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I moved this paragraph here because while I think our manifold article needs a first paragraph explaining the use and ideas behind manifolds for non-specialists, the above paragraph doesn't do the job:
--AxelBoldt
Perhaps someone could add a definition for paracompact? I assume I'm not the only one who's clueless in this regard... -- Belltower
The article says to look at the Topology Glossary. I don't think we really need a whole article on paracompactness, unless someone is keen to write one. -- Zundark, 2001 Sep 21
I'm keen to write one -- there's a lot more to the idea of paracompactness than the definition; otherwise, it would never have been defined in the first place (except maybe once, in sombody's doctoral thesis, and then forgotten) -- although it may be a while before I get to it -- there are several other articles that I'm keen to write. -- Toby Bartels 2002/05/07
There is a selflink on the page, the link to differential space goes to a redirect which goes back to — manifold. Somebody knows where it is supposed to point? Mikez 18:28, 20 Feb 2004 (UTC)
The phrase "however there is a classification simply connected manifolds of dimension ≥ 5." seems to be missing a word. Perhaps what was meant was, "however there is a classification of simply connected manifolds of dimension ≥ 5."
Also, since I came here from the Wikipedia page about General Relativity, somewhere we need a definition of _Pseudo_ Riemannian manifolds. Here I found only that for Riemannian. Or does that really belong back on the General Relativity page?
See pseudo-Riemannian manifold. Charles Matthews 08:14, 18 Mar 2004 (UTC)
Two Ck atlases are called equivalent if their union is a Ck atlas. This is an equivalence relation, and a Ck manifold is defined to be a manifold together with an equivalence class of Ck atlases.
I discovered Category:Manifold_theory was orphaned. It has one subcategory, Category:Surfaces. Should this article be in Category:Manifold_theory? Should Category:Manifold_theory be merged with or be added to Category:Differential_geometry or Category:Differential_topology, should those be added to it, or should it be deleted? -- Beland 05:17, 14 Aug 2004 (UTC)
Look at the nice, informative table in the German version. I hope somebody can translate it into English.
Andries 18:41, 27 Oct 2004 (UTC)
I remove it again, it might go to dimension (if it would be written better) Tosha 06:15, 21 Nov 2004 (UTC)
Please do not add it any more, it has little to do with concept of manifold Tosha 01:06, 22 Nov 2004 (UTC)
I agree that the current introduction by Xah Lee is poorly written and adds little to the article. That being said, I must agree with his sentiment that this article desperately needs a more gentle introduction. The first paragraph at least should be understandable to someone without any training in topology. Physics students in particular come across manifolds before ever having seen the word topology, let alone words Hausdorff, paracompact, or homeomorphic. For many of these students manifolds are their first introduction to topology. I've been meaning for a long time to write a better introduction, I just haven't had the time. In the meantime the current introduction by Xah Lee should probably be removed. -- Fropuff 16:34, 2004 Nov 22 (UTC)
To make myself clear, I do not think the manifold article has a good introduction, but this is no reason to add poorly written content to an otherwise solid article. Perhaps Fropuff should have a try at writing a better introduction (without using Hausdorff, paracompact, or homeomorphic) MathMartin 17:12, 22 Nov 2004 (UTC)
No, not all manifolds are smooth, actually. And there can be more than one smooth structure on a manifold. These are very unobvious facts. But we can't have anything in the introduction that actually contradicts the facts.
Charles Matthews 11:24, 23 Nov 2004 (UTC)
Hmmm...interesting discussion going on. I agree that Xah Lee's explanation is rather incoherent. As MathMartin has cogently observed, it's difficult to explain mathematics to laymen in an intuitive manner without introducing notions that describe something quite different. I hope we all agree that any layman explanation is something that it should be possible to build on, without having to tear down previously set ideas. The current intro looks great! Personally, I would like more mention of how patches of Euclidean space are glued together (with a note that the degrees and kinds of niceness in gluing give rise to different structures), but as it stands, I can't complain. -- C S 00:59, Dec 13, 2004 (UTC)
I, Xah Lee, hereby declare, that the original explanation i inserted for this article, roughly equivalent to the cited following, is an excellent explanation of manifold, if not the best. Further, i hereby declare all academics who say otherwise, are grave morons. Morons, i say, and i mean it.
