Talk:N-sphere/Archive 1

Archive 1

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)

stereographic Projection

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

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)

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)

This formula

$V_{n}={\pi ^{\frac {n}{2}}R^{n} \over \Gamma ({\frac {n}{2}}+1)}={C_{n}R^{n}}$

seems to be wrong to me. For example, since the gamma function is factorial on the natural numbers, n=2 gives

$V_{n}={\pi R^{2} \over 2}$ —Preceding unsigned comment added by 192.102.214.6 (talk) 16:35, 10 September 2008 (UTC)

No. Γ(n)=(n−1)!. So in particular Γ(2)=1, not 2 as in your formula. siℓℓy rabbit (talk) 19:24, 10 September 2008 (UTC)

Thanks. My bad. —Preceding unsigned comment added by 80.229.247.11 (talk) 22:46, 10 September 2008 (UTC)

Error in reduced volume formula for odd N

$C_{3}=2^{(3+1)/2}{\frac {\pi ^{(3-1)/2}}{3!!}}={\frac {4\pi }{720}}\neq {\frac {4\pi }{3}}$

Correct formula is given here: [2] —Preceding unsigned comment added by 79.97.228.230 (talk) 11:06, 10 July 2010 (UTC)

$3!!\neq 720$

$3!!=3$

This is the double factorial and not the one you're using. 74.140.199.194 (talk) 10:13, 16 July 2010 (UTC)

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 the OEIS). It's not hard to calculate with binary splitting. CRGreathouse (t | c) 04:59, 24 September 2006 (UTC)

This section in the article (cited Wed Jun 3 2009 12:40 AM)

"The non-monotonic behaviour of the numerical value of n-sphere's volume as a function of n may seem strange at first glance. However, by assigning units of length to each dimension one can see it is meaningless to compare the unit-sphere volumes in different n's, just as it is meaningless to compare a length to an area in other contexts. A meaningful comparison is obtained by using a dimensionless measure of the volume, such as the ratio of the n-sphere and its circumscribed hypercube volumes. Using this measure restores the intuitively normal behavior of a monotonic decline in the volume as the dimension increases."

seems to be bunk to me... By this logic, an equally meaningful comparison is obtained using the unit hypercube volume instead of the circumscribed hypercube, since that will again be dimensionless, and we will not see monotonic decline in volume as dimension increases. I suggest that this paragraph is removed. -- Anon (Wed Jun 3 2009 12:43 AM)

I was about to make exactly that comment. It is bunk, and moreover it's someone's unsourced opinion, and would have no place in Wikipedia even if it weren't bunkitudinal.—GraemeMcRae^talk 17:25, 2 November 2009 (UTC)

The comparison of volumes from one dimension to another, including non-integer dimensions was just inserted (and deleted) again. Do you think this is an indication that we should put something into the article itself to explain why such comparisons are inappropriate? —Quantling (talk) 20:43, 27 October 2010 (UTC)

If you can describe precisely what a 2.57-dimensional Euclidean space is and what a sphere in that space is (and give a reliable source for it), then I might be willing to allow something on that in the article. There is no need to explain in the article why we did not include one of the infinite number of things which do not belong here. JRSpriggs (talk) 20:53, 27 October 2010 (UTC)

The article is on the n-sphere and there's no way that I know to describe an n-sphere in e.g. 5.25694640486 dimensions: such fractional dimensions have their uses but describing n-spheres is not one of them.--JohnBlackburne^words_deeds 20:57, 27 October 2010 (UTC)

The non-integral dimension stuff is part of Hausdorff measure, where one does use the formula for C_n evaluated at non-integral values of n, so that's not the part I thought I'd have to explain. The reason that I now agree with you, JRSpriggs, GraemeMcRae and JohnBlackburne, that the text that was recently inserted and deleted does not belong is that it compared values C_n and C_n' for n ≠n', and that is what I now see is the "false" information. I was thinking we would add something very simple to the N-sphere article, along the lines of "Comparisons of the volume V_n(R) across different values of n are not meaningful; lengths, areas, and higher-dimensional volumes are not comparable with each other." I agree that the statement applies more generally than N-spheres, but since the statement is short, it does apply to N-spheres, and there is a chance that it will stop the volume vs. dimension edits, I am thinking that it could be worthwhile. —Quantling (talk) 21:10, 27 October 2010 (UTC)

Since this nonsense is already in the unit sphere article, and since the suggestion was made a few years ago to merge that article with this one, I wonder if it can be done, with a short section describing the numerical comparison of the "volume" and "surface area" of n-spheres of different dimensions, since these ideas are sourced in Mathworld and OEIS, and why this comparison is meaningless, even if one defines the supposedly dimensionless ratio of n-cube to n-sphere "volume". I think that whole thing would be a worthwhile undertaking.—GraemeMcRae^talk 14:13, 28 October 2010 (UTC)

I added a sentence to the article indicating the noncomparability of volumes of different dimensionality. Perhaps someone else can tackle the task of explaining why they are not comparable, even in the "dimensionless" versions of the comparisons. —Quantling (talk | contribs) 15:13, 9 November 2010 (UTC)

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)

"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)

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)

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)

Hyperspherical Volume Element Reference

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

See

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

{http://faculty.matcmadison.edu/alehnen/sphere/Apendxa/Appendixa.htm this link (—original) is incorrect/fails- corrected above}

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)

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)

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)

