# Talk:Unit sphere

WikiProject Mathematics (Rated Start-class, Mid-importance)
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Mathematics rating:
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Field: Geometry

## Introductory sentence

Articles like this really need an introductory sentence or two - after all, wikipedia isn't a textbook or a scientific journal. Granted, putting abstract mathematical concepts into non-symbols can be very, very difficult, but wikipedia is one of the few venues where non-mathematicians may come across these concepts at all, so it's worth the effort. Perhaps something could be adapted from the introduction of ball (mathematics), which mentions the unit ball. - DavidWBrooks 18:44, 19 Apr 2004 (UTC)

## Unit sphere or Unit ball?

Wrote the non-technical intro starting with "unit sphere" rather than "unit ball". In my opinion, the article should be titled "unit sphere" and "unit ball" redirected to it, rather than the other way around. Not a huge difference either way, but the rationale is: (1) the WP article on sphere is richer than the article on ball; (2) the use of the word sphere in mathematics is closer to ordinary language than the use of the word ball is. Hence it's slightly easier for the nonmathematical reader to get a handle on this (admittedly, not too difficult) subject by starting with 'sphere' and defining 'ball' in terms of it.Brian Tvedt 01:02, 23 August 2005 (UTC)

Done.--agr 15:27, 25 October 2006 (UTC)

I think this should include cartesian and spherical equations of the unit sphere as well as a parameterization.

## Formulas with radius or without?

Some of the formulas have been changed many times to include / not include a radius term. I don't think that a radius term is appropriate because it's the Unit Sphere not just the general Sphere page. For example, we no longer list that the volume of a unit sphere (in 3d) is 4/3 pi. I think that the actual values of the most common form (euclidian, 3d) are a critical piece of information for this page. The general formulas are far to general - a lot of people would appreciate a quick reference of the normal, non-general case. -- NegatedVoid (talk) 15:04, 28 August 2009 (UTC)

I'm confused about the what the subject of the article is supposed to be. I would think it would be about the sphere of radius 1, which is given as the second paragraph of the lead section and the subject of the MathWorld link given in the references. But the first sentence of the lead section and the majority of the article seems to be about the set of unit vectors is a more general space, possibly a normed vector space. Some sections of the article seem to be about unit n-spheres. Admittedly the usage of the term 'Unit sphere' in the literature is somewhat context dependent, but the article as stands seems to mix all the meanings freely without making any attempt at organizing it. There might be material for multiple articles here, on the other hand it may be better to merge some of the material into other articles.--RDBury (talk) 18:14, 19 March 2010 (UTC)

I agree with this. The title "Unit sphere" is singular, which would seem to indicate it is referring to the sphere of radius 1, as mentioned above. The lead section generally does stick to that topic, but the rest of the article certainly seems to deviate from this. I would suggest possibly retitling the article to "Unit spheres" or "Unit balls" or something of the like. Benny476 (talk) 23:41, 10 October 2015 (UTC)

### Merge with N-sphere or at least mimic it?

In the current unit sphere article we use ${\displaystyle A_{n}}$ for the hypervolume of the ${\displaystyle (n-1)}$-dimensional unit sphere (i.e., for the "area" of the surface of the ${\displaystyle n}$-dimensional unit ball). However, the example of the current N-sphere article would suggest that we use ${\displaystyle A_{n-1}}$, presumably because it represents the ${\displaystyle (n-1)}$-dimensional volume of an ${\displaystyle (n-1)}$-dimensional object. Should we adjust the current article?

Also, perhaps this article should be merged with the N-sphere article. Or at least we could use the same notation: ${\displaystyle V_{n}}$ here is ${\displaystyle C_{n}}$ there, ${\displaystyle A_{n}}$ here is ${\displaystyle S_{n-1}}$ there. What do you think? Quantling (talk) 21:20, 17 June 2010 (UTC)

## Surface area

Assuming the "surface area" of a sphere in n-space means (n-1)-dimensional Lebesgue measure, how is the surface area 2 when n=1? A "sphere" in 1-space is simply two points, each a distance of 1 from the origin. This is a finite set and therefore has measure 0, not 2. Akwdb (talk) 21:21, 22 April 2016 (UTC)

Indeed even the 1-dimensional surface of a circle has 0 measure if using a 2-dimension measure. One has to apply the correct measure. It is with the 0-dimensional measure that counts points, that the surface of the n=1 ball measures 2. —Quantling (talk | contribs) 00:41, 25 April 2016 (UTC)

## Infinite series of all nth dimensional unit sphere areas

It can be shown that the infinite sum of all nth dimensional unit sphere volumes converges to a finite value and that value can be estimated. It should be included in the article. — Preceding unsigned comment added by 97.123.64.218 (talk) 02:52, 14 November 2016 (UTC)