replies: text:  ha! well put by the previous two comments! But don't you know, precalc is what you study after prealgebra, before you get to prenumbertheory and preriemanniangeometry. you definitely need a little preriemanniangeomerty to understand manifolds. MotherFunctor 07:14, 20 May 2006 (UTC)
thread_title:  Precalculus 

message: Here's another pot-shot. According to my understanding of the latest trends and fashions in math (and you know how fashion goes...), the idea of creating a manifold by "gluing together charts from an atlas" is "old-fashioned". The new-fashioned way of doing all this is an appeal to the algebra <--> geometry duality seen, for example, in varieties and ideals, viz, the shape of the space is determined by the coordinate ring (Hilbert's Nullstellensatz and all that). So the new tools are sheaves, and its the geometry of the sheaf that expresses the shape of the space. This is the main idea driving non-commutative geometry. Admittedly, this is all very high-brow and horridly abstract, but I've now seen a half dozen books that make a big sis-boom-bah about this in the preface. Since this is the modern way, shouldn't this article at least mention this? linas 02:46, 8 March 2006 (UTC)
replies: text: Sounds interesting. Could be a topic for a new article? Do need to take a bit of care with the difference between varieties and manifolds, not all varities are manifolds. Do you know any online references for this? (alas I don't have access to a good library). --Salix alba (talk) 03:00, 8 March 2006 (UTC)
replies: text: I did see a nice discussion on the algebra-geometry duality online somewhere, but I'm afraid I misplaced it. Certainly under-grad level books on algebraic geometry certainly focus on it in the limited context of "variety"; there seem to be "bigger" parallels which I don't understand. I'm not sure, but I think I was grepping either "quantum group" or "k-theory" when I found it. Possibly look for "commutative C*-algebra" in combination with these. And no matter what, I guarentee that the above searches will turn up a goodly amount of head-scratching stuff making intriguing or even wild claims. linas 04:10, 9 March 2006 (UTC)

text: Yeah, I think I meant to say scheme (mathematics) not sheaf. I certainly don't understand the big picture. And also, I do not mean to imply that there's some proof out there that says that the topological construction of atlases and charts is a subset of a grander algebraic approach. Maybe there is some-such theorem but I wouldn't know. I'm an amateur. linas 04:10, 9 March 2006 (UTC)
thread_title:  Algebra <-->geometry 

message: *So User:Anthony Appleyard moved Manifold to Manifold (topology). Probably a bit too WP:BOLD for such a well established article, without prior discussion. The names not quite right as manifolds are not just studied in topology Manifold (mathematics) might be better.  I've a mind to move it back. Also raises the question of 350 links to the article which now go to a redirect to a disambig page. These will all need fixing. --Salix alba (talk) 15:45, 15 March 2006 (UTC)
thread_title:  Page move

message: I found some inconsistencies in the text. The definition uses the topological point of view and does not mention charts or atlases. Yet most of the rest of the article takes the differential point of view and explains at length what a chart and an atlas is. I believe that a novice reader would ask themself: Why do I need to know what a chart or an atlas is if they are not used in the definition? Of course, the answer is that they are used in the definition of a differential manifold and differential manifolds are very, very important, but, while possible, it is hard to understand that from the article. 
I think that if we added an explicit definition of a differential manifold the text would become clearer. After a precise definition is given, it should be much easier to follow the explanations in the section "Charts, atlases and transition maps". Or we could define topological manifold to be a topological space equipped with an atlas of homeomorphisms, which is sightly incorrect, but would make the rest of the text much easier to understand. What do you think? I have some difficulties with English, so I prefer not to make the changes myself.
By the way, is there a topological manifold, which cannot be equipped with a differential structure?--tzanko 21:28, 28 March 2006 (UTC)
replies: text: Differentiable manifolds are mentioned in the introduction, and there is a link there to the article on differentiable manifolds.  In the section just before the section on charts, atlases, and transition maps, this statement appears: "One of the most important is the differentiable manifold, which has a structure that permits the application of calculus."  Every topological manifold can be equipped with a differentiable structure, in higher dimensions more than one.  But there are many areas of topology that use manifolds without reference to a differentiable structure, knot theory for example. Rick Norwood 21:37, 28 March 2006 (UTC)
replies: text: What you said "every topological manifold can be equipped..." seems to contradict what I said.  -lethe talk [ +] 21:45, 28 March 2006 (UTC)
replies: text: I stand corrected.  Thanks. Rick Norwood 21:47, 28 March 2006 (UTC)

