Talk: Volume of an n-ball

Orphaned

This article is "orphaned" because most articles that would link to it probably link to the main n-sphere article instead. The "n" needs to be italicized in the title, too, but I don't know how to do that. If it bothers anyone that this article is orphaned, it could be incorporated into the n-sphere article, but the proof here is much easier to understand if it's contained in its own little article here without unrelated clutter. We could perhaps make this a subpage of n-sphere, but that's against policy. Personally I wouldn't worry too hard about this article's orphanage. Deepmath (talk) 06:35, 6 August 2009 (UTC)

What policy is it against? —Preceding unsigned comment added by 99.13.56.253 (talk) 07:43, 27 January 2010 (UTC)

Googlewashed

I think this article has been googlewashed because of the mention of something nuclear at the bottom of the article. It doesn't show up in any relevant Google searches. Fine by me. People depend way too much on search engines, and search engine companies know way too much about what people are searching for on the internet. Learn to find what you are looking for by surfing, following links, and hiding referrers, and don't listen to those n00bs who say, "Use a search engine!" Deepmath (talk) 20:00, 7 August 2009 (UTC)

came up second on my list when I googled "volume hypersphere wiki" (I like search engines!). Great article btw. RichardThePict (talk) 10:58, 4 February 2016 (UTC)

Deriving the volume of an n-ball

Is it possible to italicize the n in "n-ball" in the title of the article? 199.59.26.20 (talk) 04:17, 8 December 2009 (UTC)

Derivation via integrating two dimensions at a time.

The derivation is much shorter if one calculates the volume of an n-dimensional ball from the formula for volume of an (n-2)-dimensional ball. Base cases:

$V^{(0)}[R]=1$

$V^{(1)}[R]=2R$

Assume for the inductive argument that, for $k<n$ dimensions we have already proved that

$V^{(k)}[R]={\frac {\pi ^{k/2}R^{k}}{\Gamma (k/2+1)}}\,.$

Then we prove the formula for $V^{(n)}[R]$ via induction using integration in polar coordinates:

${\begin{aligned}V^{(n)}[R]&=\int _{0}^{R}\int _{0}^{2\pi }V^{(n-2)}[{\sqrt {R^{2}-r^{2}}}]\,r\,d\theta \,dr\\&=2\pi \int _{0}^{R}V^{(n-2)}[{\sqrt {R^{2}-r^{2}}}]\,r\,dr\\&=2\pi \int _{0}^{R}{\frac {\pi ^{(n-2)/2}}{\Gamma ((n-2)/2+1)}}\,({R^{2}-r^{2}})^{(n-2)/2}\,r\,dr\\&={\frac {2\pi ^{n/2}}{\Gamma ((n-2)/2+1)}}\left.{\frac {-(R^{2}-r^{2})^{n/2}}{2(n/2)}}\right|_{r=0}^{r=R}\\&={\frac {\pi ^{n/2}R^{n}}{\Gamma (n/2+1)}}\,.\end{aligned}}$

QED.

I am thinking we should add this (or something close) to the article. Should we also remove the current lengthy derivation? Quantling (talk) 00:32, 29 June 2010 (UTC)

Have both! And see if any steps need hints: ( Try this on a novice! )24.4.247.181 (talk) 03:02, 26 April 2012 (UTC)

Some discussion on the recursive relation $V^{(n)}[R]=\int _{0}^{R}\int _{0}^{2\pi }V^{(n-2)}[{\sqrt {R^{2}-r^{2}}}]\,r\,d\theta \,dr$ may also be helpful. - Subh83 (talk | contribs) 22:43, 24 March 2011 (UTC)

Article Quality

I have a degree in math, and one in computer science. I enjoy elegance and simplicity. This article is one of the very best I have seen on Wikipedia.24.4.247.181 (talk) 03:02, 26 April 2012 (UTC)

Mistake in surface measure formula for Lp-spheres?

The last part of this article gives a formula for the surface measure of Lp-spheres which I don't think is correct, except for the case p = 2. It takes the surface measure to be the derivative of the interior measure, with respect to the radius. This is familiar for p = 2, essentially because, in this norm, the tangent plane to a sphere at a point will be perpendicular to the direction that point to the center. However, I believe this reasoning, and indeed this formula, completely falls apart at other p. For example, consider the Manhattan metric (p = 1) in two dimensions. The interior measure of a ball of radius R will be 2^2 * 1/2 * R^2, while the surface length will be either 2^2 * sqrt(2) * R [as measured using the normal account of length] or 2^2 * 2 * R [with length measured according to the p = 1 norm]. Either way, it will not be 2^2 * 1 * R, as the provided formula claims it would be. -Chinju (talk) 22:40, 8 March 2013 (UTC)

If you say that volume and surface area should be related by:

$V_{n}(R)=\int _{0}^{R}A_{n-1}(r)\,dr,$

and if you say that surface area and volume should be proportional to a power of the radius:

$V_{n}(R)=V_{n}(1)R^{n},$

$A_{n-1}(R)=A_{n-1}(1)R^{n-1},$

then you get:

$V_{n}(1)R^{n}=V_{n}(R)=A_{n-1}(1)\int _{0}^{R}r^{n-1}\,dr=A_{n-1}(1){\frac {R^{n}}{n}},$

which leads to:

${\frac {d}{dR}}V_{n}(R)=V_{n}(1)nR^{n-1}=V_{n}(1)R^{n}\cdot {\frac {n}{R}}=A_{n-1}(1)R^{n-1}=A_{n-1}(R),$

exactly as the article claims.

I think your question is really, "Which measure are you using to measure surface area?" You say that the length of the p = 1 ball in the p = 1 norm should be 8R. But that norm (in fact any norm) gives you an n-dimensional measure, and with respect to that measure, the volume of the (n − 1)-sphere is zero. I'm not sure which measure you're using, but it's different from the one the article is using. There are many different (n − 1)-dimensional measures on Rⁿ, and the formula the article gives is only valid for one of them.

