Talk:n-sphere

hyperspherical coordinates error

In the spherical coordinates section, it says "then we may compute x1, ..., xn from r, φ1, ..., φn−1 with ..." and then goes on to give formulas for x1, ... ,xn in terms of r, φ1, ..., φn. E.g., the 2-dimensional case (polar coordinates) are given in terms of 2 angles and a radius. 2601:645:C200:2550:D90C:72D7:D8C3:D0B1 (talk) 18:48, 23 March 2023 (UTC)

There is no error. For polar coordinates, there is only one angle. In general, in dimension n, there are n coordinates (this is the definition of the dimension). So, if there are n Cartesian coordinates, there must be n sperical coordinates. As r is one of them, there must be n-1 angles. D.Lazard (talk) 22:06, 23 March 2023 (UTC)

the formulas described there only make sense for dimensions higher than 3, so for n=2 or n=3 case you'd remove the redundant cos(φn) factors AnonN10 (talk) 03:30, 4 May 2024 (UTC)

You are right. I have added footnote explaining this and containing a formula valid for n ≥ 2. D.Lazard (talk) 08:35, 4 May 2024 (UTC)

A "topological" n-sphere should probably be relegated to a section or given its own article

This article defines "n-sphere" to mean any "topological space that is homeomorphic to a standard n-sphere". This seems like a poor (unhelpful, confusing) definition for this article, which mostly focuses on metrical properties, etc. It might be better to have a separate article about using "n-sphere" to mean any homeomorphic surface, or putting it in a section about generalizations of the sphere. –jacobolus (t) 14:48, 24 August 2023 (UTC)

Please fix recurrence relation

@Shevvvv If I write the relations as "Vn=f(Sn-1)" and "Sn=g(V-1)" (to make them easier to code!) then "S1=g(V0)" but the base case given is V1. Eijkhout (talk) 16:02, 20 October 2023 (UTC)

Both $V_{0}=1$ (based on a somewhat arbitrary definition of "0-ball is a point") and $V_{1}=2$ (length of an interval of radius 1) are described. $S_{-1}$ is nonsensical, but $S_{0}$ consists of two endpoints of the interval and is therefore usually defined to be $S_{0}=2$ (though this is also kind of an arbitrary definition). If you want feel free to start with $V_{1}=2$ and $S_{1}=2\pi$ as your base cases. To be honest I don't understand why anyone needs to "code" this. –jacobolus (t) 17:27, 20 October 2023 (UTC)

Number of spherical co-ordinates

The article says

We may define a coordinate system in an ⁠$n$⁠-dimensional Euclidean space which is analogous to the spherical coordinate system defined for ⁠$3$⁠-dimensional Euclidean space, in which the coordinates consist of a radial coordinate ⁠$r$⁠, and ⁠$n-1$⁠ angular coordinates ⁠$\varphi _{1},\varphi _{2},\ldots ,\varphi _{n-1}$⁠

but if the rest of the article is talking about an ⁠$n$⁠-sphere embedded in ⁠$n+1$⁠-dimensional space, I would have thought it might be better to say

We may define a coordinate system in an ⁠$n+1$⁠-dimensional Euclidean space which is analogous to the spherical coordinate system defined for ⁠$3$⁠-dimensional Euclidean space, in which the coordinates consist of a radial coordinate ⁠$r$⁠, and ⁠$n$⁠ angular coordinates ⁠$\varphi _{1},\varphi _{2},\ldots ,\varphi _{n}$⁠

2A00:23C6:1492:7A01:3538:F088:21E4:507E (talk) 15:32, 30 January 2025 (UTC)

It might be better to move this section to a section of Spherical coordinate system. –jacobolus (t) 15:53, 30 January 2025 (UTC)

When is the Jacobian fullrank?

The article should mention when the Jacobians (of the "direct" and the inverse transformations) are fullrank (rank $n$ where $n$ is the space dimensionality).