text: This is a worthwhile item that has come up in the archives: Bishop and Goldberg's book Tensor Analysis on Manifolds () mentions a difficult theorem by Whitney, that any C1 manifold can be massaged into a real-analytic, and hence C∞ (smooth), manifold. --KSmrqT 00:45, 29 March 2006 (UTC)
replies: text:  Sounds like a manifold version of Hilbert's fifth problem.  I'll check out that book.  A section about this topic would look nice, but maybe it would go better in differential structure than here. -lethe talk [ +] 07:02, 29 March 2006 (UTC)

text: There are higher dimensional manifolds not admitting differentiable structure.  For example, I know of one in eight dimensions (Tamura), and there are more.  In addition, the existence of a triangulation in dimensions greater than four is not known and is a big open question; any smooth manifold can be given a smooth triangulation (by Munkres, Whitehead, et al).  BTW, I should mention there are non-triangulable 4-manifolds.  So the question of how special dimension four is, is really still open.  --C S (Talk) 17:21, 13 April 2006 (UTC)
replies: text: Ah neat.  So it's open whether higher D manifolds are triangulable?  I remember asking my algebraic prof that, after he invoked without justification a theorem of Rado that all surfaces are, and he muttered some stuff that made it sound like he didn't know.  Why do you bring up triangulation here though?  Is existence of a triangulation related to the existence of a smooth structure?  Or just another structure whose existence becomes unclear in high dim? -lethe talk [ +] 17:29, 13 April 2006 (UTC)
replies: text: Well, that theorem of Rado is pretty simple to prove, relatively speaking, compared to dimension 3, so let's give your prof a break :-)  I'm a little surprised he didn't know though, but I would guess this is more known among older topologists harking back to the golden age of topology, rather than the younger ones.  Triangulations are very closely related to smooth structures.  The theorem I mentioned shows that every smooth structure arises from smoothing a triangulation.  Now, there are different ways to smooth, so different smooth structures can arise.  However, in dimensions up to four, it's known you can always smooth a triangulation.  But I believe that's unknown for higher dimensions (not sure).  One way to show uniqueness of smooth structure for dimension 2 and 3 is to study smoothings of triangulations.  --C S (Talk) 02:17, 14 April 2006 (UTC)
thread_title:  Topological vs. Differential 

message: (Off topic)
"1:1 At the first God made the heaven and the earth. 
1:2    And the earth was waste and without form; and it was dark on the face of the deep: and the Spirit of God was moving on the face of the waters. 
1:3    And God said, Let there be light: and there was light. 
1:4    And God, looking on the light, saw that it was good: and God made a division between the light and the dark, 
1:5    Naming the light, Day, and the dark, Night. And there was evening and there was morning, the first day."
The above is from the Basic English Bible; a version of the Bible written using only the 800 Basic English words.  I got curious, after my previous post, about how close my esitmate of 8000 basic English words was.  I still think most people know 8000 words, but it is interesting how much you can say with just 800. Rick Norwood 15:05, 2 April 2006 (UTC)
thread_title:  Basic English 

message: *I removed this paragraph from the intro earlier, because I didn't think it was necessary.
:A technical mathematical definition of a manifold is given below. To fully understand the mathematics behind manifolds, it is necessary to know elementary concepts regarding sets and functions, and a working knowledge of calculus and topology would be helpful.
*Rick Norwood put it back, but I'm afraid it is still superfluous.
Ben Standeven 01:43, 11 April 2006 (UTC)
thread_title:  Pointless paragraph in intro 