I think it would be good for the article to clarify which measure it's using. I will look into this. Ozob (talk) 01:15, 10 March 2013 (UTC)

Some basic integration shows that surface area for parameterized surfaces behaves correctly under dilation. Hausdorff measure also behaves correctly. It seems the problem is that:

$V_{n}(R)\neq \int _{0}^{R}A_{n-1}(r)\,dr.$

Now, my intuition was that an n-ball could be decomposed as a disjoint union of (n − 1)-balls, and that therefore the volume of the n-ball was the integral of the areas of the (n − 1)-balls. It seems that's not true when p = 1, as you've pointed out. But the formula in the article works not just for p = 2 but also for p = ∞. So this has left me rather confused; I think the above integral should be correct after including a correction factor, but I have no clue what the correction factor should be or what it should represent.

For the moment I've removed the incorrect paragraph from the article. Ozob (talk) 02:26, 10 March 2013 (UTC)

It seems that what I'm trying to use is the coarea formula. If f : R^m → Rⁿ is Lipschitz, m > n, and A is a Lebesgue measurable set in R^m, then:

$\int _{A}|\operatorname {Jac} _{n}(f)(x)|\,d{\mathcal {L}}^{m}(x)=\int _{\mathbf {R} ^{n}}{\mathcal {H}}^{m-n}(A\cap f^{-1}(y))\,d{\mathcal {L}}^{n}(y),$

where ${\mathcal {L}}$ denotes Lebesgue measure and ${\mathcal {H}}$ denotes Hausdorff measure (Federer, 3.2.11).

For our current problem, we've been calling the dimension of the domain to be n not m, and we want the dimension of the codomain to be 1. The function f will be the p-norm, and the set A will be the p-ball of radius R. Then the coarea formula tells us:

$\int _{B_{p}^{n}(R)}|\operatorname {Jac} _{n}(\lVert x\rVert _{p})|\,d{\mathcal {L}}^{m}(x)=\int _{0}^{R}{\mathcal {H}}^{n-1}(S_{p}^{n-1}(r))\,d{\mathcal {L}}^{1}(r),$

where $B_{p}^{n}(R)$ is the n-ball of radius R in the L^p norm and $S_{p}^{n-1}(r)$ is the (n − 1)-sphere of radius r in the L^p norm. This is exactly the formula I was claiming above ... except for the presence of the Jacobian. When p is 2 or ∞, the Jacobian is almost everywhere equal to 1. When p is 1, it's almost everywhere equal to ${\sqrt {n}}$, which explains your observation above. I don't see an easy way to compute it for other values of p, and consequently I don't see an easy way to compute the integral. (It wouldn't give the desired result even if we knew what it was, because the left-hand side is no longer the volume. We would need a slight generalization to integrals of functions, specifically theorem 3.2.12 in Federer, with the function g being the inverse of the absolute value of the Jacobian. If we could integrate g over an L^p-sphere of radius 1 then we would know the correction factor.)

I remember being suspicious of this claim when I rewrote the article exactly because of the reason you mentioned (the radial direction is no longer normal to the surface); but I checked it for p = ∞ and it worked, and I had the naive and incorrect decomposition argument above to back me up. I guess I should have checked p = 1! Ozob (talk) 03:20, 10 March 2013 (UTC)

Reflections on definitions and notation

(@Ozob) Hi - I wrote some changes recently which you reversed. Sorry you didn't like them. What I was trying to do was make this excellent material more accessible to a wider audience. Not everyone has the same level of expertise, and I wanted to help make it useful to the professional as well as to the student and interested layman. Some of these fascinating details do not seem to handily appear elsewhere.

(1) Opening with a more formal definition gives the reader an opportunity to orient themselves before going on to all the heady formulae. If anything, I was concerned that the definition should be even more precise.

(2) It seemed like a good idea to clearly distinguish early-on between ball and sphere, especially for the benefit of novices, many of whom have always thought of these terms as synonyms. I suspect this is even more true for n-ball and n-sphere.

(3) Transitioning to the notation S_n, rather than A_n, for the measure of surface area -- exactly matches the usage in related articles, where S_super_n, S_n, and S_n(r) respectively denote: the SET of the n-sphere itself, the measure of the unit n-sphere, and the measure of an n-sphere of radius r. This provides the reader a seamless transition among related articles.

btw: The matter of "2 k!" versus "2(k!)" was a good catch. It was something I had briefly considered -- although the former style is perferred for most mathematical publications with which I deal. When writing for Wikipedia, I always have to keep in mind it is for "everyone" -- not just experts. (Ziki42 (talk) 02:37, 8 December 2013 (UTC))

I agree that the reader should have a precise understanding of what's going on everywhere in the article. But I wasn't sure that the level of detail you provided right there was helpful. If the reader knows what an n-ball is, then specifying its radius and center is unnecessary; and if the reader doesn't know what an n-ball is, then I don't think that specifying its radius and center helps.

I thought that the definition given in the first sentence was enough to distinguish a ball from a sphere. But it might need amplification, and I wouldn't object too much to having another sentence or two in there. Maybe a formula, since that's completely unambiguous.

Generally, Wikipedia articles aren't required to be consistent with each other, but we shouldn't recklessly avoid consistency when it's easy to achieve. However, I still don't like the notations $S_n$ and $S_n(r)$ for the surface area. I would rather change related articles to use $A_n$ rather than change this one to use $S_n$. If there were a standard notation used in a lot of books, then I'd be willing to adopt that, too.

Finally, often when I encounter a sticky point in exposition or notation, I think to myself, "How would my mother interpret this?" If she would say, "Huh?" then I try to make it clearer. (I can't carry this too far, because she isn't a mathematician. But for some reason this principle works for me.) Ozob (talk) 04:05, 8 December 2013 (UTC)

What about this elegant formula?

I believe this article should mention this beautiful formula:

$V_{n}(R)={\frac {2^{n}{\frac {\pi }{2}}^{\lfloor {\frac {n}{2}}\rfloor }R^{n}}{n!!}}\,$

It is much simpler than the other formulas in the article involving $\Gamma ({\frac {n}{2}}+1)$ and ${\sqrt {\pi }}^{n}$.

The factor of $2^{n}$ has a geometrical meaning: a $n$-ball can be divided into $2^{n}$ congruent pieces (in two dimensions they are the four quadrants, in three dimensions they're the eight octants).

