Talk:N-sphere

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[edit] Surface Area

I think there is something wrong with the surface area equation it should be: S_n = V_n * n / R^n

http://mathworld.wolfram.com/Hypersphere.html —Preceding unsigned comment added by 129.97.4.239 (talk) 15:49, 17 October 2007 (UTC)

[edit] stereographic Projection

Would somebody explain what the section "stereographic Projection" is all about? PAR 21:44, 25 October 2005 (UTC)

[edit] merge to sphere

I propose to merge this stuff into sphere. A hypersphere is a sphere. Something of which the points are all equidistant to some special point called the origin. It's dimension is of little interest for most things I think. --MarSch 14:05, 26 October 2005 (UTC)

Looking at the links to the sphere article, I would estimate 90 percent are referring to a 2-D sphere. I think 99% of people looking for info on a sphere are thinking of a 2-D sphere. I don't think the separation is that bad as long as "sphere" links quickly and easily to "hypersphere". If it is merged, it should absolutely have its own separate section, with the first part of the article devoted to 2-D spheres exclusively. No one should have to ponder the meaning of an "N-dimensional sphere" until all the information on the 2-D sphere has been presented except for perhaps a short sentence somewhere at the end of the introduction. PAR 15:07, 26 October 2005 (UTC)
If Wikipedia were only used by mathematicians I would agree. However, most people expect to see an article on an "ordinary" 2-dimensional sphere in the sphere article. Some time ago I proposed (see Talk:Sphere#Split article?) splitting the sphere material and the n-sphere material into two separate pages with the n-sphere article discussing the general case (and hypersphere redirecting there). I started a draft article at User:Fropuff/Draft 3 for the n-sphere article, but I got sidetracked and never finished it. Having a seperate n-sphere article leads to a higher clarity of presentation since you don't need to first discuss the 2-sphere case. If you like this proposal feel free to use anything from my draft article. -- Fropuff 15:09, 26 October 2005 (UTC)
Disagree with merging. The aim of this article is to given a serious math view on the sphere and its generalizations to higher dimensions. The sphere article is aimed at people who wonder what that round thing is all about. Oleg Alexandrov (talk) 06:17, 27 October 2005 (UTC)
Okay, I like Fropuff's idea, although I don't like titles with arbitrary variable names in them, so I would prefer hypersphere over n-sphere, even though it is a bit inaccurate. So ideally there is the sphere article about the 2-sphere and there is the hypersphere article about the sphere, independent of dimensional bias. How about moving things around so that info on the 2D sphere is at 2-sphere and sphere is about the general sphere?--MarSch 10:40, 27 October 2005 (UTC)

[edit] Error in Volume formula

Having wasted over 30' on trying out the volume formula in Matlab for a project of mine, I finally gave up and crossed-referenced it. To my surpise, there is a much simpler way to calculate the volume by applying the other formulas. These can be found at [1]

But is there an error? PAR 08:14, 18 November 2005 (UTC)
  • I think there is a recurrence relationship for the surface S(n) and volume V(n) of an n-D sphere of radius R; namely,
S(n+2)=2πRV(n), with V(0)=1, and n=0,1,2,3,...
and
V(n)=RS(n)/n, with S(1)=2, and n=1,2,3,...
209.167.89.139 15:54, 15 September 2006 (UTC)
I put these in the article. PAR 22:31, 15 September 2006 (UTC)

[edit] The volume vs dimension curve

An anon posted a question in the article. I'm moving it here slightly refrasing to give context:

Is there any theories about, or implications to the change in direction of the unit sphere volume curve as a function of dimension?" (It increases to a max for n=5.2569464, and then decreses.)

I don't know myself. But in my geeky opinion, maxima in curves are cool ;-). Shanes 01:20, 3 January 2006 (UTC)

sources? pictures?

Not that I am one to be able to determine whay, but Mathworld gives a maxima at n~7 and Wiki says n~5. Anyone dare figure out who is right? 00:03, 12 April 2006 65.78.1.68

Mathworld gives the dimension for which the "surface area", not the hypervolume, is maximal; the dimension for which the "surface area" is maximal is indeed about 7.