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 2^N. Now, consider the unit simplex at each of the 2^N 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 ($2^{n}$) increases while the volume of a unit n-box does not, does not mean that the volume of the $2^{n}$ snips increases. In fact, for "linear snips", it decreases, as Silly rabbit has shown above. You need to show that $2^{n}$ times the volume of each "curvy snip" overwhelms the volume of the unit n-box, not just that $2^{n}$ overwhelms it. PAR (talk) 16:12, 6 April 2008 (UTC)

Concerning corner snips, note that with $R=1$ the box volume is $2^{n}$. The number of corners is also $2^{n}$. Each corner is √n from the center, and is very far outside the sphere for large $n$. For $n=1$ the 1-sphere and 1-cube are identical line segments. As $n$ increases the volume of each curved snip increases, to a limit of unity. For large $n$ the "curved snip" surrounding the corner with all coordinates plus one is basically the whole unit cube with positive coordinates; the sphere occupies only a tiny fraction of this unit cube. GOR42 (talk) 16:08, 8 July 2008 (UTC)

Indeed, except that if one isn't careful with the snipping, then it is possible to remove more than you had to begin with. The naive approach of snipping off the simplices bounded on one side by a tangent plane to the sphere and on the other side by the corner of the cube turns out to remove parts of the cube more than once, which is clearly not allowed without some explicit accounting. The only workable way to use linear snips (that I know of) is the one I discuss above, and that way fails to cut out the sphere since the resulting volume does not converge to zero. siℓℓy rabbit (talk) 17:48, 8 July 2008 (UTC)

Here is one way to see that the limit of the ratio R(n) of [the volume of the unit radius n-ball Dⁿ] to [the volume of the side-2 n-cube Qⁿ] approaches 0 as n → ∞. Note that the projection of Dⁿ to the plane spanned by any two of the n coordinates (i.e., just forget the other n-2 coordinates) is a D².

Hence if n = 2k, we know that Dⁿ lies inside of (D²)^k, i.e., D^2k ⊂ (D²)^k. Hence vol(Dⁿ) ≤ (vol(D²))^k = π^k. And of course vol(Qⁿ) = 2^2k = 4^k. Thus R(2k) ≤ π^k / 4^k = (π / 4)^k. In short, R(2k) ≤ (π/4)^k.

And if n = 2k+1, we can take the cartesian product of each side of the inclusion D^2k ⊂ (D²)^k with the interval [-1,1], showing that D^2k+1 ⊂ (D²)^k x [-1,1]. Hence R(2k+1) ≤ (2⋅π^k) / (2⋅4^k) = (π / 4)^k. In short, R(2k+1) ≤ (π/4)^k.

Note that k = floor(2k/2) = floor(2k+1)/2). And so, whether n is odd or even, we always have R(n) ≤ (π/4)^floor(n/2). Since 0 < π < 4, this means that R(n) → 0 as n → ∞.Daqu (talk) 07:12, 1 September 2013 (UTC)

                   *                    *                    *

A question: People discuss above whether or not "any polytope inscribed inside the n-cube tends to zero". But what does this even mean, for "any" polytope?

It's clear what it means in the cases of the three families of regular polytopes that exist in all dimensions: The simplex, the cube, and the "cross-polytope" -- the dual of the cube (which in 3D is just the octahedron).

But for an arbitrary n-dimensional polytope, what do its higher-dimensional versions even mean? Just curious.Daqu (talk) 03:55, 2 September 2013 (UTC)

More of a question for the help desk I think. The n-cube itself is a polytope and when inscribed in itself the ratio of the volumes is 1. For cross polytopes my calculations give the ratio as 1/n!. It's not clear how an n-symplex would be "inscribed" in an n-cube. For n=2 I get .46 as the largest possible ratio. --RDBury (talk) 06:50, 2 September 2013 (UTC)

I intended the question for the people who wrote: "It is clearly false that any polytope inscribed inside the N-cube tends to zero . . .", − and for that matter, also "The Volume of any polytope inscribed inside the inscribed n-ball goes to zero in the large n-limit", so I can evaluate their reasoning. I have no idea what connection the help desk might have with answering this question. But my point is that, in general, there is no clear meaning one can give to generalizing to higher dimensions an arbitrary polytope inscribed in the n-cube.

You are certainly right, RDBury, that there are many ways that a regular n-simplex can be inscribed in a (say unit) n-cube. But one could always take for each n the one of maximum volume and ask what fraction of the volume of a (side-1) n-cube this n-simplex takes up, and what happens to that number as n →∞.Daqu (talk) 16:06, 4 September 2013 (UTC)

Williams Reference - N-Spheres and Archimedes

I believe Williams' proof is wrong and the reference should be removed. GOR42 (talk) 16:08, 8 July 2008 (UTC)

Done! I should have removed it to begin with on the basis of WP:SELFPUB, but I hadn't examined the source. Thanks, siℓℓy rabbit (talk) 05:15, 8 July 2008 (UTC)

I noticed that for negative integer dimensions, the following formula,

$V_{n}={\pi ^{\frac {n}{2}}R^{n}*Gamma(1-{\frac {n}{2}})}$,

is equivalent to V(+d) = 1/V(-d). Boris. —Preceding unsigned comment added by 161.209.206.1 (talk) 21:26, 17 September 2008 (UTC)

Uniform point picking from an N-ball

Since N-Ball redirects here, I think it is an appropriate place to include the procedure for uniform point-picking from an n-ball. The internet seems woefully devoid of this procedure, and it's fairly simple and useful. Just pick a uniformly random point on the n-sphere (which is already explained in the article), generate a random value X uniform on [0,1], take the nth root of X, and multiply it into the random point from the n-sphere. This is a generalization of the procedure described on MathWorld (http://mathworld.wolfram.com/DiskPointPicking.html), and even on MathWorld the generalization is not given anywhere.