It also hints us that ${\frac {\pi }{2}}$ is perhaps more fundamental than $\pi$. — Preceding unsigned comment added by 86.195.251.245 (talk) 04:31, 19 October 2014 (UTC)

But we do mention it (for odd dimensions) immediately below the other formula. And in even dimensions it's ambiguous — there are two incompatible ways of defining the double factorial when n is even. —David Eppstein (talk) 06:39, 19 October 2014 (UTC)

While we could have this article adopt a convention on the meaning of the double factorial of an even integer, I think that would be a bad idea. Most of the derivations in the article suggest (at least to me) that in some sense, the volume really is given by the gamma function (or maybe the beta function). Double factorials only enter when we try to get exact values of the relevant special functions. They're not really important to the volume formula itself, so it would be misleading to the reader to treat them as anything other than a computational convenience. As such, it's best to stick to using them only where they have established, unambiguous meanings.

Also, the formula proposed above could just as well be used to say that 2π is more fundamental than π, since the numerator would become $2^{n{\bmod {2}}}(2\pi )^{\lfloor n/2\rfloor }R^{n}$. Ozob (talk) 14:41, 19 October 2014 (UTC)

Merge with N-sphere wiki?

I suggest merging this with the n-sphere wiki, since it already has a section on the volume and surface areas of n-spheres. The n-sphere page also contains some material not referenced on this wiki, and vice versa. OneMoreLevel (talk) 23:08, 2 June 2015 (UTC)

I think this article is too long to comfortably fit into the n-sphere article. Also, a quick glance didn't find me anything in the n-sphere article that was about volumes or surface areas that wasn't already here. What did you have in mind? Ozob (talk) 02:11, 3 June 2015 (UTC)

My own feeling is that per Wikipedia:Summary style the redundancy should be reduced by trimming what's in the n-sphere article to the essentials (why do the recurrences and other relations need to be there? just give the formulas) rather than merging the two articles. —David Eppstein (talk) 03:11, 3 June 2015 (UTC)

Formula for the generalized l_p unit ball

To my best knowledge, the formula for $B_{p_{1},\dots ,p_{n}}$ should have $|x_{i}|^{p_{i}}$ not $|x_{i}|^{1/p_{i}}$. Is that correct? 212.158.158.94 (talk) 15:46, 18 June 2015 (UTC)

Yes, it is. I've fixed the article. Ozob (talk) 01:52, 19 June 2015 (UTC)

Relation with surface area

Seems to me that this equation is false for the case of 3-dimensions in which $A(R)=4\pi R^{2}$ and $V(R)={\tfrac {4}{3}}\pi R^{3}$. The exponents come out wrong.

$A_{n}(R)\propto R^{n-1}$ and $V_{n}(R)\propto R^{n}$. It would seem to me, that at least in 3-dimensional space, that:

${\frac {\mathrm {d} V_{n}(R)}{\mathrm {d} R}}=A_{n}(R)$

and this would be the case with any of the integer dimension, n. 50.47.104.180 (talk) 04:45, 9 July 2018 (UTC)

So I marked it {{dubious}}. 50.47.104.180 (talk) 04:45, 9 July 2018 (UTC)

Perhaps you're confused by the fact that the boundary of an n-ball is not an n-sphere, it's an (n-1)-sphere. The formula you give is the area for a 2-sphere, not a 3-sphere. 3-spheres have volume, not area. —David Eppstein (talk) 04:53, 9 July 2018 (UTC)

So this is the case?:

$A_{n}(R)\propto R^{n-1}$ and $V_{n+1}(R)\propto R^{n}$

to get this?:

${\frac {\mathrm {d} V_{n+1}(R)}{\mathrm {d} R}}=A_{n}(R)$

50.47.104.180 (talk) 05:01, 9 July 2018 (UTC)

Yes. For instance, a 2-ball (a disk in the plane, the interior of a circle, a 2-dimensional thing) has area $V_{2}(R)=\pi R^{2}$ and its boundary, a 1-sphere (a circle in the plane, a 1-dimensional thing) has perimeter $A_{1}(R)=2\pi R$. Or even more simply a 1-ball (an interval on the real line, a 1-dimensional thing) has length $2R$ and its boundary (two points in the plane, a 0-ball, a 0-dimensional thing) has cardinality 2. —David Eppstein (talk) 05:29, 9 July 2018 (UTC)

At first glance, the units of equation 2 seem to be off by one. Does it make more sense to index in the same way as the rest of the article, i.e. $A_{n}\propto R^{n-1}$? What's sacrificed in brevity would be recovered with notational clarity. — Preceding unsigned comment added by 162.220.196.186 (talk) 00:05, 30 October 2021 (UTC)

The subscript indicates the number of dimensions of the object in question. The boundary of a normal 3-dimensional ball is 2-dimensional so we talk about that the surface area of that boundary as A₂(R) = 4πR², and so on. That the 2-dimensional boundary is often talked about in the context of that 3-dimensional ball is not enough reason to mess with the subscript. —Quantling (talk | contribs) 18:32, 31 October 2021 (UTC)

Same user, different place. As written, the dimension-indexed equation reads "the boundary of an n+1 ball ($A_{n}$) is a [more complicated] coefficient times $R^{n}$" where the re-indexing happens in text and not in the equation, i.e. $A_{n+1}$. The volume-indexed equation would read "the boundary of an n-ball is a coefficient times $R^{n-1}$. Volume-indexing matches the rest of the article, requires less text, matches the dimensional expectations of many and shows a simpler equation. I suspect reasons to re-index outweigh the fore-mentioned, so let's add those to the article. Perhaps fractional-dimensional objects break down or the other equation gets ugly. Perhaps there's a long-standing tradition started by somebody famous.... It's clearly confused more people than stupid old me. :-)

$A_{n}={\frac {2\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}\right)}}R^{n-1}$ — Preceding unsigned comment added by 174.16.111.107 (talk) 19:32, 1 November 2021 (UTC)

For better or worse, it's long-standing tradition to use the dimension of the object in question. Reasons include that the aforementioned 2-sphere can be represented as the boundary of a 3-ball, but it doesn't have to be. It can be defined in other ways too, can be embedded in other than 3-dimensional space, etc. The fact that it is 2-dimensional is a constant over all these variations. —Quantling (talk | contribs) 01:52, 2 November 2021 (UTC)

The alternative is even more confusing in a wider variety of contexts. With the current convention, you can say e.g. “glue together two 2-dimensional balls along their boundaries to make a 2-dimensional sphere”, “project the 2-dimensional sphere onto a 2-dimensional plane”, or “slice a 3-ball up in three different ways: as 2-balls lying in parallel planes, as coaxial 2-dimensional cylinders, or as concentric 2-spheres”, or “locally a 3-manifold looks like either the 3-sphere, the Euclidean 3-space, or the hyperbolic 3-space”, Etc. –jacobolus (t) 02:03, 2 November 2021 (UTC)

