Wikipedia:Featured picture candidates/N-sphere

N-sphere
Voting period ends on 13 Sep 2010 at 19:37:48 (UTC)
 * Reason:The image is flashy (so it is eye-catching?), but the meat of the nomination is found in the EV. Get ready to go to Geometry 101.

We have formulas for spheres to describe their volume, or content. For a circle, its content = 2*pi*r^2. For a sphere, its content = 4/3*pi*r^3. We in grade school have learned formulas for the 0, 1, 2, and 3 dimensions of a sphere, but topologists, or literally translated, studiers of structure, have taken the thought a step further.

Take a look at this section, and follow the math or skip to the "This leads to the recurrence relations" part for Vn = ... What this section says, in other words, is that topologists have given a function for computing content of spheres larger than 3D (or r, or n, or what have you), like 4, 5, 6 to infinity. This is where things get freaky.

One would assume that as the dimension of a sphere gets larger, the larger it's content gets, right? Indeed, stick r=10 in 2*pi*r^2 and 4/3*pi*r^3, and the later is much larger. But this is true only for small values of r. Keep reading the article.

The article concludes with a beautiful limit equation, but anecdotally anyone can see for themselves that for r (or n, or dimensions) larger than 5, the content of the sphere gets smaller and smaller (because the formula is recursive). Something that gets smaller as parts of it gets larger? This is unexpected.

I give it a 2/10 on eye-catchyness, but the EV on this is phenomenal, and I give it a 10/10. It makes one wonder: could the the 11th dimension be all around us, but we don't see it because its tiny?
 * Articles in which this image appears:N-sphere
 * FP category for this image:Sciences, mathematics
 * Creator:Claudio Rocchini, with derivative work by Pbroks13

 “What consenting mathematicians get up to behind closed  doors is their business, but please don't do it in public.”
 * Support as nominator --Gut Monk (talk) 19:37, 4 September 2010 (UTC)
 * Comment This nom is going to need an exceedingly good, clear caption to make the subject matter more easily comprehended. Currently, it does not. To start with, it would be nice to have a description of what the three smaller images at the bottom represent (4, 5 & 6 dimensions?). The caption should also explain what the bigger one means; apparently it is a superimposition of the three. And if it is a superimposition of three spheres with a different number of dimensions, what is the significance of the superimposition and how does it help us to understand multi-dimensional spheres? And one final thought: the n-sphere article launches very quickly into abstruse, specialty lingo. If this image and its linked article are to be the subject of Today’s Featured Picture on the Main Page for a day, it should, IMO, have a lead paragraph that is more suited for a general-interest readership. As it is, right between the first and second paragraphs of the lede is this   $$S^n = \left\{ x \in \mathbb{R}^{n+1} : \|x\| = 1\right\}.$$   Without a more approachable lede in the article, the first thing that comes to mind is a statement made by one of our more notable editors, Dweller in May 2008:
 * Greg L (talk) 21:46, 4 September 2010 (UTC)
 * "This nom is going to need an exceedingly good, clear caption to make the subject matter more easily comprehended."