We need a reference to be able to add this to an article. You describe it as if it were original research, which we can't include, but it seems very likely to me that this is described already in some published source that we can cite. —David Eppstein (talk) 00:07, 14 April 2010 (UTC)

Certainly I have been unable to find any references to this procedure on the internet other than the MathWorld disk point-picking article. It is "original research," I had to generate the proof myself. It's obvious that the distribution is radially symmetric and it's fairly easy to show that it generates the correct probability distribution over radii. However, I suspect that you're correct and the procedure has been described elsewhere. Finding that proper reference is the trick.

The correctness of the procedure is obvious enough; perhaps it could be added without a citation. Quantling (talk) 13:47, 14 April 2010 (UTC)

Try Generating random points in a ball, Sanjeev Banerjia;Rex A. Dwyer; DOI: 10.1080/03610919308813149; Communications in Statistics - Simulation and Computation, Volume 22, Issue 4 1993 , pages 1205 - 1209. Penguian (talk) 15:41, 20 April 2010 (UTC)

n-sphere vs n-dimensions

Hi. I'm a newbie here.

I find it confusing that an n-sphere is in (n+1) dimensional space.

Then the letter "n" is used in the formula for the volume. V = ... R^n.

I think a different letter, say perhaps k or m, should be used for other formulas that are currently using "n".

If n-spheres go n=0, n=1, n=2, etc... then n is defined as which type of n-sphere you are using. The letter should not be re-used as the number of dimensions.

Does anyone else agree?

- TKinsman —Preceding unsigned comment added by Tkinsman (talk • contribs) 02:08, 22 August 2010 (UTC)

There is a good reason for calling the surface of an (n+1)-dimensional ball an n-sphere, namely that it comes up most often in contexts where we're concerned only with the n degrees of freedom in that surface rather than with what lies within or above it. But I agree that the split usage in this article is needlessly confusing. —Tamfang (talk) 02:15, 23 August 2010 (UTC)

So, could we change the "n-dimensions" to another letter? How do people come to a concensus on Wikipedia?

Tkinsman (talk) 09:33, 23 August 2010 (UTC)

We don't want to change all the n's to n+1 because the formulas would be a mess. Maybe the best would be to specify that the formulas are for an (n-1) sphere, and that the reason for doing that is that the formulas are simpler. Next best would be to use maybe m=n-1 and say the formulas are for an m-sphere. This conversation we are having right now is how people come to a consensus on Wikipedia. PAR (talk) 12:19, 23 August 2010 (UTC)

Topologically and diffeomorphically, the n-sphere, which is an n-dimensional Riemannian manifold, can be embedded as other than the surface of an (n + 1)-ball. Though, I too find the surface of an (n + 1)-ball to be the most "natural" such embedding, I nonetheless think it is important to keep the article in terms of n itself. Otherwise, the casual reader may be lead to believe that, for the n-sphere, there is something necessary about the (n + 1)-ball. Perhaps more importantly, mathematicians have agreed to refer to, e.g., the earth's surface as a "2-sphere" rather than the "surface of the 3-ball," and I don't think Wikipedia is the place to second guess that. Quantling (talk) 14:01, 23 August 2010 (UTC)

As for the original question, I'll expand on the above comment by Quantling: Euclidean space Rⁿ⁺¹, like the sphere of radius 1 in Rⁿ⁺¹ that is centered at the origin — called Sⁿ — are each topological spaces that are "locally Euclidean". A space X is locally Euclidean when every one of its points has a neighborhood that is topologically equivalent to an open set in some Euclidean space. When those Euclidean space(s) are of a constant dimension k, then X is said to be a manifold of dimension k. In the case of the Euclidean space Rⁿ⁺¹, obviously every point of it has a neighborhood topologically equivalent to an open set in itself -- by definition. And so it is a manifold of dimension n+1. In the case of the sphere Sⁿ, by the use of stereographic projection (and many other ways) it can be easily be shown to locally Euclidean of dimension n. That is the reason for the numerical designations.

In topology and geometry, a manifold can be isolated from any space within which it may be originally defined. So the sphere Sⁿ need not be thought of as a subset of Rⁿ⁺¹, but instead may be thought of as a space unto itself. Or, it can also be a submanifold of any manifold of dimension ≥ n+1. (For instance, it's an equator of an equator of Sⁿ⁺². Sⁿ could just as well be thought of as a submanifold of Rⁿ⁺⁷³.) For this reason, it's better to designate the dimension of a manifold by the "stuff it's made of" rather than by which manifold it happens to be a subset of in any one way of thinking about it.Daqu (talk) 03:33, 11 September 2013 (UTC)

hypersphere area element

I removed the hypersphere area element paragraph because the area element should be a sum of forms of penultimate rank. But it wasn't. Also, shouldn't it be expressed in hyperspherical coordinates rather than Cartesian coordinates? —Quantling (talk | contribs) 13:32, 22 March 2011 (UTC)

definition of n-sphere as a topological space without requirement of embedding

It is fine to introduce the n-sphere as an embedding in $\mathbb {R} ^{n+1}$ space. But it can have an independent definition as a topological space without the embedding. An n-sphere is a n-manifold such that if a single point from it is removed, it becomes homeomorphic to $\mathbb {R} ^{n}$. Or in other words, an n-sphere is a topological equivalent of $\mathbb {R} ^{n}\cup \{\infty \}$. This is the basis for stereographic projection. (Reference: James W. Vick, "Homology theory", pp. 40). I think it will be good to add this definition. - Subh83 (talk | contribs) 21:19, 24 March 2011 (UTC)

Why don't you add it? I think it should come after the embedding idea, since the embedding is more intuitive for a new reader.PAR (talk) 04:59, 25 March 2011 (UTC)

Just did.- Subh83 (talk | contribs) 06:32, 25 March 2011 (UTC)

square root of pi

Copied from [3].--Salix (talk): 17:16, 16 April 2011 (UTC)

There is a small error in the article about the n-sphere. it states that (1/2)! is the square root of pi, in factt it should be a half of the square root of pi, namely about 0.886226.