None of what's written precludes volume-indexing in my mind, but I'm not an expert. There's a convention that the community finds useful and it's clearly contrary to intuition of a sizeable number of readers. I believe the passage needs edification and I'm ill-equipped to explain why we choose dimension-indexing. I'd happily edify the difference between dimensional and volume indexing for those of us with a different intuition, but since I invented these terms, I doubt they are in common usage which undercuts the value. — Preceding unsigned comment added by 162.220.196.186 (talk) 17:17, 2 November 2021 (UTC)

Just to be clear, you want the article to be something like:

Relation with surface area

Let A_n − 1(R) denote the surface area of the (n − 1)-sphere of radius R in n-dimensional Euclidean space. The (n − 1)-sphere is the boundary of the n-ball of radius R, and the surface area and the volume are related by:

$A_{n-1}(R)={\frac {d}{dR}}V_{n}(R)={\frac {n}{R}}V_{n}(R).$

Thus, A_n − 1(R) inherits formulas and recursion relationships from V_n(R), such as

$A_{n-1}(R)={\frac {2\pi ^{n/2}}{\Gamma {\bigl (}{\tfrac {n}{2}}{\bigr )}}}R^{n}.$

Trivially, there are also formulas in terms of factorials and double factorials.

I think that might arguably be an improvement to legibility. –jacobolus (t) 19:39, 2 November 2021 (UTC)

I believe that the request is that the subscript of A be incremented, as in:

$A_{n}(R)={\frac {d}{dR}}V_{n}(R)={\frac {n}{R}}V_{n}(R).$

I am against that. —Quantling (talk | contribs) 22:45, 2 November 2021 (UTC)

The version Quantling is against is intuitive to me. I'm assuming that the entire field finds dimension indexing more intuitive/clear/easy, but it's just an index. A Wikipedia article cannot change the conventions of a well developed field. I advocate a better explanation of why there are different indices for boundaries and volumes. Context, history, convention, deeper applications, etc... I believe that will clear up the index-switching confusion. — Preceding unsigned comment added by 162.220.196.186 (talk) 23:40, 2 November 2021 (UTC)

Using $A_{n}(R)$ to mean the $(n-1)$-dimensional area seems certain to cause significant confusion. I would be opposed to that. But switching to $A_{n-1}(R)$ so that the dimension $n$ of the space in which the sphere is embedded stays consistent with the rest of the article seems like an okay idea, marginally clearer than the current version. –jacobolus (t) 02:11, 3 November 2021 (UTC)

something like: Topologists||Geometers conventionally index boundaries by boundary dimension, not the dimensionality of the corresponding ball. For example, the boundary of a 3-ball is given by $A_{2}$ because it needs 2 dimensions. They borrow this notation from the n-sphere, a related but distinct object which represents the boundary of an n-ball. This distinction is important because ... — Preceding unsigned comment added by 162.220.196.186 (talk)

That section currently starts with "Let A_{n − 1}(R) denote the surface area of the (n − 1)-sphere of radius R in n-dimensional Euclidean space. The (n − 1)-sphere is the boundary of the n-ball of radius R, ..." which covers some of what you are asking for. For anything that you still find lacking, perhaps you could insert a clause into one of those two sentences? Would it help to replace "n-ball" with "n-dimensional ball" and similarly "(n − 1)-sphere" with "(n − 1)-dimensional sphere"? —Quantling (talk | contribs) 17:30, 6 November 2021 (UTC)

Apologies, it seems I missed those. Great edits. Truly. Last question: do geometers consider $n$-balls within $m$-spaces such that $n<m$? The first sentence indicates that a $(n-1)$-sphere must be in $n$-space. Perhaps something like "... in any Euclidean space with dimension $n$ or greater."? — Preceding unsigned comment added by 162.220.196.186 (talk) 17:57, 8 November 2021 (UTC)

Delete

I strongly suggest deleting this article, because it is so poorly written.2600:1700:E1C0:F340:B9E5:859B:2EA8:5A2F (talk) 02:53, 18 October 2018 (UTC)

The illustration "graphs of volumes and surface areas of the n-sphere in (n+1)-dimensional space" is off by 1

This graph misstates the dimensions of n-spheres on the x-axis, erroneously increasing their dimension by 1.50.205.142.35 (talk) 14:49, 7 December 2019 (UTC)

It is a ball-versus-sphere problem; I have fixed the caption. --JBL (talk) 16:05, 7 December 2019 (UTC)

There is another problem with the otherwise very informative image at the top of the article. The very last value given for volumes, for a 15-dimensional ball, is wrong. It says 15 pi^7 / 118771, but it should say 256 pi^7 / 2027025. These fractions are almost equal as decimals but very different as numbers. It is confusing because the value that is given does not follow the formulas. Troglo (talk) 09:22, 28 April 2020 (UTC)

perhaps worth being explicit about the definition of 'volume'

Conventionally, volume is defined in units of an n-cube of side length 1, while the size of an n-ball is defined by its radius, with a “unit” sphere or ball having radius 1. Both the choice of basic volumetric shape and the choice of standard size for various shapes are arbitrary (in the sense that we could choose different conventions if we liked), and the combination of conventions leads to formulas (e.g. the ones in this article) which mix up different phenomena.

In particular, if we instead consider a unit-diameter ball, then we remove a power of 2 from the formula. If we use the volume of a standard n-simplex as a basic volumetric unit instead of an n-cube, we remove a factorial. Alternately, we could continue to consider unit-radius balls, but make the basic volumetric unit the volume of an n-orthoplex. This again removes the factorial and power of 2 from the formulas, leaving behind only the part of the formula which is specifically related to understanding spheres/balls per se. –jacobolus (t) 12:59, 16 October 2021 (UTC)

(And for that matter, the definition of $\pi$ is a bit arbitrary; arguably the proper constant is ${\tfrac {1}{2}}\pi$.)

