This is an important concept in mathematics with many incarnations. It allows for a simple geometric interpretation of gluing paper together. It has been attempted to make this article accessible to non-mathematicians and reserve the technical details mostly for specific incarnations such as differentiable manifold and topological manifold. Therefore I would be very interested in finding out whether we succeeded in this at all. All other comments are also welcome of course. -- MarSch 11:26, 20 October 2005 (UTC)
Wow, very well done. I really appreciate the work done to make this comprehensible to non-math people (like me). A few comments, from someone who had absolutely no idea what a manifold was before reading the article:
This is a nice piece of work! CDC (talk) 22:22, 21 October 2005 (UTC)
Tony 02:49, 22 October 2005 (UTC)
I agree, this is a well done article. I like the second and third person here, the you's and we's, they help to make the material more accessible to the casual reader.
The pictures rock, but the article needs a hook in the lead paragraph. I left out the wiki formatting and links, but how about something like this for a lead?
"While many people think of x's and y's and long schoolroom afternoons when they think of mathematics, many mathematicians think about shapes and surfaces. But our normal three dimensional space of width, height and depth is often inadequate for reasoning about mathematical problems. Over time, a variety of ideas have converged on the idea of a manifold as a way to help think about surfaces.
An everyday example of this sort of thinking is the common street map, which uses a two dimensional drawing to represent features of the earth, whose surface is three dimensional. Indeed, much of the terminology of manifolds is inspired by map-making or cartography, we speak of an atlas of charts which can be pieced together as a patchwork to describe a manifold." (by User:Dethomas -- MarSch 10:27, 23 October 2005 (UTC))
The current opening paragraph doesn't engage the casual reader, while the second paragraph,if he gets that far, dumps a load of complexity on his head. The first paragraph of the Introduction is a much better answer to the reader's implicit "Why are manifolds important?" question.
Any lead paragraph needs to tell the reader why the material is worth his time. So what are manifolds and why should the reader care? Because they aid in mathematical reasoning about surfaces, shapes and spaces in ways that everyday three space is ill-suited to do. Let's just say so. I'm quite willing to let the "long schoolroom afternoons" go, but most readers need a boost to get over the "eek, math!!" threshold. I think the street map analogy or something of that nature places the reader in familiar territory (as do the words "chart" and "atlas") while opening the door to the depth of detail in the article. But that's just my opinion.
Dethomas 00:24, 24 October 2005 (UTC)
Imagine you have a few sheets of paper and some glue. The paper is of a special high-quality kind that can be strechted and molded into whatever shape you want and it never tears. You could cover the Earth with just two such sheets. One strechted over the North pole all the way down to Antarctica and another stretched over the South pole all the way up to Greenland, with a bit of glue at the tropics where they overlap. You have just proved that the surface of the Earth is a paper manifold!
Not really, not for a lead or opening paragraph. A good lead convinces the reader, in a few sentences, that the material is worth his time and effort. My sense is you are being too literal (pedantic?) with the "earth's surface is two dimensional" mathematical idea, and hence missing the point. To the the target audience, the non-technical reader, the idea that the earth's surface has height, width, and depth but we commonly represent it with flat, two dimensional chucks of paper called "maps" is understandable and nonthreatening. By analogy, we can proceed from the familiar concept of a street map to the unfamiliar concept of a manifold, draw the reader into the article, and let the Introduction section do it's job.
The current (Oct 29) lead is too big, too wooden and does little to draw the non-technical reader into the body of the article. So I'm still suggesting something like this:
"Our mundane notions of width, height and depth are often inadequate for reasoning about some types of mathematical problems. Over time, a variety of ideas have converged on the idea of a manifold as a way to help mathematicians think about surfaces and related topics. An everyday example of this sort of thinking is the common street map, which uses a two dimensional drawing to represent features of the earth, whose surface is actually three dimensional on the scale of daily experience. Indeed, much of the terminology of manifolds is inspired by map-making or cartography. We speak of an atlas of charts, which can be pieced together as a [[[patchwork]]] to describe a manifold."