I do not know how this equation editor works so cannot correct it myself. Anthony C Robin — Preceding unsigned comment added by 213.106.253.19 (talk • contribs)

It doesn't say (1/2)!=√π; it says Γ(1/2)=√π. Not the same thing. —Tamfang (talk) 19:44, 16 April 2011 (UTC)

Limiting behaviour section

I just reverted this as I have two concerns. First it is unclear what the limit is. E.g. each of the V_n is an n dimensional value. So if the radius is 1 metre the sequence is 1, 2m, 3.14m², 4.19m³. So is the limit measured in metres, square metres or what? I would say they are not directly comparable and so no limit exists. Given this I think we really need the source of this, to see what it is based on. It may be clearer then exactly what the result is.--JohnBlackburne^words_deeds 23:48, 11 July 2011 (UTC)

The content of the n-sphere as a fraction of the circumscribed hypercube tends to zero; this is meaningful and, I think, worth mentioning. On the other hand, the language that you removed was sloppy as you point out. —Tamfang (talk) 21:14, 22 July 2011 (UTC)

Yes it converges to zero as anyone able to follow the algebra can see, but including it based on that is OR as it requires more that trivial arithmetic. So it could still do with a reference, which would also clarify how meaningful or notable it is. --JohnBlackburne^words_deeds 21:28, 22 July 2011 (UTC)

values for V₀ and S₀

the article states:

This leads to the recurrence relations: $V_{n}={\frac {2\pi R^{2}}{n}}V_{n-2}$ $S_{n}={\frac {2\pi R^{2}}{n-1}}S_{n-2}.$

I think the values for V₀, V₁, S₀, and S₁ should be added

Just granpa (talk) 14:45, 8 January 2012 (UTC)

V₀ = 1r⁰

V₁ = 2r¹

V₂ = πr²

S₀ = 2r⁰

S₁ = 2πr¹

Just granpa (talk) 15:02, 8 January 2012 (UTC)

S₁ = 2πr is already in the section, I'm not sure the $V_{0}$ the volume of inside $S_{-1}$ sphere is well defined. For the 0-dimensional spheres V₁ = 2r is fine but S₀ = 2 requires some explanation as what a 0-dimensional measure is.--Salix (talk): 22:12, 8 January 2012 (UTC)

S₀ is the 2 endpoints at either end of a 1 dimensional sphere.

Just granpa (talk) 22:47, 8 January 2012 (UTC)

I mostly agree with Salix on this, the V0, S1 and S0 values work for the recurrence, so you might take these as defined values, but the idea of a 0-volumn is rather questionable and a -1-volumn even more so. The entire section is unreferenced though so it's a bit difficult to nit pick.--RDBury (talk) 14:10, 9 January 2012 (UTC)

There is nothing odd about a 1-dimensional volume. I have no idea why you would think there was. I accept that V₀ is wrong though.

Just granpa (talk) 20:49, 9 January 2012 (UTC)

I was one off in the indices for S so ignore what I said about S1. The idea of a 0-volumn is still questionable.--RDBury (talk) 14:59, 10 January 2012 (UTC)

Actually, now that I think about it some more V0 might not be meaingless after all. Certainly there is no circle of points a distance r from the origin (hence it would be meaningless to talk about a surface area) but the volume within that distance still includes the one single point at the origin. Hence V0=1.Just granpa (talk) 17:26, 10 January 2012 (UTC)

What do you mean by:

(The 0-sphere consists of two points, at −R and +R; its 0-volume is the number of points, S0 = 2.)

the volume of the 0-sphere (line segment) is 2r.

the 2 points at +R and -R represent the surface area of teh 0-sphere.

I'm not sure what is meant by 0-volume.

Just granpa (talk) 21:27, 10 January 2012 (UTC)

The 1-ball of radius R centered at the origin is the line segment from −R to +R on a 1-dimensional number line. Its boundary is a 0-sphere, which has points at -R and +R. If you are brave you might take a look at Hausdorff measure to grapple with what volume means when the number of dimensions is not a positive integer. In this case, I think it is correct to conclude that the 0-sphere has a 0-volume of 2, because with any positive radius no matter how small it is always possible to find two 1-balls of that radius whose union contains the two points (and the same cannot be said if you restrict yourself to one 1-ball and use a very small radius). —Quantling (talk | contribs) 21:49, 10 January 2012 (UTC)

The 1-dimensional volume of a 1-dimensional sphere is teh 1-dimensional length between the 2 points that are a distance r from the origin. Therefore the 1-dimensional volume would be 2r. When speaking of a 0-dimensional sphere its confusing. First of all in 0-dimesions the only thing that exists is a single point. There are no points a distance r from taht solitary point. However that solitary point is within a distance r from the origin and therefore counts as belonging to the volume of the 0-dimensional sphere. That agrees with the math which says that the 0-dimensional sphere hase volume of 1r^0 Just granpa (talk) 22:55, 10 January 2012 (UTC)