It's clear from the examples given, which are easy to check by hand, that the max is between 5 and 6. --agr 11:20, 12 April 2006 (UTC)
It's 5.2569464048605767801328... (sequence A074455 in OEIS). It's not hard to calculate with binary splitting. CRGreathouse (t | c) 04:59, 24 September 2006 (UTC)

[edit] Indexing

The wikipedia articles seem (at first glance) to be stunningly consistent as to the definition of an n-sphere (being the surface of an n+1-ball). However, as the MathWorld article states, in the wild there is no consistency: various authors are about equally likely to say a 1-sphere is a circle or two points of a segment (maddeningly, sometimes both in the same article). This should be noted in the main article; I'm not sure how. --128.2.203.167

The usage given in Wikipedia is pretty much universal throughout mathematics. Actually, I've never seen any mathematician use n-sphere to refer to an (n-1)-manifold - can you cite an example? We shouldn't copy mistakes from MathWorld (which is full of them). --Zundark 09:57, 10 March 2006 (UTC)
Well, in all fairness, I have seen Coxeter use the alternate indexing, but otherwise I agree with Zundark. -- Fropuff 18:43, 10 March 2006 (UTC)
Spheres are perfectly good manifolds in the absence of balls. Just because you can embed an n-sphere in Euclidean (n+1)-space is no reason to call it an (n+1)-sphere, so no mathematician in their right mind will. So Coxeter must have made a typo or had an overzealous editor or whatever. --MarSch 17:37, 12 April 2006 (UTC)
Seems like a mess. 3-sphere is talking about a 3D surface in 4-space, while this article says a 3-sphere is an ordinary sphere in 3-space. See also Talk:3-sphere#n-sphere_in_n-space.3F. Apparently an n-sphere is the surface, and an n-ball is the interior??? Tom Ruen 03:14, 6 October 2006 (UTC)
Ugh, reading again. I took The term n-sphere is used for a sphere of dimension n' to mean n-sphere is embedded in n-space, rather confusing, and later while being misled it says by example an ordinary sphere in three dimensions is a 2-sphere, denoted by ; the 1-sphere being a circle, and the 0 -sphere is the end points of an interval.. So an (n-1)-sphere bounds an n-ball which has n-dimensional volume. Apparently a sphere only has area and a ball IN the sphere has volume??? Yucky yuck indexing! Tom Ruen 03:20, 6 October 2006 (UTC)
Anyone feel free to improve, but I felt is necessary for serious clarification, which I attempted to do with the Warning. I'm still not satified, but better than it was. Tom Ruen 03:51, 6 October 2006 (UTC)
It wouldn't be so confusing if we just make an effort to be consistent. As far as I'm aware 99% of the mathematics community uses the term n-sphere to mean a sphere in n+1 dimensional space. I don't think undo attention should be given to the other convention. A simple remark is sufficient. The big warning box seems like serious overkill to me. -- Fropuff 04:09, 6 October 2006 (UTC)
You don't consider it confusing to talk of n-balls not having an n-sphere surface? Tom Ruen 05:21, 6 October 2006 (UTC)
Of course not. A 3-ball has a volume and a 2-sphere has an area. The boundary of a n-manifold is an (n−1)-manifold. This seems perfectly natural to me. -- Fropuff 05:29, 6 October 2006 (UTC)
Well language IS confusing when coming from different purposes. SPHERE and BALL are english words, both imply something in 3D. I wouldn't expect a 2-sphere to mean the surface covers a 3-ball volume because I see both as meaning solids. In polyhedra, is a dodecahedron a surface of 12 pentagons or the volume of space enclosed by the pentagons? Well, it depends on what you're interested in! Apparently you say 99% use 2-sphere=sphere because they're interested in topology. That doesn't make it less confusing to people looking at the volume! Tom Ruen 05:57, 6 October 2006 (UTC)
Okay, clear enough for me now, being stupid as I am. HOWEVER strange over half the article refers to the volume which has nothing apparently to do with the hypersphere itself! Tom Ruen 06:35, 6 October 2006 (UTC)

OK, admittedly I hadn't read this discussion last time I put that note in. We can't simply ignore the issue though. I've put a subsection describing the two "conventions" back in. 129.16.97.227 11:08, 20 June 2007 (UTC)

[edit] "hypermeridian" - possible neologism

The word "hypermeridian" seems to be a neologism. All Google results for "hypermeridian(s)" originate from Wikipedia. --Ixfd64 03:23, 20 December 2006 (UTC)