Manifold explanation by Xah Lee: Manifold, like polytope, is a generic name for certain concept of arbitrary dimensions. In 2-D, we have curves. For example, take out a piece of paper and scribble on it with a pencil in one stroke. What you have drawn, is a curve, or plane curve. The curves themselves are of 1-dimension, but they sits in a 2-dimensional plane. Now imagine the path of a fly. That would be a curve in space, called space curve. The curve itself is still one dimensional, but sits now in a 3 dimensional space. But now in 3-dimensions, we can have something else beside curves, namely surfaces. The paper you took out earlier, would be a flat surface called a plane, which is a 2-dimensional object, sitting in 3-d space. You can bend the paper or roll up into a cylinder and its still a surface in space. There can be lots of surfaces. A soap bubble, is a specific surface called sphere or hemisphere. A flag is a surface. A open umbrella is a surface. The gist here is that in 3-D space we can have 1-D objects like curves, or 2-D objects like surfaces. As with regular solids of higher dimensions, mathematician's imagination faculties thought about curves and surfaces in higher dimensions. Basically, in n-dimension you can have "curves/surfaces thing" that has dimensions less then n. For example, in 3-D we've seen the 2-D surfaces or the 1-D curves, and in 2-D plane we can have 1-D curves or 0-D dots. Therefore, in 4-dimensional space, we can have not only 0-D dots or 1-D curves or 2-D surfaces, but we can also have "curve/surface-like thing" that is 3-dimensional, sitting in a 4-dimensional space. The general name for these curve/surface-like "things" in arbitrary dimensions is called _MANIFOLD_. Thus, a curve is just a 1-dimensional manifold. A surface is a 2-dimensional manifold.
The above is exerpted from originally a newsgroup posting, archived here: http://xahlee.org/UnixResource_dir/writ/mathterms.html (will needs to be turned into an essay later)
Now, there are few mathematicians here. I have a advice for you: acquaint diverse fields, and get an overview of the history of mathematics, and step outside of your abode and understand social sciences and history of literature of human animals. Then you may perecive the quality of my manifold explanation. (baring a few grammatical errors or non-standard phrasings)
Xah Lee 17:36, 2004 Dec 13 (UTC)
gentlemen, have a look at the Curves page: http://en.wikipedia.org/wiki/Curve a simple idea of length is chalked up by moronic math academicians to metric space with incomprehensible formulas. Xah Lee 20:11, 2004 Dec 30 (UTC)
here and now i'll explain the moronicity of the current introduction.
First, assume our audience is someone who, is a math student, and has already painstakingly finished 2 years of undergraduate math courses of USA standard, including multivariable calculus, and as well as first courses in linear algebra and differential equations. A commendable task that less than one percent of college-educated elite can say for themselves, and perhaps one in a million among the literate of the world.
Now, in my introduction of manifold, any non-moron who are ignorant of calculus will however understand fully, and come away with an appreciation of the imagination of mathematics and the depth of mathematicians.
Now, in the current introduction, it gives:
* near every point of the space, we have a coordinate system; or * near every point, the environment is like that in Euclidean space of a given dimension.
Now, what is meant by these clauses? Coordinate system?? Euclidean space??? laymen would have no clue. College-going sophomores share the befuddlement.
Some academicians obsess about certain minute correctness or rigor. These academicians, when pressed, would know zilch of alternative representations of the object under different systems of foundation. They write what they wont because all their life they have submerged in mediocre conventional math texts of this era of associated styles and thoughts. And their ignorance of the whole establishment of mathematics and its movements and their professorship status exasperated and habituated a haughtiness to them.
To our readers, the current introduction with Coordinate Systems and Euclidean Space is as good as none. They are mere guises of an underlying formal system sans the actual jargon. They fail to capture the meaning of manifold that captured the (great and real) mathematician's imaginations. The current into is good only to math professors who ALREADY KNEW what manifold is. It's a masturbation. (the vast majority of math texts are all like that.)
Mathematicians in general are a class of specialized morons. They know only an extremely narrow specialization, like that of calculator to numbers, a plumber to pipes. They know almost nothing or understands nothing outside of a tiny math cavity or academic bureaucratics.
The concept of a manifold and huge number of other math subjects are something a child can appreciate, but mathematicians actively prevented that. This in part, unbeknownst to themselves, subconsciously a protection mechanism of their own status. And this is in general is true of scholars in other fields.