The only thing that has a value of 2 is teh surface area of a 1-dimensional sphere. These are the 2 points that are distance r frmo the origin on teh 1-dimensional line segment. You seem to be calling that a volume for some reason I cant fathom. Just granpa (talk) 22:57, 10 January 2012 (UTC)

From the Hausdorff measure article, the "The zero dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite.". The 0-sphere is a set with just two points so its the zero dimensional Hausdorff measure is 2. 0-volumn, is just short hand for zero dimensional Hausdorff measure.--Salix (talk): 23:31, 10 January 2012 (UTC)

but you keep saying volume when you mean surface area. It is S0 that equals 2 not V0. Just granpa (talk) 23:49, 10 January 2012 (UTC)

"0-volume" is the same as "number of points"

"1-volume" is the same as "length"

"2-volume" is the same as "area"

"3-volume" is the usual three-dimensional concept of "volume"

"4-volume" is often called "hypervolume" though I've seen the latter term also used for higher-dimensional volumes.

Put that with the fact than an n-ball's "surface" is an (n-1)-sphere, and maybe it all makes sense? —Quantling (talk | contribs) 19:28, 11 January 2012 (UTC)

of course 1-volume is length but so is the surface area of a 2-dimensional sphere. One is a volume and one is a surface area. both are 1-dimensional.

'Number of points' is 0-dimensional. It can refer to the volume of a 0-dimensional sphere or to teh surface area of a 1-dimensional sphere. Just granpa (talk) 19:49, 11 January 2012 (UTC)

S_n-1 as a function of V_n-1

the surface area of an n-dimensional cube (S_n-1 in the article) is equal to 2n * the volume of an n-1 dimensional cube (V_n-1 in the articlee) of the same size.

Anyone know what the equivalent formula for spheres would be?

Just granpa (talk) 18:20, 8 January 2012 (UTC)

Use the ratio of the two formulae?

$S_{n}(R)={\frac {2\,\pi ^{\frac {n+1}{2}}}{\Gamma ({\frac {n+1}{2}})}}R^{n}$

$V_{n}(R)={\frac {\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}}+1)}}R^{n}$

I compute a ratio of

${\frac {2{\sqrt {\pi }}\,\Gamma ({\frac {n}{2}}+1)}{\Gamma ({\frac {n+1}{2}})}}\sim {\sqrt {2\pi n}}$

—Quantling (talk | contribs) 19:48, 10 January 2012 (UTC)

What are you some kind of magician? I spent days trying to figure that out.

I even asked on a major science forum and teh math expert there couldnt answer it. Just granpa (talk) 20:36, 10 January 2012 (UTC)

It approaches that value as n goes to infinity but for small values of n its a bit off. Just granpa (talk) 20:50, 10 January 2012 (UTC)

Since you know that is close to the right answer could you plug that back into the forumula and calculate the correction factor? Just granpa (talk) 20:52, 10 January 2012 (UTC)

The left-hand side of my last formula is exact; no corrections are required. The right-hand side, ${\sqrt {2\pi n}}\,,$ is asymptotically correct for large n. We could go to the trouble of finding next-order terms for that asymptotic expression, and they will give you more digits of precision for large n, but not necessarily for (and probably not for) small n. —Quantling (talk | contribs) 22:00, 10 January 2012 (UTC)

I suppose you could write the values of the Gamma functions in terms of factorial and a rational multiple of $\Gamma (1/2)$, and that would give an expression that has about n/2 factors multiplied together, but nicely each factor would be devoid of Gamma functions and pi's. —Quantling (talk | contribs) 22:05, 10 January 2012 (UTC)

Actually its surprisingly accurate even for small values of n. Just granpa (talk) 22:44, 10 January 2012 (UTC)

To get the correction factor

I plugged (gamma[1+n/2] / gamma[0.5+n/2]) / √(n/2) into wolframalpha

Note the 'divided by' rather than the 'equals to'

It gave (sqrt(2) (1+n) ((2+n)/2)!)/(sqrt(n) (2+n) (0.5+0.5 n)!)

http://img860.imageshack.us/img860/8449/50742415.gif

Just granpa (talk) 23:01, 10 January 2012 (UTC)

n-spherical harmonics?

Does anyone know if the construction of an L^2 basis using ultra-spherical polynomials described in the section (N-sphere#Spherical_volume_element) is (essentially) the same as the one using Legendre functions described in the article on spherical harmonics (Spherical_harmonics#Higher_dimensions)? Are the products of ultra-spherical polynomials eigenfunctions of the Laplacian? Steven J Haker (talk) 00:32, 18 January 2013 (UTC)

Why was this removed?

Today I mentioned in the article the fact that, amusingly, the sum of all even-dimensional balls' volumes is e^π.

Within a few hours, someone had undone this part of my revision, pointing out that "you can't sum volumes across dimensions", as well as questioning my use of the word "amusing".

I don't think it's a bad thing to mention mathematical coincidences. That is how several people won Fields medals for explaining John Conway's observation of "monstrous moonshine".

And you certainly *can* sum volumes from across dimensions, since volumes are positive real numbers, and it is well-known that they can be added. (Even though the sum might be infinity. Which it is not in this case.)

And since I don't know what significance this fact may have -- if any -- I called it "amusing". By which I merely meant intriguing. This word is widely used with reference to mathematical facts that may or may not have significance.

