Talk:Hypersphere

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Contents 1 stereographic Projection 2 merge to Torus 3 merge to sphere 4 Error in Volume formula 5 The volume vs dimension curve 6 Indexing 7 "hypermeridian" - possible neologism

[edit] stereographic Projection

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

[edit] merge to Torus

I popose merging this article with the article on a torus, because the best representation of a hyersphere is a torus, according to this page: [[1]].Nschoem 18:44, 4 April 2007 (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 [2]

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)

[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)