You can take this verbatim as the lead, or you can bless it and I'll put it in, or somebody can write a new lead, but if we are trying to reach non-technical readers, we should ditch the current opening paragraphs if favor of something Joe Everyman can read without passing out.
Dethomas 05:39, 30 October 2005 (UTC)
I thought I did make specific remarks, and offered specific remedies to what I saw as essential problems. But that's just an opinion.
In any event, an effective lead paragraph draws in the casual reader. If a trivial example does the job, use it. I think it would be more helpful if you remembered the article is aimed at a non-technical audience, that the rigor of a textbook is inappropriate, and give me the benefit of the doubt, as opposed to disrespect of your contempt.
The phrase "you can nevertheless bring a lot of mathematics from the plane over to the sphere' is precisely what you what to tell the reader in a lead, and precisely what my examples have suggested. Read what's in the comment, not what you wish was in the comment.
Or not. Your burden.
Dethomas 18:05, 1 November 2005 (UTC)
Good to see this moving forward. A number of goals agreed in the talk archives still could use some work. One lingering concern is the lack of a link, say in Other types and generalizations of manifolds, to the original and still important type, the Riemannian manifold. -- KSmrq T 22:36, 23 October 2005 (UTC)
My comments hardly deserve their own subsubheading, but that seems to be the way. I suggest moving some of the technical details in the introduction into paragraphs in the lead. As a mathematician, I wanted to engage the mathematics more quickly, and the lead doesn't really allow me to do that. In fact, I'll go so far as to say that there should surely be technical details in the first sentence of the article. It's an article about mathematics, so the content should be about mathematics. NatusRoma 05:41, 3 November 2005 (UTC)
I personally think, as the first leading paragraph stands for an introduction, it might be better to merge the "introduction" section into up above the TOC. Other things are pretty good. However, I don't know if it can go through in case you put it up for FAC -- not everybody's interested in this difficult mathematical thing. Deryc k C. 08:14, 9 November 2005 (UTC)
YES. You MUST pu this on FAC! This is basically more interesting than shoe polish! More seriously, some comments:
Vb 15:01, 11 November 2005 (UTC)
This is an important concept in mathematics with many incarnations. It allows for a simple geometric interpretation of gluing paper together. It has been attempted to make this article accessible to non-mathematicians and reserve the technical details mostly for specific incarnations such as differentiable manifold and topological manifold. Therefore I would be very interested in finding out whether we succeeded in this at all. All other comments are also welcome of course. -- MarSch 11:26, 20 October 2005 (UTC)
Wow, very well done. I really appreciate the work done to make this comprehensible to non-math people (like me). A few comments, from someone who had absolutely no idea what a manifold was before reading the article:
This is a nice piece of work! CDC (talk) 22:22, 21 October 2005 (UTC)
Tony 02:49, 22 October 2005 (UTC)
I agree, this is a well done article. I like the second and third person here, the you's and we's, they help to make the material more accessible to the casual reader.
The pictures rock, but the article needs a hook in the lead paragraph. I left out the wiki formatting and links, but how about something like this for a lead?
"While many people think of x's and y's and long schoolroom afternoons when they think of mathematics, many mathematicians think about shapes and surfaces. But our normal three dimensional space of width, height and depth is often inadequate for reasoning about mathematical problems. Over time, a variety of ideas have converged on the idea of a manifold as a way to help think about surfaces.
An everyday example of this sort of thinking is the common street map, which uses a two dimensional drawing to represent features of the earth, whose surface is three dimensional. Indeed, much of the terminology of manifolds is inspired by map-making or cartography, we speak of an atlas of charts which can be pieced together as a patchwork to describe a manifold." (by User:Dethomas -- MarSch 10:27, 23 October 2005 (UTC))
The current opening paragraph doesn't engage the casual reader, while the second paragraph,if he gets that far, dumps a load of complexity on his head. The first paragraph of the Introduction is a much better answer to the reader's implicit "Why are manifolds important?" question.