Here is a table:

Dimension	Unit-radius ball / n-cube	Unit-diameter ball / n-simplex or Unit-radius ball / n-orthoplex
0	$1$	$1$
1	$2$	$1$
2	$\pi$	$1\left({\frac {\pi }{2}}\right)$
3	${\frac {4\pi }{3}}$	$2\left({\frac {\pi }{2}}\right)$
4	${\frac {\pi ^{2}}{2}}$	$1\cdot 3\left({\frac {\pi }{2}}\right)^{2}$
5	${\frac {8\pi ^{2}}{15}}$	$2\cdot 4\left({\frac {\pi }{2}}\right)^{2}$
6	${\frac {\pi ^{3}}{6}}$	$1\cdot 3\cdot 5\left({\frac {\pi }{2}}\right)^{3}$
7	${\frac {16\pi ^{3}}{105}}$	$2\cdot 4\cdot 6\left({\frac {\pi }{2}}\right)^{3}$
8	${\frac {\pi ^{4}}{24}}$	$1\cdot 3\cdot 5\cdot 7\left({\frac {\pi }{2}}\right)^{4}$
9	${\frac {32\pi ^{4}}{945}}$	$2\cdot 4\cdot 6\cdot 8\left({\frac {\pi }{2}}\right)^{4}$
10	${\frac {\pi ^{5}}{120}}$	$1\cdot 3\cdot 5\cdot 7\cdot 9\left({\frac {\pi }{2}}\right)^{5}$
11	${\frac {64\pi ^{5}}{10395}}$	$2\cdot 4\cdot 6\cdot 8\cdot 10\left({\frac {\pi }{2}}\right)^{5}$
$n$	${\begin{aligned}&{\frac {2^{n}}{n!!}}{\Bigl (}{\dfrac {\pi }{2}}{\Bigr )}^{\left\lfloor n/2\right\rfloor }\\&\ ={\frac {\pi ^{n/2}}{\operatorname {\Gamma } \left({\frac {n}{2}}+1\right)}}\end{aligned}}$	${\begin{aligned}&(n-1)!!{\Bigl (}{\dfrac {\pi }{2}}{\Bigr )}^{\left\lfloor n/2\right\rfloor }\\[10mu]&\ =\pi ^{(n-1)/2}\operatorname {\Gamma } \left({\tfrac {n}{2}}+{\tfrac {1}{2}}\right)\end{aligned}}$

The recurrence is $V_{0}=1,\,V_{1}=1,\,V_{n}=(n-1){\frac {\pi }{2}}V_{n-2}$.

–jacobolus (t) 13:35, 16 October 2021 (UTC)

I would not add any of this to the article because conventionally, volume is defined in units of an n-cube of side length 1, while the size of an n-ball is defined by its radius, with a unit ball having radius 1. — Preceding unsigned comment added by Quantling (talk • contribs) 16 October 2021 (UTC)

Sorry, I am not arguing we should add this table. Only pointing out that the assumptions implicit in prevailing conventions lead to misinterpretation/confusion about the source of particular parts of a formula. The power of 2 here is the most significant blemish. Someone who looks at File:hypersphere volume and surface area graphs.svg is going to think there is something special about dimension 5, but really the graph here should be a consistent decline from 1 onward; the powers of two arbitrarily inserted into the numerator from the mismatch between "diameter" 1 cubes vs. diameter 2 balls/spheres creates a grossly misleading impression. But it is also misleading if readers come away with the idea that spheres get "smaller" as the dimension grows. It must be understood that volume is the ratio of a particular shape to a standard cube, but the choice of cube as a unit is arbitrary, and if a different shape is chosen as a reference we can easily come to the opposite conclusion that spheres keep getting "bigger" without bound as the dimension grows. It is also kind of interesting to note that the volume of spheres is approximately halfway (on log scale) between the volume of cubes and the volume of simplices. –jacobolus (t) 04:20, 17 October 2021 (UTC)

Comparing quantities of different dimensions, such as asking whether 4 square feet is bigger than 3 cubic feet is all nonsense, IMHO. Who should care one wit whether the 4 is bigger than the 3 when we are talking square feet vs. cubic feet? Likewise, the shape of the curve of V_n(R) as a function of n is pretty darn pointless, IMHO. It does not bother me that the shape of that curve changes when one talks about unit radius vs. unit diameter, because the shape is not mathematically interesting in the first place.

In contrast, if you want to write something about the relative size of a hyperball to its inscribed hypercube or its circumscribed hypercube, that might be interesting. You could even talk about the relative hyperareas of their surfaces. Please give it a go if that catches your interest. —Quantling (talk | contribs) 17:09, 18 October 2021 (UTC)

Wikipedia is not in the business of deciding what is and is not the correct measurement of volume. You can find the formula for the volume of an n-ball in terms of the radius in many sources. You cannot say the same for the volume in terms of the diameter; nor can you say the same for the result of rescaling by the volume of a unit simplex. Adding this content would be against policy.

For what it's worth, my opinion is that 2π is a more fundamental constant than either π or π/2. Ozob (talk) 06:29, 25 November 2021 (UTC)

For what it's worth, my opinion is that π is a more fundamental constant than 2π. A1E6 (talk) 12:02, 25 November 2021 (UTC)

No, really. Stop arguing about the "fundamentality" of the constants. It's ill-defined. A1E6 (talk) 12:07, 25 November 2021 (UTC)

Missing subtopic: Hyperbolic geometry

This article is mostly about Euclidean geometry, but it also has some material about balls in L^p norms. It does not have any information about the volume of balls in hyperbolic geometry, however. Maybe it should? They behave quite differently from Euclidean balls (growing exponentially with radius rather than polynomially) and if not here I don't know where I'd go to find formulas for their volume. (There are formulas in https://math.stackexchange.com/questions/1445278/what-is-the-volume-of-the-sphere-in-hyperbolic-space/3652822 but that's not a reliable source so we'd have to look for a better source with the same material.) —David Eppstein (talk) 00:54, 30 October 2021 (UTC)

There is also some relevant material at Spherical cap. –jacobolus (t) 02:40, 30 October 2021 (UTC)

Dimension maximizing the volume of a fixed-radius ball

I am in favor of deleting the section "Dimension maximizing the volume of a fixed-radius ball" as was just done ... except that for reasons that escape me, people want to make the comparison from one dimension to another. (In fact, a graphic showing how different dimensions compare for the radius of R = 1 still remains in the article!) I recommend that we either put in a brief section to satisfy the demand for it -- such as that which was just removed -- or put in an explanation as to why we aren't including the section. —Quantling (talk | contribs) 03:09, 11 November 2021 (UTC)

This is one arbitrary problem among many tangentially related (at best) problems that might be considered by someone learning about volume of disks and balls. It might make an amusing exercise for a high school calculus class, but I see no relevance to any other problem (elsewhere in mathematics or in the world), and not really much relevance to this article either. I think it should be skipped (which is why I deleted it). –jacobolus (t) 06:01, 11 November 2021 (UTC)

But the picture was meticulously prepared and interactive. As there are only 2 metric quantities that can describe an n-ball, namely radius and diameter, maybe the guardians of truth of this article respectfully consider putting 2 pictures: one for the unit radius and another one for the unit diameter? Guswen (talk) 22:35, 22 November 2021 (UTC)

A beautiful and arduously created image of meaninglessness is still meaningless. —David Eppstein (talk) 06:33, 23 November 2021 (UTC)

Meaning (or meaninglessness) is subjective. And imposing one's meaning to others can be fatal, as teaches the history of the human kind.

