Talk:Unit sphere

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)

What is this article about?

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 $A_{n}$ for the hypervolume of the $(n-1)$-dimensional unit sphere (i.e., for the "area" of the surface of the $n$-dimensional unit ball). However, the example of the current N-sphere article would suggest that we use $A_{n-1}$, presumably because it represents the $(n-1)$-dimensional volume of an $(n-1)$-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: $V_{n}$ here is $C_{n}$ there, $A_{n}$ here is $S_{n-1}$ there. What do you think? Quantling (talk) 21:20, 17 June 2010 (UTC)

I am going to make the edit to change the subscript to match the dimension. —Quantling (talk | contribs) 22:18, 10 December 2021 (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)

Unit spheres of a graph

It may be useful to add somewhere that there is a concept of "unit spheres" for graphs. This is not really my field but an expert could even add a specific page in Wikipedia. MClerc (talk) 09:02, 23 February 2023 (UTC)

Do you mean drawing a circle of radius 1 on graph paper? Do you mean all nodes connected by an edge to a specified node in a graph (computer science)? If not these, what do you mean? Thanks! —Quantling (talk | contribs) 14:27, 23 February 2023 (UTC)

Meaning of "interior"

The article currently discusses "open ball" and "closed ball" where "open" and "closed" are used in their topological sense. (Here the topology is that defined by the metric or other norm.) However, despite what the article could be interpreted to mean, the open ball is not the (topological) interior of the sphere for several common cases. In Euclidean space, for example, the (topological) interior of the sphere is the empty set — any open set that contains a point of the sphere necessarily includes points not belonging to the sphere.

I agree there are levels of understanding and the use of "interior" is meant in a non-topological sense in the article. However, because we are using "open" and "closed" in a topological sense ... if we can come up with a rephrasing that doesn't make the statement false when "interior" is also interpreted in the topological sense, that would be better, yes? —Quantling (talk | contribs) 23:02, 20 November 2023 (UTC)

You're right, we should rephrase that; sorry for any confusion. It's the interior (in a precise topological sense) of the closed ball, and the interior (in a more casual sense) of a sphere. My impression is that most of the time unadorned "unit ball" means the open unit ball, and exceptions are pretty easy to tell from context. –jacobolus (t) 03:44, 21 November 2023 (UTC)

@Quantling I switched "interior" to "inside"; is that clear enough? –jacobolus (t) 05:04, 21 November 2023 (UTC)

Yes "inside" is good. Or if you prefer it, "enclosed by". If we care about distinguishing and promoting open ball over closed ball -- and I'm not sure that we need to make the distinction in this article — but if we do, should the word "strictly" also be used, as in "strictly inside"?

A quibble: "It's the interior (in a precise topological sense) of the closed ball" is true for Euclidean space, but not necessarily in more general topological cases including some metric spaces (which we discuss briefly in the article). For example, when the entire metric (topological) space is only the unit closed ball subset of Euclidean space then its topological interior is itself, rather than all points strictly less than distance 1 from the center. —Quantling (talk | contribs) 14:39, 21 November 2023 (UTC)