That's important for such applications as preserving dimensions. VictorPorton (talk) 05:23, 8 March 2025 (UTC)

Not every mathematical expression needs to be as narrow as possible

It's fine to have mathematical expressions be about 20 ems wide, as in this example, from the stable version of this article:

The ⁠$0$⁠-ball is sometimes defined as a single point. The ⁠$0$⁠-dimensional Hausdorff measure is the number of points in a set. So $$V_{0}=1.$$

A unit ⁠$1$⁠-ball is a line segment whose points have a single coordinate in the interval ⁠$[-1,1]$⁠ of length ⁠$2$⁠, and the ⁠$0$⁠-sphere consists of its two end-points, with coordinate ⁠$\{-1,1\}$⁠. $$S_{0}=2,\quad V_{1}=2.$$

A unit ⁠$1$⁠-sphere is the unit circle in the Euclidean plane, and its interior is the unit disk (⁠$2$⁠-ball). $$S_{1}=2\pi ,\quad V_{2}=\pi .$$

The interior of a 2-sphere in three-dimensional space is the unit ⁠$3$⁠-ball. $$S_{2}=4\pi ,\quad V_{3}={\tfrac {4}{3}}\pi .$$

In general, ⁠$S_{n-1}$⁠ and ⁠$V_{n}$⁠ are given in closed form by the expressions $$S_{n-1}={\frac {2\pi ^{n/2}}{\Gamma {\bigl (}{\frac {n}{2}}{\bigr )}}},\quad V_{n}={\frac {\pi ^{n/2}}{\Gamma {\bigl (}{\frac {n}{2}}+1{\bigr )}}},$$ where ⁠$\Gamma$⁠ is the gamma function.

@Sbb is edit warring to change the last line to their preferred

$${\begin{aligned}S_{n-1}&={\frac {2\pi ^{n/2}}{\Gamma {\bigl (}{\frac {n}{2}}{\bigr )}}},\\V_{n}&={\frac {\pi ^{n/2}}{\Gamma {\bigl (}{\frac {n}{2}}+1{\bigr )}}},\end{aligned}}$$

on the grounds that their small phone screen renders a scrollbar for the stable version.

In my opinion reducing information density in this way is misguided and unnecessary. It doesn't match the several previous lines, and this expression is, regardless, fairly narrow and much narrower than several other mathematical expressions on the same article, such as

$$J_{n}={\begin{pmatrix}c_{1}&-rs_{1}&0&0&\cdots &0\\s_{1}c_{2}&rc_{1}c_{2}&-rs_{1}s_{2}&0&\cdots &0\\\vdots &\vdots &\vdots &&\ddots &\vdots \\&&&&&0\\s_{1}\dotsm s_{n-2}c_{n-1}&\cdots &\cdots &&&-rs_{1}\dotsm s_{n-2}s_{n-1}\\s_{1}\dotsm s_{n-2}s_{n-1}&rc_{1}\dotsm s_{n-1}&\cdots &&&{\phantom {-}}rs_{1}\dotsm s_{n-2}c_{n-1}\end{pmatrix}}.$$

or

$${\begin{aligned}&\int _{0}^{\pi }\sin ^{n-j-1}\left(\varphi _{j}\right)C_{s}^{\left({\frac {n-j-1}{2}}\right)}\cos \left(\varphi _{j}\right)C_{s'}^{\left({\frac {n-j-1}{2}}\right)}\cos \left(\varphi _{j}\right)\,d\varphi _{j}\\[6pt]&\qquad ={\frac {2^{3-n+j}\pi \Gamma (s+n-j-1)}{s!(2s+n-j-1)\Gamma ^{2}\left({\frac {n-j-1}{2}}\right)}}\delta _{s,s'}\end{aligned}}$$

–jacobolus (t) 01:10, 18 August 2026 (UTC)

As an aside, a lot of your other trivial changes are (harmless but) completely unnecessary. For example, it doesn't accomplish anything to change every instance of \ldots to \dotsc, \cdots to \dotsb, etc.; One set of names comes from LaTeX and the other from the amsmath package, but they render exactly the same way, and which names to use is a matter of personal preference; My impression is that the amsmath package defined the \dotsc, etc. versions to be "semantic" rather than presentational names, in case some author or editor might hypothetically want one or another of them to look different than the usual rendering, but in practice nobody does that and it's in any event impossible in Mediawiki LaTeX. From what I understand the only ones that actually render differently are \dotsi which have slightly different space intended for the use between integral signs, but these are quite rarely used compared to the others. It's especially silly to change the bare \dots to some other version, if it was working correctly, since amsmath uses this name to automatically pick the right set of dots for the context.

Making this kind of "cosmetic" markup change mostly just creates watchlist spam and wastes other editors' attention for no practical benefit. –jacobolus (t) 01:29, 18 August 2026 (UTC)