[edit] Hypersphere image

Unlike the Tesseract image, I cannot see how the picture on the hypersphere page could possibly be a hypersphere. A photograph taken of a four dimensional sphere would look exactly like a circle if it wasn't for light, just like a photograph of a sphere would look exactly like a circle if there was no light. I am sorry if you found the previous sentence hard to understand since a photograph can't be taken without light, but if no one knows why that picture is up there within the next day or so, I will go ahead and remove it.--eskimospy (talk • contribs • count) 01:55, 15 May 2007 (UTC)

Look at the picture in the sphere article. Does that look like a circle? I doesn't because it shows two sets of lines, parallels and meridians, that form a coordinate system for the sphere. Well, hyperspheres have three sets of lines, parallels and two orthogonal sets of meridians. That is what the hypersphere picture shows. Maybe a better explanation is called for, but the image is correct, so please leave it alone. --agr 02:31, 15 May 2007 (UTC)
Yes, it does look like a circle. Notice the outline is a circle. The outline of the hypersphere is not a circle. For that picture to be a four dimensional sphere, it would have to be made of other three dimensional spheres (and three dimensional spheres only) as a three dimensional sphere is made up of two dimensional circles (the lines in the picture of the three dimensional sphere are circles).--eskimospy (talk • contribs • count) 22:55, 15 May 2007 (UTC)
The image in this article is not an entire hypersphere. It's a Stereographic projection of one pole of the hypersphere. A portion of the surfaces, which is three dimensional, is flattened out and fills space. You can also file a hypersphere with circles. There are three orthogonal sets, instead of two on an ordinary sphere. The projections of those lines are what is being shown here. --23:47, 15 May 2007 (UTC)
Okay, I see what you mean.--Eskimospy 03:48, 16 May 2007 (UTC)

[edit] Move to n-sphere

This article really needs to be moved to n-sphere. Hyperspheres generally refer to spheres of dimension > 2; the term also carries the strong suggestion that the spheres are embedded as hypersurfaces, which is inappropriate in many contexts. The argument against (mentioned above) is very weak: the use of a variable "n" in the title. In fact n is the defacto standard letter for the dimension of a manifold in general, and for spheres in particular. It is almost as canonical as the zeta in zeta-function. I would just move the page, but n-sphere has one edit, so an admin is needed. Geometry guy 21:49, 30 October 2007 (UTC)

I completely agree with Geometry guy here. This article should be about spheres in general dimensions not just n>2. Besides which the hypersphere terminology is rather uncommon in my experience. If there are no objections in the next few days I will go ahead and make the move. -- Fropuff 05:55, 31 October 2007 (UTC)
For what it's worth, there was an argument last year over at Talk:Measure polytope regarding hypercube vs. n-cube. It would seem that hypercube won out, though I still disagree with that decision. -- Fropuff 06:12, 31 October 2007 (UTC)

No comments or objections, so I went ahead with the move. -- Fropuff 23:45, 5 November 2007 (UTC)

[edit] Hyperspherical Volume Element Reference

Hey I'm brand new to this so maybe somebody can edit:

See

http://faculty.matcmadison.edu/alehnen/sphere/Apendxa/Appendixa.htm

See Theorem A7 for a possible reference to:

3rd last line, under the heading Hyperspherical coordinates --> the n-sphere volume element

Thanks,

Matt —Preceding unsigned comment added by 129.97.22.48 (talk) 03:22, 7 November 2007 (UTC)

[edit] Vanishing space

Little notice has been taken of the significance, in terms of cosmology,of the manner in which an increase in dimensionality decreases volume (content) in relation to surface area.The circle, sphere, etc, gets smaller and smaller in comparison with the smallest regular box which it can be placed in (square, cube, etc), as it gets rounder and more corners are cut off. By the 11th dimension, this corner waste is 99.91%; so that a hendecaglome which exactly fits in a hendecacube box, occupies less than a thousandth the space, yet touches the all 22 decacube sides. Colcestrian 21:48, 8 November 2007 (UTC)

[edit] Double factorials

Why are all formulas for the hypervolumes given in terms of the obscure double factorial, instead of the ubiquitous factorial? IMHO, they should be given primarily with factorials, with a side note that they get a minor simplification with double factorials. Albmont (talk) 17:37, 25 March 2008 (UTC)