 * Comment Now we're talking!!  This is my suggested caption, but this also is part of the reason I've nominated the candidate: this can make a fantastic nomination with precise adjustments.  Suggested caption:"As the dimension of a sphere approaches infiniti, its volume, or content, approaches zero."  Improve how?
 * "To start with, it would be nice to have a description of what the three smaller images at the bottom represent (4, 5 & 6 dimensions?)."
 * Here I'll disagree. How does it matter?  While the article needs work, the picture, in itself, I think, is eye-catching enough to call attention to the article.
 * "What is the significance of the superimposition and how does it help us to understand multi-dimensional spheres?"
 * I don't know. Researching...
 * " If this image and its linked article are to be the subject of Today’s Featured Picture on the Main Page for a day, it should, IMO, have a lead paragraph that is more suited for a general-interest readership."
 * Needs work. Yup. Gut Monk (talk) 22:27, 4 September 2010 (UTC)
 * Augmented quote"Furthermore, the n-sphere article quickly launches into abstruse, specialty lingo. If the image and its linked article are to be the subject of Today’s Featured Picture on the Main Page for a day, it should, IMO, have a lead paragraph that is more suited for general readership."
 * We have a week to flesh it out. Fleshing it out... Gut Monk (talk) 22:44, 4 September 2010 (UTC)
 * Support Perhaps the article needs a paragraph at the start for a more general readership. The notation should stay there though - for the mathematically literate a concise definition is actually kind of important. I should remind that this is FPC though. Noodle snacks (talk) 00:32, 5 September 2010 (UTC)
 * Mmm added a section. Gut Monk (talk) 02:14, 5 September 2010 (UTC)
 * Yeah, that doesn't work - you are using an interesting fact about n-spheres as the article lead, rather than explaining what it is. This is going to be more confusing for your average reader. I'm sorry to be a pendant, but you also need to be really careful with terminology. There are also plenty of other more minor issues with the edit, to do with tone, the MS, etc. I'm also not sure that your layman would know what to expect with the content of a one or zero dimensional sphere would be either. I've incorporated some of your text into your article. Noodle snacks (talk) 03:12, 5 September 2010 (UTC)
 * Questions I went to American public school, so for some of this math stuff I ask you to speak slowly =) (eh, it's really not too bad). Anyway, is PEMDAS in effect for the two formulas listed in the Reason? Using it I got that the former formula was larger than the latter (628 0 versus 4180). Is it fair to say that the image basically shows different 3-dimensional shapes that can be formed inside a larger 3-dimensional shape? Is the basic meaning of that then that the more dimensions used to measure a shape the more restriction there is on its possible size? Because of conflicting coordinate points? I can't claim to be a mathematician, but dimensions seem like theoretical pure math to me, and there are a lot of math-related articles on Wiki heavy with jargon. As for Greg L, this is what gets mathematicians off, it's just sick! xD As for the picture itself the file seems to be far below the minimum dimension criteria. --I'ḏ ♥ One 12:25, 5 September 2010 (UTC)
 * r = 10, circle = 2*pi*r^2 = 2*3.14*10*10 = 628, sphere = 4/3*pi*r^3 = 4/3*3.14*10*10*10 = 4186.66
 * "Is it fair to say that the image basically shows different 3-dimensional shapes that can be formed inside a larger 3-dimensional shape?" Yes
 * "Is the basic meaning of that then that the more dimensions used to measure a shape the more restriction there is on its possible size?" Ahh, yes...no?  I'm starting to agree with Greg on the concern on wording.  I'll keep working the caption and hopefully the edit (and future nomination) will make it much more clear.  Gut Monk (talk) 19:49, 5 September 2010 (UTC)
 * Comment #2 The caption begins nicely now, but it hasn’t ended in my opinion. The caption should explain what each picture pane means. Right now, we have something reminiscent of a Spirograph and a teaser of a tag line. I took a look at Gut’s first-cut effort at making the article’s lede more approachable for a general-interest audience. Though it needed some cleanup for tone (it was addressing the reader in the first-person), I think he was on the right track. Greg L (talk) 17:02, 5 September 2010 (UTC) P.S. The word “dimensions” has multiple meanings. Besides “axial coordinates”, it can also mean the “size” of something. So the current caption will read to many as follows: “Did you know, as the size of something gets larger, its "volume" gets smaller? I would suggest something along the lines of this: Mathematicians often analyze topology (the shape of objects) that exists in a greater number of spatial dimensions than the three we can point to. As the number of spatial dimensions in which a sphere exists increases, its volume decreases. Too often our mathematics and physics-related articles suffer from “show-off-itis” where contributing editors regurgitate material from textbooks onto the pages of Wikipedia; it makes dueling editors feel smart in many cases but simultaneously makes readers feel dumb and defeats the whole purpose of educating. This is easily betrayed when arcane formulas are ralphed onto the pages and aren’t immediately followed up with an explanation of what each term in the formula means. Technical articles should always start out using simple plain-speak and build from thereon; that’s the really hard part of technical writing—believe me, I know. The n-sphere article, IMO, is suffering dearly from this phenomenon and is in an extreme state of flux. I suggest this nom be withdrawn until the article is stable and has a paragraph or two in the lede that makes the subject matter approachable for a general-interest readership.  Greg L (talk) 17:22, 5 September 2010 (UTC)
 * "Too often our mathematics and physics-related articles suffer from “show-off-itis” where contributing editors regurgitate material from textbooks onto the pages of Wikipedia; it makes dueling editors feel smart in many cases but simultaneously makes readers feel dumb and defeats the whole purpose of educating." Tell it like it is brother!
 * This duelling editor crap is absolute nonsense (AGF?). When the reader has background understanding in the appropriate areas, the notation that is there is necessary, useful and straightforward. Sure, there could be a few more words here and there, but not at the cost of dumbing it down. Noodle snacks (talk) 07:18, 6 September 2010 (UTC)
 * It’s not an “AGF” issue; it’s an issue of “understanding human nature.” Dueling editors all wide-eyed to show what they learned in college happens all the time on Wikipedia. It’s easy to *understand* level-200 math and science and regurgitate it onto Wikipedia’s pages. Youngsters fresh out of college capable of transplanting stuff from their textbooks to Wikipedia are a dime a dozen. *Explaining* the material to make it understandable takes extremely good technical writing skills that volunteer contributors to Wikipedia seldom have. This reality shouldn’t come as such a shock. “AGF” does not require that other editors admire your edits and contributions as much as you do. Greg L (talk) 20:15, 6 September 2010 (UTC)
 * Oppose For reasons stated in “P.S.”, above. And as I ponder the four-pane image and the larger superposition at top, I can not escape the ever-developing realization that there is nothing to learn from the superposition; it does not illustrate a trend or make a pattern of nature suddenly apparent. It appears this is no more illuminating than having a superposition of Chicago, Los Angeles, and Tokyo: a jumble that does not reveal a pattern or trend and therefor does not illuminate. If the superposition does make something apparent that was once hidden, it desperately needs a cogent caption explaining its magic, which escapes me at the moment. Right now, the image, caption, and the linked article amounts to this: “Do you see the above jumble? Well, there is no rational explanation for how the superposition illustrates anything other than ‘jumble’. And going to the article won’t shed any light on the issue because the text there is even more incomprehensible.”  Greg L (talk) 17:40, 5 September 2010 (UTC)
 * Right now you have a 3 space projected on a 3 space that is orthogonal in 4D. That seems too abstract to me.  I'm going to try a 2 space projection on 2 space, because that way we at least can visualize it.  Make it bigger per Heart One, too.
 * Withdraw in 24 hours A very cool concept, but I now agree that the article is in flux. I'll give this nomination 24 hours for further brainstorming opportunity then close it. Gut Monk (talk) 19:47, 5 September 2010 (UTC)
 * Oppose Firstly, you're right, GM, somehow an extra ^1 got worked into my answer, but I don't see a reason for why this inanimate nomination should be so undersized, there's a bunch of online graph makers. --I'ḏ ♥ One 13:02, 6 September 2010 (UTC)
 * Comment It says "as the dimensions of something gets larger, its 'content' gets smaller". "Something" is a vague word and as applied to the n-cube, this is just false.  Furthermore, where it says "only true for small values of r ", that's wrong; it should say only true of small values of the dimension.   "r" is not the dimension; it's the radius.  And obviously, as the dimension grows, rn either grows or shrinks according as r > 1 or r < 1.  The statement should be about what happens when r = 1 and the dimension grows. Michael Hardy (talk) 19:00, 6 September 2010 (UTC)
 * ...and the area of a circle is not 2&pi;r2. Michael Hardy (talk) 19:04, 6 September 2010 (UTC)
 * Comment. The proposed image caption has nothing to do with the image.  The image shows the stereographic projections of the 3-sphere onto the 3-dimensional Euclidean space.  The colored curves are the images of certain circles (parallels, meridians, and hypermerideans) under the stereographic projection.  This has nothing to do with "volume getting smaller" as the dimension increases.   Sławomir Biały  (talk) 01:12, 7 September 2010 (UTC)