Xah Lee 08:18, 2005 Jan 5 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 | Archive 3 | → | Archive 5 |
I moved this paragraph here because while I think our manifold article needs a first paragraph explaining the use and ideas behind manifolds for non-specialists, the above paragraph doesn't do the job:
--AxelBoldt
Perhaps someone could add a definition for paracompact? I assume I'm not the only one who's clueless in this regard... -- Belltower
The article says to look at the Topology Glossary. I don't think we really need a whole article on paracompactness, unless someone is keen to write one. -- Zundark, 2001 Sep 21
I'm keen to write one -- there's a lot more to the idea of paracompactness than the definition; otherwise, it would never have been defined in the first place (except maybe once, in sombody's doctoral thesis, and then forgotten) -- although it may be a while before I get to it -- there are several other articles that I'm keen to write. -- Toby Bartels 2002/05/07
There is a selflink on the page, the link to differential space goes to a redirect which goes back to — manifold. Somebody knows where it is supposed to point? Mikez 18:28, 20 Feb 2004 (UTC)
The phrase "however there is a classification simply connected manifolds of dimension ≥ 5." seems to be missing a word. Perhaps what was meant was, "however there is a classification of simply connected manifolds of dimension ≥ 5."
Also, since I came here from the Wikipedia page about General Relativity, somewhere we need a definition of _Pseudo_ Riemannian manifolds. Here I found only that for Riemannian. Or does that really belong back on the General Relativity page?
See pseudo-Riemannian manifold. Charles Matthews 08:14, 18 Mar 2004 (UTC)
Two Ck atlases are called equivalent if their union is a Ck atlas. This is an equivalence relation, and a Ck manifold is defined to be a manifold together with an equivalence class of Ck atlases.
I discovered Category:Manifold_theory was orphaned. It has one subcategory, Category:Surfaces. Should this article be in Category:Manifold_theory? Should Category:Manifold_theory be merged with or be added to Category:Differential_geometry or Category:Differential_topology, should those be added to it, or should it be deleted? -- Beland 05:17, 14 Aug 2004 (UTC)
Look at the nice, informative table in the German version. I hope somebody can translate it into English.
Andries 18:41, 27 Oct 2004 (UTC)
I remove it again, it might go to dimension (if it would be written better) Tosha 06:15, 21 Nov 2004 (UTC)
Please do not add it any more, it has little to do with concept of manifold Tosha 01:06, 22 Nov 2004 (UTC)
I agree that the current introduction by Xah Lee is poorly written and adds little to the article. That being said, I must agree with his sentiment that this article desperately needs a more gentle introduction. The first paragraph at least should be understandable to someone without any training in topology. Physics students in particular come across manifolds before ever having seen the word topology, let alone words Hausdorff, paracompact, or homeomorphic. For many of these students manifolds are their first introduction to topology. I've been meaning for a long time to write a better introduction, I just haven't had the time. In the meantime the current introduction by Xah Lee should probably be removed. -- Fropuff 16:34, 2004 Nov 22 (UTC)
To make myself clear, I do not think the manifold article has a good introduction, but this is no reason to add poorly written content to an otherwise solid article. Perhaps Fropuff should have a try at writing a better introduction (without using Hausdorff, paracompact, or homeomorphic) MathMartin 17:12, 22 Nov 2004 (UTC)
No, not all manifolds are smooth, actually. And there can be more than one smooth structure on a manifold. These are very unobvious facts. But we can't have anything in the introduction that actually contradicts the facts.
Charles Matthews 11:24, 23 Nov 2004 (UTC)
Hmmm...interesting discussion going on. I agree that Xah Lee's explanation is rather incoherent. As MathMartin has cogently observed, it's difficult to explain mathematics to laymen in an intuitive manner without introducing notions that describe something quite different. I hope we all agree that any layman explanation is something that it should be possible to build on, without having to tear down previously set ideas. The current intro looks great! Personally, I would like more mention of how patches of Euclidean space are glued together (with a note that the degrees and kinds of niceness in gluing give rise to different structures), but as it stands, I can't complain. -- C S 00:59, Dec 13, 2004 (UTC)
I, Xah Lee, hereby declare, that the original explanation i inserted for this article, roughly equivalent to the cited following, is an excellent explanation of manifold, if not the best. Further, i hereby declare all academics who say otherwise, are grave morons. Morons, i say, and i mean it.