I agree with the spirit of your suggestion. Besides which, n!! could easily be read as (n!)!.
Can you confirm how it should be written in terms of simple factorials?
The article at the moment has:
(For even n, \Gamma\left(\frac{n}{2}+1\right)= \left(\frac{n}{2}\right)!; for odd n, \Gamma\left(\frac{n}{2}+1\right)= \sqrt{\pi} \frac{n!!}{2^{(n+1)/2}}, where n!! denotes the double factorial.)
Out of interest, most of the other language Wikis don't mention the odd case at all, except for Polish (which I think actually has a slightly nicer layout for the formula), Czech and Thai.
—DIV (128.250.80.15 (talk) 02:06, 28 April 2008 (UTC))

The odd formula can be rewritten as \sqrt\pi \frac{n!!}{2^{(n+1)/2}} = \sqrt\pi\frac{n!}{\frac{n-1}2!\,2^n}, I think. —David Eppstein (talk) 02:47, 28 April 2008 (UTC)

[edit] Intuitive proof: why the n-sphere Volume goes to zero in the large n limit

To make a sphere from the unit n-box, we must snip the corners. With n large, we have a lot of snipping to do, eventually we are snipping everything. This must mean the number of corners beats the dimension. Let us examine this. 2-box has 4-corners, 3-box has 8-corners, 4-box has 16-corners, n-box has 2n-corners. So even though the volume is multiplied by another dimension, the number of snips goes as the power of dimension. Ergo, snips win!

(I will let someone else add this if they think it applicable.)

This must be true, therefore, for any inscribed polytope.

I have just noticed Colcestrian noted the property of the snips, but I have added the explicit reason why the snips win. Colcestrian also mentions cosmological implications. For a given object of constant mass, if the volume goes to zero, we form a black-hole in n-space simply by increasing the dimension. This is strange to a General Relativist - black holes without curvature.

I thought there may be a quantum cosmological implication. However, I am convinced now that as n increases, the 3-space increases even as the n-volume decreases. This is a simple consequence of taking n-3 derivatives of the n-ball volume to get the 3-volume. So you cannot form a 3-dimensional black hole by increasing the number of dimension. This leads to an even stranger conclusion. An n-black hole would look quite normal in 3-space, for n>>3.

Bbharim (talk) 09:37, 1 April 2008 (UTC)

But each snip takes a smaller fraction of the volume as the dimension increases. You have to account for that as well. Also, please put comments at the end of the page, not at the beginning, thanks. PAR (talk) 09:29, 3 April 2008 (UTC)
Indeed not. If you just snip off the simplex at each of the corners, the volume of the resulting polytope does not tend to zero. You need to snip off some of the curvy bits as well. So this explanation is quite misleading. silly rabbit (talk) 12:00, 3 April 2008 (UTC)
After much thought I stand. The Volume of any polytope inscribed inside the inscribed n-ball goes to zero in the large n-limit, including strangely, any inscribed n-box. (No curvy bit snips needed). This stuff is highly counter-intuitive. The correct intuitive understanding is the 2n number of snips. This overwhelms all other considerations, and applies (in a general sense) to all shapes. Bbharim (talk) 09:12, 6 April 2008 (UTC)
I agree that the volume of any polytope inscribed inside the unit N-ball tends to zero, but this has nothing to do with my post above. It is clearly false that any polytope inscribed inside the N-cube tends to zero, which is what needs to happen in order for your corner-snipping argument to work. My interpretation of your argument is the following: Consider the N-cube of side length 2 (so that the unit N-sphere can be inscribed in it). The total volume of the cube is 2N. Now, consider the unit simplex at each of the 2N corners. The volume of this simplex is
\frac{1}{N!}.
So that, the total deleted volume is
\frac{2^N}{N!}.
But by Stirling's formula,
\frac{2^N}{N!}\to 0
as N\to\infty. So this snipping procedure cannot possibly exhaust the volume of the N-cube in the limit. It is likely that you are imagining a different way to do the snipping, but I would like to see the details of the calculation. silly rabbit (talk) 13:26, 6 April 2008 (UTC)
Silly Rabbit is right. Just because the number of snips (2n) increases while the volume of a unit n-box does not, does not mean that the volume of the 2n snips increases. In fact, for "linear snips", it decreases, as Silly rabbit has shown above. You need to show that 2n times the volume of each "curvy snip" overwhelms the volume of the unit n-box, not just that 2n overwhelms it. PAR (talk) 16:12, 6 April 2008 (UTC)