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The graphic titled "Graphs of volumes (V) and surface areas (S) ..." contains the valuable information consisting of the exact values of volumes of unit spheres and unit balls for many low dimensions.

Unfortunately, this graphic is so poorly labeled that it is virtually unreadable.

No axes are labeled, or if they are, the labeling is incomprehensible.

One does not even find numbers for dimensions along the x-axis (except in a few cases).

I hope someone can produce a graphic that contains the same information, but much more clearly marked. 2601:200:C000:1A0:E0B5:5C8B:ACD0:5BEA ( talk) 18:54, 25 September 2022 (UTC) reply

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) reply

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) reply

I just made some edits to the 'volume and surface area' section to make it more useful as a quick-reference for the formulas. Hopefully that's okay. I haven't edited much on Wikipedia before, but it was so hard to figure out what I was looking for that I was inspired to fix it up. If anyone objects, please make it better instead of reverting to what it was before, which was very hard to follow. Shevvvv ( talk) 21:07, 11 July 2023 (UTC) reply

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) reply

@ 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) reply

Both ${\displaystyle V_{0}=1}$ (based on a somewhat arbitrary definition of "0-ball is a point") and ${\displaystyle V_{1}=2}$ (length of an interval of radius 1) are described. ${\displaystyle S_{-1}}$ is nonsensical, but ${\displaystyle S_{0}}$ consists of two endpoints of the interval and is therefore usually defined to be ${\displaystyle S_{0}=2}$ (though this is also kind of an arbitrary definition). If you want feel free to start with ${\displaystyle V_{1}=2}$ and ${\displaystyle S_{1}=2\pi }$ 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) reply