Any lead paragraph needs to tell the reader why the material is worth his time. So what are manifolds and why should the reader care? Because they aid in mathematical reasoning about surfaces, shapes and spaces in ways that everyday three space is ill-suited to do. Let's just say so. I'm quite willing to let the "long schoolroom afternoons" go, but most readers need a boost to get over the "eek, math!!" threshold. I think the street map analogy or something of that nature places the reader in familiar territory (as do the words "chart" and "atlas") while opening the door to the depth of detail in the article. But that's just my opinion.
Dethomas 00:24, 24 October 2005 (UTC)
Imagine you have a few sheets of paper and some glue. The paper is of a special high-quality kind that can be strechted and molded into whatever shape you want and it never tears. You could cover the Earth with just two such sheets. One strechted over the North pole all the way down to Antarctica and another stretched over the South pole all the way up to Greenland, with a bit of glue at the tropics where they overlap. You have just proved that the surface of the Earth is a paper manifold!
Not really, not for a lead or opening paragraph. A good lead convinces the reader, in a few sentences, that the material is worth his time and effort. My sense is you are being too literal (pedantic?) with the "earth's surface is two dimensional" mathematical idea, and hence missing the point. To the the target audience, the non-technical reader, the idea that the earth's surface has height, width, and depth but we commonly represent it with flat, two dimensional chucks of paper called "maps" is understandable and nonthreatening. By analogy, we can proceed from the familiar concept of a street map to the unfamiliar concept of a manifold, draw the reader into the article, and let the Introduction section do it's job.
The current (Oct 29) lead is too big, too wooden and does little to draw the non-technical reader into the body of the article. So I'm still suggesting something like this:
"Our mundane notions of width, height and depth are often inadequate for reasoning about some types of mathematical problems. Over time, a variety of ideas have converged on the idea of a manifold as a way to help mathematicians think about surfaces and related topics. An everyday example of this sort of thinking is the common street map, which uses a two dimensional drawing to represent features of the earth, whose surface is actually three dimensional on the scale of daily experience. Indeed, much of the terminology of manifolds is inspired by map-making or cartography. We speak of an atlas of charts, which can be pieced together as a [[[patchwork]]] to describe a manifold."
You can take this verbatim as the lead, or you can bless it and I'll put it in, or somebody can write a new lead, but if we are trying to reach non-technical readers, we should ditch the current opening paragraphs if favor of something Joe Everyman can read without passing out.
Dethomas 05:39, 30 October 2005 (UTC)
I thought I did make specific remarks, and offered specific remedies to what I saw as essential problems. But that's just an opinion.
In any event, an effective lead paragraph draws in the casual reader. If a trivial example does the job, use it. I think it would be more helpful if you remembered the article is aimed at a non-technical audience, that the rigor of a textbook is inappropriate, and give me the benefit of the doubt, as opposed to disrespect of your contempt.
The phrase "you can nevertheless bring a lot of mathematics from the plane over to the sphere' is precisely what you what to tell the reader in a lead, and precisely what my examples have suggested. Read what's in the comment, not what you wish was in the comment.
Or not. Your burden.
Dethomas 18:05, 1 November 2005 (UTC)
Good to see this moving forward. A number of goals agreed in the talk archives still could use some work. One lingering concern is the lack of a link, say in Other types and generalizations of manifolds, to the original and still important type, the Riemannian manifold. -- KSmrq T 22:36, 23 October 2005 (UTC)
My comments hardly deserve their own subsubheading, but that seems to be the way. I suggest moving some of the technical details in the introduction into paragraphs in the lead. As a mathematician, I wanted to engage the mathematics more quickly, and the lead doesn't really allow me to do that. In fact, I'll go so far as to say that there should surely be technical details in the first sentence of the article. It's an article about mathematics, so the content should be about mathematics. NatusRoma 05:41, 3 November 2005 (UTC)
I personally think, as the first leading paragraph stands for an introduction, it might be better to merge the "introduction" section into up above the TOC. Other things are pretty good. However, I don't know if it can go through in case you put it up for FAC -- not everybody's interested in this difficult mathematical thing. Deryc k C. 08:14, 9 November 2005 (UTC)
YES. You MUST pu this on FAC! This is basically more interesting than shoe polish! More seriously, some comments:
Vb 15:01, 11 November 2005 (UTC)