message: It's a little awkward responding to a question raised in an edit summary! In response to my "typo, fine tuning" summary, lethe's summary says "This is a weird typo, Kmsrq (also, isn't mathbf preferred in math formulas?))". So first, that should be "KSmrq". The typo is "give"→"given". Font selection for TeX has evolved, and I'm not sure what's now preferred. Looks like mathbf, the historic choice, is the common Wikipedia choice, so I shouldn't have changed it. --KSmrqT 11:39, 14 April 2006 (UTC)
replies: text: The wierd typo I was referring to was changing B to Bold as the name of the unit ball.  I assume that was a typo?  You don't actually want to name your unit ball "Bold"?  You have to admit, that is kinda weird.  As for tex, I don't care that much.  I was just fixing the typo and so changed that as well. -lethe talk [ +] 16:22, 14 April 2006 (UTC)
replies: text:  It was a very logical typo, just forgetful. Typically mathematical italics are letterspaced farther apart than ordinary italics, and I was trying to see if there was also any difference with the boldface. (Didn't see any.) You see, "B" for ball and "B" for bold; obvious, eh? Sorry to leave the debris from my experiment; thanks for catching it. --KSmrqT 16:33, 14 April 2006 (UTC)
replies: text: By the way, I have been having persistent problems getting your name right ever since I met you.  Does it stand for something or is it short for something?  That might help me remember. -lethe talk [ +] 18:08, 14 April 2006 (UTC)
replies: text:  Yes, it stands for something; but if I tell you what, then I lose my anonymity. I'd like to switch to just KS. --KSmrqT 22:49, 14 April 2006 (UTC)
replies: text: KS, I can do that. From now on in my head I shall call you Kansas.  -lethe talk [ +] 22:56, 14 April 2006 (UTC)

text: I've always assumed KSmrq is a dragon, a relative of the one Bilbo met, Smaug, greatest and dreadfullest of all calamities : ) Rick Norwood 21:02, 14 April 2006 (UTC)
replies: text: Now try saying KSmrq out loud three times fast. -lethe talk [ +] 21:51, 14 April 2006 (UTC)
thread_title:  reply to lethe edit summary 

message: There's lots of examples of what a manifold is, but no examples of what isn't a manifold.  Maybe a simple counter example would go a long way to helping people (me) understand?
81.103.232.63 23:21, 20 April 2006 (UTC)
replies: text: The figure 8 counterexample is in the article now.  An idealized umbrella with a one dimensional handle and a two dimensional top would not be a manifold.  Should we add more examples? Rick Norwood 23:26, 20 April 2006 (UTC)

text: I thought perhaps the non-examples of the long line and the line with two origins should be in here, so show why we like paracompactness and Hausdorffness.  For another nonexample, no indiscrete space can be a manifold, and discrete spaces can only be 0-dimensional manifolds if they're countable. -lethe talk [ +] 01:05, 21 April 2006 (UTC)
replies: text: Sounds like good material for the article to me. Rick Norwood 12:36, 21 April 2006 (UTC)

text: I support adding good counterexamples. A link to physics' string theory might be a fun way to show a practical way being a manifold is nice. How about a T-junction of (infinitely thin) walls? Or the cubic real algebraic curve y2 = (x+1)(x2−1), which has an isolated point, thus is not an n-manifold? Or y2 = x3, to remind us that a curve with a cusp can be, surprise, a smooth manifold? Some time back the elementary part of the article mentioned a string dangling from a sphere; it's a natural. --KSmrqT 18:06, 21 April 2006 (UTC)
thread_title:  What is not a manifold? 

message: I am seeking to contact the Manifold page curator.
Thanks.
Sincerely,
gmjaet@sbcglobal.net
GeMiJa 16:35, 21 April 2006 (UTC)
replies: text: Yes?  We're listening... What do you want?  Talk pages are how you contact the authors of articles at wikipedia, and other interested parties, and you've arrived here.  So what can we help you with?  -lethe talk [ +] 16:52, 21 April 2006 (UTC)
thread_title:  Contact Request:  Manifold Page Curator 

message: In mathematics, a manifold is an abstract multi-dimensional space where one cannot meaningfully make a choice of a coordinate system.  In contrast to Euclidean space , one cannot distinguish between a straight line and a curve, nor define the distance between two points.   Another difference is that many manifolds possess a complicated spatial structure when viewed as a whole. There are finitary manifolds that are nonetheless unbounded ; other manifolds possess holes, regions where a circle cannot be continuously shrunk to a point.
The  surface of Earth is an example of a manifold; locally it seems to be flat, but viewed as a whole it is round.   However, since manifold geometry aims to be a minimalist theory, the surface of a sphere and the surface of an ellipsoid cannot be distinguished as manifolds.
A manifold can be constructed by "gluing" separate Euclidean spaces together; for example, a world map can be made by gluing many maps of local regions together, and accounting for the resulting distortions.  Alternatively,  a manifold can be defined by a system of equations.  However, as a geometric entity in and of itself, a manifold exists independently of any ambient space.
Rmilson 13:07, 24 April 2006 (UTC)
thread_title:  draft revision of first paragraph