And it's just a fact that unit radius n-ball attains maximum volume for n=5 and maximum surface for n=7, while unit diameter n-ball attains maximum volume for n={0, 1} and maximum surface for n={2, 3}. Why would one want to hide it? Even if the ultimate relevance of those facts (in mathematics or in the world) remains to be researched, you hinder this research by hiding them. Guswen (talk) 15:47, 23 November 2021 (UTC)

In fact there is a mathematical theory of meaningfulness. See R. D. Luce, "Dimensionally invariant numerical laws correspond to meaningful qualitative relations", Phil. Sci. 1978, http://philpapers.org/rec/LUCDIN. It includes not comparing quantities that scale in different ways from each other. —David Eppstein (talk) 17:00, 23 November 2021 (UTC)

Why do you refer to mathematical meaningfulness, why we discuss your subjective opinion about the fact that unit radius n-ball attains maximum volume in n=5 and maximum surface in n=7 dimensions is meaningless? I do not share your opinion. Am I alone? I doubt that. Someone has devoted his work and passion to create this arduously created image. Guswen (talk) 17:27, 23 November 2021 (UTC)

Guswen, you're not alone. And David, the article http://philpapers.org/rec/LUCDIN is about dimensions in physics. The Wikipedia article is about volume in n-dimensional Euclidean space which is a real number defined by an integral. A1E6 (talk) 17:33, 23 November 2021 (UTC)

A real number defined by an integral with a scale factor, the radius/diameter/other defining measure of the ball. It makes no sense, and is formally counter to Luce's theory, to compare values that scale as $r^{p}$ with other values that scale as $r^{q}$, whether you choose to do so by fixing $r=1$ or by whatever other arbitrary way of eliminating that scale factor you choose. —David Eppstein (talk) 17:50, 23 November 2021 (UTC)

A definite integral over some interval $I$ is a linear map from the space of all real-valued integrable functions on $I$ to $\mathbb {R}$. In the case of

$V_{n}(R)=\int _{-R}^{R}V_{n-1}\!\left({\sqrt {R^{2}-x^{2}}}\right)dx,$

$I=[-R,R]$. Luce's theory belongs to physics, not math. A1E6 (talk) 18:03, 23 November 2021 (UTC)

Fix $r=\{-1,-i,i\}$ and assume $p,q$ is complex and see what happens. Guswen (talk) 18:06, 23 November 2021 (UTC)

$r=1$ is clearly different.Guswen (talk) 20:28, 23 November 2021 (

There are many other loopholes, like that, I see but, having a limited time, as everyone, unable to explore within a simple framework I propose. Guswen (talk) 22:18, 23 November 2021 (UTC)

The Number Of The Beast is 666 and as a real value number of radians we can compute that cos(666) = 0.99984437405678246306.... While mathematically coherent (to some extent), this is sufficiently relevant for exactly sin(666π) Wikipedia articles. —Quantling (talk | contribs) 00:34, 24 November 2021 (UTC)

I don't see your point. cos(666)=0.999844374056783, while sin(666*pi)=-1.312994683264207e-13. Guswen (talk) 08:39, 24 November 2021 (UTC)

For the record, $\sin(666\pi )=0$. A1E6 (talk) 11:27, 24 November 2021 (UTC)

True :) I must have been slightly brainwashed this morning :) And Matlab was not very supportive :) Guswen (talk) 14:16, 24 November 2021 (UTC)

For the record, cos(666) = 0.99984437405678246306955025055120548486968039390258.... —Quantling (talk | contribs) 14:31, 24 November 2021 (UTC)

Wow! What a news! Guswen (talk) 06:49, 26 November 2021 (UTC)

The removed picture was very informative and, most importantly, was reflecting the content of this article (namely, the function $n\mapsto V_{n}(R)$). Also, we often want to maximize some function (the volume in this case). And balls are often described using radius. I don't think it is comparable to $0.99984437\ldots$. If you disagree, you can delete all the formulas using radius because the radius is an arbitrary measure, or delete the Unit circle article because $1$ is, again, "arbitrary". The picture is still in the N-sphere article, anyway. A1E6 (talk) 01:03, 24 November 2021 (UTC)

I vote against removing the formulas and article mentioned by @A1E6:. My objection is to comparisons of n-dimensional volume to m-dimensional volume for n ≠ m. —Quantling (talk | contribs) 16:22, 24 November 2021 (UTC)

When I was writing about removing the formulas, I was describing an intentionally absurd scenario which would be in accordance with someone's way of thinking. And I'm really tired of explaining the fact that we compare $V_{n}(R)$'s and $V_{n}(R)$'s are real numbers. A1E6 (talk) 19:54, 24 November 2021 (UTC)

@Quantling: If you will, you can go to the N-sphere article, remove the picture and give your reasons. I hope, though, that this would alert other people who would be willing to explain to you that your reasoning is not valid. A1E6 (talk) 20:07, 24 November 2021 (UTC)

@David Eppstein: You may not like (or find disturbing?) the fact that n-balls have zero volumes and surfaces in negative, even dimensions. But it's not very polite to call someone crank. Don't you think? Guswen (talk) 21:38, 24 November 2021 (UTC)

I do not understand well what is the object of this discussion. It seems to be whether some WP:Original research is worth to be kept or removed from the article. Should I recall that all original research must be removed, and all results must have been discussed in reliable sources. In particular, it seems that the title of this thread refers to original research.