I think at minimum this issue should be discussed on this Talk page before that (true, relevant, and clearly stated) statement is removed.Daqu (talk) 03:29, 28 August 2013 (UTC)

I think at minimum you need to find a reliably published source explaining the relevance of this factoid to the geometry of spheres before it is included. —David Eppstein (talk) 04:41, 28 August 2013 (UTC)

I would state it more strongly than David has. Many simple series sum to expressions with simple closed forms, and "coincidences" of this nature are simply too numerous to give much attention to individually. Though this case can be phrased in the context of multidimensional volumes with a particular choice of unit for the radius and a somewhat arbitrary choice of how to scale this unit to higher dimensions, at best it is an amusing coincidence, because the observation can essentially not be interpreted meaningfully. And even if it did have some more meaningful interpretation, its relevance in the context would still be nil in my opinion. So while it might have some relevance in an article devoted to mathematical "coincidences" (which would have the potential to grow to gigabytes in size), it probably does not belong in an article that is intended to define and describe n-spheres. The "monstrous moonshine" example is more notable in that it is about serious mathematicians paying attention to something; it is not notable for the mathematical detail itself. — Quondum 12:18, 28 August 2013 (UTC)

The fact certainly is easy to interpret as a sum of infinite-dimensional volumes, by just taking the cartesian product of each sphere with the infinite-dimensional cube [0,1]^∞. What exactly leads you (Quondum) to believe this fact is not related to "serious mathematicians paying attention to something" ?Daqu (talk) 15:27, 28 August 2013 (UTC)

You'll have to be more clear by what you mean by cartesian product, which I do not understand to apply in the way that you use it.

I would reinterpret the sum as a series formed by the ratios of the n-volumes of the n-spheres to the n-cubes, with a fixed chosen radius/side ratio. This makes sense as a sequence of (dimensionless) properties of geometric figures, which one could choose to sum. Once could concoct endless such examples though.

As to mathematicians paying this particular example attention, I can only reiterate what the others have said: establish this for this example through a notable reference. Wikipedia has inclusion criteria guidelines. — Quondum 19:40, 28 August 2013 (UTC)

I am the 'someone' and have answered already here, but would also add that we don't add facts as editors find them 'amusing'. Saying so is editorialising, i.e. adding the opinion of the editor to the article which is never done. Stick to the facts as reported in reliable sources.--JohnBlackburne^words_deeds 13:31, 28 August 2013 (UTC)

The fact that the volume of a unit n-sphere approaches zero as n increases certainly has significance in geometry and probability theory. The fact that it approaches zero fast enough so that the sum of the volumes converge to a constant is a stronger result, at least worthy of note. Having said the sum is a constant, readers with enough of a mathematical bent to get this far in our article might wonder what that constant happens to be. I submit that if that constant were hard to compute or exhibit in closed form, it would have a name and and likely an article of it own here, irrespective of the the fact that it sums volumes of different dimension. WP:OTHERSTUFF notwithstanding, in an encyclopedia that features an article Zero is an even number, we can at least mention the fact that the sum is e.--agr (talk) 11:43, 29 August 2013 (UTC)

You seem to be missing a crucial point that has already been made: volumes in different dimensions are not comparable nor summable, until one has established a figure in each dimension against which to compare it as a ratio. Is this the circumscribed (or even inscribed) n-cube, the n-simplex, some other polytope or the inscribed equivalents? Oh, oops, the case being suggested here does not even appear as any of these, it is 2^–n times the circumscribed n-cube. Okay, if we allow that, then why not k^–n times any of the figures, for some other abitrary k? And the n-volume ratio for the inscribed figures does not go to zero – it diverges. So which sum was it you were referring to again? I apologize for coming across as patronizing, but it appears to be necessary in making the point that this is merely one example in a sea of equivalent choices, none of which has any distinguishing merit other than how we choose to extend our units to higher dimensions, and to standardize a sphere's sizes by its radius. I, for one, would argue that the most natural way of extending units to higher dimensions would be in terms of an n-sphere, so the sum of ratios would be infinity. — Quondum 16:24, 29 August 2013 (UTC)

There is a natural comparison object in Rⁿ: the unit n-cube, which has volume 1. That is how volume is normally defined. The reason volumes in different dimensions are not comparable is that lower dimensional objects have measure 0 in Rⁿ. None the less, the volume of a unit n-sphere is a real number for each n, and those numbers form a well-defined, infinite sequence. The behavior of infinite sequences is generally of interest and one way they are studied is to sum them in a series. If the sum is finite, that says something about how fast the sequence approaches zero. Going from a sequence to a series is one of the most basic moves in mathematics.--agr (talk) 18:36, 29 August 2013 (UTC)

Actually, if we're focused on "amusingness", then it's better to sum the volumes of the even dimensional balls of radius R. This sum is given by

$V_{\operatorname {even} }(R)=e^{\pi R^{2}}.$

If we sum the odd dimensional spheres, then we get

$S_{\operatorname {odd} }(R)=V_{\operatorname {even} }'(R)=2\pi Re^{\pi R^{2}}$

is just the derivative of the "total volume" with respect to R, so a rather useful mnemonic for remembering both of these facts. Somewhat harder to remember is that the total volume of the odd dimensional balls of radius R is

$V_{\operatorname {odd} }(R)=2e^{\pi R^{2}}\int _{0}^{R}e^{-\pi x^{2}}\,dx.$

and the total surface area of the even dimensional spheres of radius R is

$S_{\operatorname {even} }=V'_{\operatorname {odd} }(R).$

These two results can be put together. Let V(R) be the sum of the volumes of the R-balls in all dimensions. Then

$V(R)=2\int _{0}^{\infty }e^{-\pi x^{2}}e^{2\pi Rx}\,dx$

(note also that this is just a Laplace transform). If S(R) is the sum of the surface measures of the n-spheres in all dimensions, then

$S(R)=V'(R)=4\pi \int _{0}^{\infty }xe^{-\pi x^{2}}e^{2\pi Rx}\,dx.$

--Sławomir Biały (talk) 12:01, 16 September 2013 (UTC)

I also object to the removal of the sum of spheres of different volumes--- It is clearly relevant that the volume of the n-sphere goes to zero as n goes to infinity. But saying that the the infinite sequence has a finite sum is an even stronger statement than saying the sequence goes to zero, and the sum's striking value will make it easy to remember this fact! So for consistency's sake, one should either put the value of the sum back or remove the weaker statement that the n-volume goes to zero as n-> infinity.