Manifold explanation by Xah Lee: Manifold, like polytope, is a generic name for certain concept of arbitrary dimensions. In 2-D, we have curves. For example, take out a piece of paper and scribble on it with a pencil in one stroke. What you have drawn, is a curve, or plane curve. The curves themselves are of 1-dimension, but they sits in a 2-dimensional plane. Now imagine the path of a fly. That would be a curve in space, called space curve. The curve itself is still one dimensional, but sits now in a 3 dimensional space. But now in 3-dimensions, we can have something else beside curves, namely surfaces. The paper you took out earlier, would be a flat surface called a plane, which is a 2-dimensional object, sitting in 3-d space. You can bend the paper or roll up into a cylinder and its still a surface in space. There can be lots of surfaces. A soap bubble, is a specific surface called sphere or hemisphere. A flag is a surface. A open umbrella is a surface. The gist here is that in 3-D space we can have 1-D objects like curves, or 2-D objects like surfaces. As with regular solids of higher dimensions, mathematician's imagination faculties thought about curves and surfaces in higher dimensions. Basically, in n-dimension you can have "curves/surfaces thing" that has dimensions less then n. For example, in 3-D we've seen the 2-D surfaces or the 1-D curves, and in 2-D plane we can have 1-D curves or 0-D dots. Therefore, in 4-dimensional space, we can have not only 0-D dots or 1-D curves or 2-D surfaces, but we can also have "curve/surface-like thing" that is 3-dimensional, sitting in a 4-dimensional space. The general name for these curve/surface-like "things" in arbitrary dimensions is called _MANIFOLD_. Thus, a curve is just a 1-dimensional manifold. A surface is a 2-dimensional manifold.
The above is exerpted from originally a newsgroup posting, archived here: http://xahlee.org/UnixResource_dir/writ/mathterms.html (will needs to be turned into an essay later)
Now, there are few mathematicians here. I have a advice for you: acquaint diverse fields, and get an overview of the history of mathematics, and step outside of your abode and understand social sciences and history of literature of human animals. Then you may perecive the quality of my manifold explanation. (baring a few grammatical errors or non-standard phrasings)
Xah Lee 17:36, 2004 Dec 13 (UTC)
gentlemen, have a look at the Curves page: http://en.wikipedia.org/wiki/Curve a simple idea of length is chalked up by moronic math academicians to metric space with incomprehensible formulas. Xah Lee 20:11, 2004 Dec 30 (UTC)
here and now i'll explain the moronicity of the current introduction.
First, assume our audience is someone who, is a math student, and has already painstakingly finished 2 years of undergraduate math courses of USA standard, including multivariable calculus, and as well as first courses in linear algebra and differential equations. A commendable task that less than one percent of college-educated elite can say for themselves, and perhaps one in a million among the literate of the world.
Now, in my introduction of manifold, any non-moron who are ignorant of calculus will however understand fully, and come away with an appreciation of the imagination of mathematics and the depth of mathematicians.
Now, in the current introduction, it gives:
* near every point of the space, we have a coordinate system; or * near every point, the environment is like that in Euclidean space of a given dimension.
Now, what is meant by these clauses? Coordinate system?? Euclidean space??? laymen would have no clue. College-going sophomores share the befuddlement.
Some academicians obsess about certain minute correctness or rigor. These academicians, when pressed, would know zilch of alternative representations of the object under different systems of foundation. They write what they wont because all their life they have submerged in mediocre conventional math texts of this era of associated styles and thoughts. And their ignorance of the whole establishment of mathematics and its movements and their professorship status exasperated and habituated a haughtiness to them.
To our readers, the current introduction with Coordinate Systems and Euclidean Space is as good as none. They are mere guises of an underlying formal system sans the actual jargon. They fail to capture the meaning of manifold that captured the (great and real) mathematician's imaginations. The current into is good only to math professors who ALREADY KNEW what manifold is. It's a masturbation. (the vast majority of math texts are all like that.)
Mathematicians in general are a class of specialized morons. They know only an extremely narrow specialization, like that of calculator to numbers, a plumber to pipes. They know almost nothing or understands nothing outside of a tiny math cavity or academic bureaucratics.
The concept of a manifold and huge number of other math subjects are something a child can appreciate, but mathematicians actively prevented that. This in part, unbeknownst to themselves, subconsciously a protection mechanism of their own status. And this is in general is true of scholars in other fields.
Xah Lee 08:18, 2005 Jan 5 (UTC)