For clarity, I rewrote the lead for making it a true summary of the content of the article. I suggest to reorganize the whole article in the same spirit. In particular, it is ridiculous to devote so much space and table entries for saying in a confuse way that if $V=V_{n}R^{n},$ then $R=V^{1/n}/V_{n}^{1/n}.$ D.Lazard (talk) 21:44, 24 November 2021 (UTC)

The primary object of this discussion is whether the picture [1] should be kept or removed from this article.

The secondary object of this discussion is whether we accept the fact that n-balls have zero volumes and surfaces in negative, even dimensions which follows from the recurrence relation $V_{n}(R)={\frac {2\pi }{n}}R^{2}V_{n-2}(R)$ already disclosed in this article. Guswen (talk) 23:05, 24 November 2021 (UTC)

No one was talking about issues related to original research. That is not the object of this discussion whatsoever. A1E6 (talk) 02:26, 25 November 2021 (UTC)

That picture does not describe anything of encyclopedic value. The subject of what dimension maximizes the volume of a fixed-radius ball is a curiosity of no real importance. If you want to prove me wrong, don't just contradict me; find some Wikipedia:Reliable Sources to the contrary.

Also, regarding the supposed "secondary object" of this discussion: There is no such thing as an n-ball in negative dimension. There's a formula which can be evaluated at negative n, but it doesn't describe anything geometric. Ozob (talk) 06:34, 25 November 2021 (UTC)

Again, this is your subjective opinion that "There is no such thing as an n-ball in negative dimension". Though you're obviously right that it doesn't describe anything geometric. Guswen (talk) 10:20, 25 November 2021 (UTC)

Find me some reliable sources that discuss balls in negative dimensions and we can discuss this further. Ozob (talk) 16:40, 25 November 2021 (UTC)

I found an article that discusses negative dimensions. The author is a contributor here, so I don't know whether that's an issue. —Quantling (talk | contribs) 17:53, 30 November 2021 (UTC)

No one was talking about issues related to original research. This is exactly the problem: people discuss the content of the article without considering Wikipedia policy, and it is now clear that the two objects of this discussion are both WP:OR, as no source are provided for them. D.Lazard (talk) 09:20, 25 November 2021 (UTC)

The picture [2] created by User:Cmglee on 5 February 2018 can hardly be even close to WP:OR. It merely shows volumes and surfaces of unit radius n-balls, known perhaps since XVIII century, when Leonhard Euler discovered Gamma function. The picture is also present at n-sphere article. Guswen (talk) 10:17, 25 November 2021 (UTC)

SVGs created by the Wikipedia community are original research? A1E6 (talk) 11:51, 25 November 2021 (UTC)

Displaying the graph of a function is not original research (per WP:CALC). But such a graph is aimed to illustrate the study of this function (here the function $n\mapsto V_{n}$), and this study may be original research. This seems to be the case here. D.Lazard (talk) 15:20, 25 November 2021 (UTC)

Following this logic, the whole article is problematic itself. A1E6 (talk) 15:30, 25 November 2021 (UTC)

No, because nearly everything in the article is in reliable sources. As far as I'm aware, there are no reliable sources for the study of the function $n\mapsto V_{n}$, so no such study can be included in the article. But I would change my opinion completely if there were a reliable source which studied this function.

Until reliable sources are provided, there's no reason to continue this discussion. Ozob (talk) 16:40, 25 November 2021 (UTC)

@Ozob: Here are the reliable sources: 1) https://www2.karlin.mff.cuni.cz/~slavik/papers/p-balls.pdf, 2) D. J. Smith, M. K. Vamanamurthy: How small is a unit ball?, Math. Mag. 62(1989), 101–107, 3) https://mathworld.wolfram.com/Hypersphere.html A1E6 (talk) 17:40, 25 November 2021 (UTC)

In particular, in (1), see It is well known that the volume of the n-dimensional unit ball is maximal in dimension five, where it begins to decrease, and tends to zero as $n\to \infty$... A1E6 (talk) 17:46, 25 November 2021 (UTC)

Wonderful! I now have no objection to putting this fact into the article. Ozob (talk) 02:34, 26 November 2021 (UTC)

And do you agree with including the picture in the article? A1E6 (talk) 02:51, 26 November 2021 (UTC)

No. The text is too big and the illustration too dense. I don't think it looks good. Ozob (talk) 01:58, 27 November 2021 (UTC)

It is nice to have some kind of illustration if possible. But I think it would be more helpful to show a diagram of dimensions 1–3 explicitly comparing a unit-radius ball to a unit-side cube, instead of a chart implying that balls of fractional dimensions are meaningful or that n-volume can be meaningfully compared to m-volume. –jacobolus (t) 17:23, 25 November 2021 (UTC)

Would it be helpful? Volume of a unit-side n-cube is 1 (also in negative, fractal, and - in general - complex dimensions), surface is 2*n. Guswen (talk) 18:17, 25 November 2021 (UTC)

The point would be to graphically show the units of volume being compared. –jacobolus (t) 00:29, 27 November 2021 (UTC)

Gentlemen, Let's just delete this whole speculative, wacky article and eventually close this futile discussion. n-sphere article seems sufficient. Guswen (talk) 17:36, 25 November 2021 (UTC)

Whether you call it m-volume or n-volume, it is the Lebesgue measure and a real number. If you imply that these cannot be compared, I disagree. Indeed, unlike in the case of vaguely stating that the comparison is "meaningless", it can be proved that they can be compared. A1E6 (talk) 23:05, 25 November 2021 (UTC)

I think we all agree that we are discussing real numbers and that the real numbers are totally ordered. The sense in which it is "meaningless" is that one can't assign consistent units across different n. That makes the question of which n achieves the maximum a purely academic one. But, as your sources above demonstrate, it is nevertheless a question which has been studied. A quick mention of this question in this article looks appropriate to me. Ozob (talk) 02:34, 26 November 2021 (UTC)

Fortunately, we're not in physics and are not leaving the realm of real numbers when discussing volume. A1E6 (talk) 03:02, 26 November 2021 (UTC)

You keep repeating this "physics" thing as if somehow mathematics is purely a realm of manipulation of symbols, that any correct manipulation of symbols is equally as interesting and meaningful as any other, and that dimensional analysis is somehow an impure thing that must be cast aside when we do mathematics. None of those positions is a good one. —David Eppstein (talk) 08:43, 26 November 2021 (UTC)