173.79.1.186 (talk) 19:43, 23 December 2016 (UTC)

So find a reliably published source saying so. I'm not particularly troubled by the cross-dimensional issue: clearly it's meaningful to consider shapes like square+circle, cube+cylinder+sphere, etc, and take a limit, but I repeat what I said three years ago: we need a reference explaining why this limit has some mathematical significance before we can include it. —David Eppstein (talk)

Manifold definition?

In the introduction there is the statement: "For n ≥ 2, the n-spheres are the simply connected n-dimensional manifolds of constant, positive curvature.".

Isn't this also true for n=1? (Ziki42 (talk) 23:38, 7 December 2013 (UTC))

No. For n=1 (i.e., a circle) it is not simply-connected (π₁(S¹)≠0). Moreover, the intrinsic curvature (which is what is meant by "curvature of a manifold") of a circle is zero. - Subh83 (talk | contribs) 17:40, 11 December 2013 (UTC)

Recursive formula

I took a glance at the recursive formula in the box and realized that s2 is 4pi and not 2pi x v1 = 2pi x pi =4pi^2. Either I'm right or I'll be right back to strike this off--cha 21:55, 26 January 2014 (UTC) The fact is, I've seen these kinds of mistakes on Wikipedia before, but then again that was a long time agocha 22:15, 26 January 2014 (UTC)

Arccosinus or arctangent?

In "Spherical coordinates" section the hyperspherical coordinates are results of arccosinus function. In some other sources there is arccotangent instead:

[4]

Whose formulas are correct and whose are an error?

--VictorPorton (talk) 22:59, 17 April 2014 (UTC)

You can use either. In the simplest 2D case given x any y you can either calculate r then use that to work out the angle using arccos or arcsin, or you can work out the angle first as arctan (y / x).

So neither way is correct, both can be generalised. The way in the article is better though, I think. It ties in better to how the Cartesian coordinates are calculated from spherical coordinates, and so minimises the number of operations (sin and cos only, not tan) which need to be considered.--JohnBlackburne^words_deeds 23:40, 17 April 2014 (UTC)

I've added alternate formulas

I've added alternate formulas to "Spherical coordinates" section (with arc-cotangent instead of arc-cosine). --VictorPorton (talk) 15:42, 19 April 2014 (UTC)

let a million flowers bloom?

Please be aware that Rgdboer has started a new article at Hypersphere, which formerly redirected here. —Tamfang (talk) 09:57, 22 February 2015 (UTC)

Restrictive definition

The term "n-sphere" generally seems to be used to mean an n-dimensional Riemannian manifold of constant positive curvature (with minor restrictions that assure certain global properties). That such a manifold can always be embedded in a Euclidean space of dimension n+1 does not mean that it is defined as living in such a space. This article has clearly been written from a perspective that confuses the n-sphere with the hypersphere. I'm hesitant to start the edit change, because this is likely to be controversial, but I hope at least to sensitize others that the article seems to forget that n-spheres occur in contents that do not involve an embedding in a Euclidean space. —Quondum 22:22, 23 July 2016 (UTC)

I have started with a change of the lead. Perhaps I should have waited longer for comment; instead the changes can serve as an illustration of the change as a basis for comment. —Quondum 17:51, 25 July 2016 (UTC)

Simpler derivation

I like the derivation of the volume of the n-sphere that proceeds by simply exponentiating both sides of the integral of a Gaussian, reexpressing the volume element as dr r^(n-1) times the volume of the n-sphere. There is no need for fancy coordinate systems. This is how Hormander does it in his book.

173.79.1.186 (talk) 19:47, 23 December 2016 (UTC)

the the

@Loraof: This edit added a "the the" grammar error. I think that it should be "to the", but I'd rather leave the correction to someone who understands the material! -- John of Reading (talk) 09:20, 23 October 2017 (UTC)

Fixed. Thanks! Loraof (talk) 14:42, 23 October 2017 (UTC)

(-1)-sphere

I think it might be worth mentioning something like this in the article:

The definition can be extended to n=-1: A (-1)-sphere is the surface of a point, which is the empty space. Many of the definitions and recurrences that apply for n>=0 also work for n=-1, but some do not. The (-1)-sphere, being trivial, is of little interest in geometry and topology, but the fact that it can be defined is of some interest in category theory and metamathematics.

I don't actually know that the (-1)-sphere is of little interest in geometry and topology. If I'm wrong about that, the article should instead be amended to discuss the (-1)-sphere everywhere relevant.

But I'm pretty sure I could find cites from categorists using the (-1)-sphere as an example of "backward generalizing" or "negative thinking" outside category theory and/or mentioning it in connection with the empty space in Top. --173.228.85.220 (talk) 03:57, 28 October 2017 (UTC)

Minor amendment to derivation

The derivation of one of the recursion formulas (suggested amendment below; I'm having trouble aligning the equal signs) does not appear consistent dimensionally. May I suggest writing out the radius more explicitly before equating it to 1 as in the case of a unit-sphere?

We can also represent the unit (n + 2)-sphere as a union of tori, each the product of a circle (1-sphere) with an n-sphere. Let $r^{2}+R^{2}=A^{2}$ and $r=A\cos \theta$, so that $R=A\sin \theta$, $dR=A\cos \theta \,d\theta$ and $ds=Ad\theta$. Then,

${\begin{aligned}S_{n+2}A^{n+2}=\int _{0}^{\pi /2}S_{1}rS_{n}R^{n}\,ds=\int _{0}^{\pi /2}S_{1}A\cos \theta \ S_{n}R^{n}\,Ad\theta =S_{1}A\int _{0}^{\pi /2}S_{n}R^{n}A\cos \theta \,d\theta \\=S_{1}A\int _{0}^{A}S_{n}R^{n}\,dR=S_{1}A\int _{0}^{A}S_{n}R^{n}\,dR\\=2\pi AV_{n+1}A^{n+1}=2\pi V_{n+1}A^{n+2}\end{aligned}}$

In the case of a unit-sphere, $A=1$, thus $S_{n+2}=2\pi V_{n+1}$.

172.9.11.103 (talk) 05:05, 16 November 2017 (UTC)

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Sampling from uniform distribution over interio of ball

this statement is simply not true : "With a point selected uniformly at random from the surface of the unit (n - 1)-sphere (e.g., by using Marsaglia's algorithm), one needs only a radius to obtain a point uniformly at random from within the unit n-ball. If u is a number generated uniformly at random from the interval [0, 1] and x is a point selected uniformly at random from the unit (n - 1)-sphere, then u1/nx is uniformly distributed within the unit n-ball."

because sampling with radius from uniform distribution returns more point from near the center of n-sphere, example in R :

n=10000 r=1

odl=function(x,y,r,kolo) return(sum(((kolo[,1]-x)^2 + (kolo[,2]-y)^2) < r^2))

kolo=data.frame(x=runif(n)*2-1, y=runif(n)*2-1)

kolo = kolo / sqrt(rowSums(kolo^2))*runif(n)

plot(kolo, cex=.1) — Preceding unsigned comment added by 95.160.100.54 (talk) 17:01, 19 June 2018 (UTC)

The illustration "graphs of volumes and surface areas of the n-sphere in (n+1)-dimensional space" is off by 1

This graph misstates the dimensions of n-spheres on the x-axis, erroneously increasing their dimension by 1.50.205.142.35 (talk) 14:49, 7 December 2019 (UTC)

I don't follow. It seems to me the graph says V2=pi and S1=2pi, consistent with the "Examples" section. —Kusma (t·c) 15:05, 7 December 2019 (UTC)

It's a sphere-versus-ball problem: the caption says n-sphere, but it should be n-ball. Edit: I have fixed it. --JBL (talk) 16:02, 7 December 2019 (UTC)

"4-sphere" listed at Redirects for discussion

A discussion is taking place to address the redirect 4-sphere. The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 May 9#4-sphere until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 (talk) 14:42, 9 May 2020 (UTC)

Very bad method in even modestly high dimensions

The subsection Uniformly at random on the (n-1)-sphere of the section Generating random points ends with this paragraph:

"An alternative given by Marsaglia is to uniformly randomly select a point x = (x1, x2,... xn) in the unit n-cube by sampling each xi independently from the uniform distribution over (–1,1), computing r as above, and rejecting the point and resampling if r ≥ 1 (i.e., if the point is not in the n-ball), and when a point in the ball is obtained scaling it up to the spherical surface by the factor 1/r; then again 1/rx is uniformly distributed over the surface of the unit n-ball."

This is a particularly bad method in even modestly high dimensions, simply because the fraction of points in the n-dimensional cube [-1,1]ⁿ that lie inside the (n-1)-sphere of radius = 1 becomes extremely small. For instance, for n = 6, the 6-dimensional ball has only about 8% of the volume of the surrounding cube. In 8 dimensions it's only about 1.5%. In 10 dimensions it's about 1/4 of one percent. In 15 dimensions it's less than 1/50,000 of the volume.50.205.142.50 (talk) 16:03, 18 June 2020 (UTC)

Formula for S different in two places

In one place, the article says S_n-1 = n pi^(n/2)/Gamma(n/2+1), but in the blue box summary it says S_n(R) = 2 pi^((n+1)/2) R^n/Gamma((n+1)/2). These don't match because one has an n in the denominator and the other has a 2 in the place where the n was.--John Lawrence (talk) 19:46, 15 July 2020 (UTC)

Moment of inertia

What is the moment of inertia of an n-sphere? Just granpa (talk) 08:59, 3 January 2021 (UTC)

For a 3D sphere I get ${\text{Density}}\cdot \int _{0}^{r}2\pi x\cdot 2{\sqrt {r^{2}-x^{2}}}\cdot x^{2}dx={\text{Density}}\cdot {\frac {8\pi r^{5}}{15}}\quad$ (Link)

For a 4D hypersphere I get ${\text{Density}}\cdot \int _{0}^{r}4\pi x^{2}\cdot 2{\sqrt {r^{2}-x^{2}}}\cdot x^{2}dx={\text{Density}}\cdot {\frac {\pi ^{2}r^{6}}{4}}\quad$ (Link)

Just granpa (talk) 00:59, 7 January 2021 (UTC)

Error in the "Spherical Coordinates" section?

The final term in the arccot set of equations seems to be incorrect.

I think it should be

phi(n-1) = arccot[x(n-1) / x(n)] — Preceding unsigned comment added by 71.178.237.32 (talk) 01:15, 13 February 2022 (UTC)