This is not about manipulation of symbols. This is about n-balls. Guswen (talk) 10:42, 26 November 2021 (UTC)

@David Eppstein: Finding/not finding the mathematical fact interesting is your subjective opinion. Also, the notability of the result can be supported with sources, which I provided when Ozob disputed the notability. A1E6 (talk) 13:05, 26 November 2021 (UTC)

Arbitrary scalars can be compared in some purely formal sense, but when you talk about a "n-volume" you are implicitly talking about a scalar ratio of two n-blade-valued quantities. Comparing such scalar ratios of different-dimensional blades abstracted from their geometric context is meaningless without some additional information. There is no need for “physics” to talk about dimensional quantities; this is purely geometric/mathematical. When you compare e.g. an area of a circle to the volume of a sphere as scalars ("real numbers"), you are "really" (i.e. meaningfully) comparing the volume of a unit-height cylinder to the volume of the sphere. But just comparing the two numbers as scalars without somehow first matching their dimension is almost entirely arbitrary and useless. –jacobolus (t) 00:29, 27 November 2021 (UTC)

@Jacobolus: When I'm talking about the n-volume, I'm implicitly talking about the Lebesgue measure. A1E6 (talk) 13:53, 27 November 2021 (UTC)

It doesn’t matter how formal or informal you make your definition of n-volume. Regardless, it is not inherently meaningful to compare length (1-volume) with area (2-volume) or 4-volume with 7-volume. –jacobolus (t) 18:56, 27 November 2021 (UTC)

There can occasionally be some sense in comparing different dimensional measures, when it is recognized that the comparison depends on a scale factor and the result is used to set a "natural" scale for an object. For instance, in equable shape, a comparison between area and perimeter sets up the question of what scale factor to use to make the two equal. Similarly, I could imagine situations where some kind of optimization criterion depends on terms with different scaling behavior and the optimum is found by finding a scale at which the contributions from different terms balance. But no such purpose has been suggested or sourced here, and even the topic of equable shapes, although notable, is I think only dubiously of interest mathematically, more notable from its common use as an exam question. —David Eppstein (talk) 19:46, 27 November 2021 (UTC)

The article equable shape mentioned by David is using physically defined quantities like metres (these are defined with respect to light which does not exist in mathematics), rendering the arguments therein invalid in terms of a pure mathematical setting.

"It doesn't matter how formal or informal you make your definition of n-volume" – I disagree. The definition is fundamental for deciding whether n-volumes can be compared to each other. Instead of using vague terms like "inherently meaningful", I rely on strict mathematical definitions. A1E6 (talk) 20:21, 27 November 2021 (UTC)

Your obsession with labeling the mathematics of length and area as "physics" is bordering on crankery. Physical units are used in that article only to make the content more accessible; they are not in any way central to the underlying concepts. Equable shapes are perfectly well defined as a purely mathematical and even purely number theoretical concept (i.e. which integer lengths of triangles have equal areas and perimeters); the whole point of this discussion is that being well defined mathematically is not enough to create a purpose for a concept. —David Eppstein (talk) 20:44, 27 November 2021 (UTC)

In no way did I label the mathematics of length and area as physics.

"Equable shapes are perfectly well defined as a purely mathematical and even purely number theoretical concept" – I didn't say they were not perfectly defined. It seems that your reply is a huge misunderstanding between us. A1E6 (talk) 20:50, 27 November 2021 (UTC)

You did, when you dismissed equable shape as "using physically defined quantities" and therefore as somehow irrelevant. —David Eppstein (talk) 20:52, 27 November 2021 (UTC)

I really didn't. The definition of equable shape is fine. I opposed to the following statement: An area cannot be equal to a length except relative to a particular unit of measurement. (which was present in the article) when we talk about length and area purely from a mathematical perspective where we do not have things such as metres. In a physical setting, the statement is fine. A1E6 (talk) 20:56, 27 November 2021 (UTC)

That statement is correct even from a purely mathematical perspective. Mathematically, length is usually defined as Lebesgue measure on the real line or as the 1-dimensional Hausdorff measure in a metric space. Area, on the other hand, is usually defined as the product Lebesgue measure on the real plane or the 2-dimensional Hausdorff measure in a metric space. All these measures require a choice of scale factor. For Lebesgue measure on the real line, we set this scale factor using the points 0 and 1; for Lebesgue measure on the plane, we use the square defined by (0, 0) and (1, 1); for Hausdorff measure, we use the metric.

But if we look at R or R² purely as topological groups, and if we look at a metric space purely as a topological space, then we lose the information we used to set the scale. As a topological group, R has an origin (the identity element), but all the other points are equivalent (formally, for any other two points, there is an automorphism of topological groups which sends one point to the other). Consequently there is no natural choice of Haar measure. R² has an even more severe problem because it does not have a canonical product structure. Even if you know the location of the origin and one other point, you don't know the rectangle they determine because you don't know which way the edges point. To fix a Haar measure you need something extra (like a third point or a choice of product structure).

It follows that length and area are, in the most general case, incomparable. They aren't actually real numbers; their values are elements of different $\mathbf {R} _{+}^{\times }$-torsors; one for lengths and one for areas. The extra data that you need to make them real numbers can be characterized in many different ways; one of those ways is a choice of a particular unit of measurement. Ozob (talk) 21:44, 27 November 2021 (UTC)

@Ozob: In the article, $n\mapsto V_{n}(R)$ is a function from $\mathbb {N} \to \mathbb {R}$ characterizing the volume. What choice of a particular unit of measurement has been made? A1E6 (talk) 22:00, 27 November 2021 (UTC)

The product Lebesgue measure. It has a standard unit cube whose volume is defined to be 1. Ozob (talk) 22:19, 27 November 2021 (UTC)

And that is the standard definition of the volume, making it a real number and making it comparable. A1E6 (talk) 22:26, 27 November 2021 (UTC)

Ah, so I think we agree! There is no natural normalization of the Haar measure on an n-dimensional real vector space, but if we choose units by fixing a basis, then everything is comparable. We both believe this, right? Ozob (talk) 23:51, 27 November 2021 (UTC)

But we also need extra information (not as much as a basis) just to define the concept of a ball. That choice is already adequate to normalize the Haar measure, although maybe using a ball to normalize the measure to be the formula given in this article (rather than just saying that the measure of the ball is one) is a bit unnatural. —David Eppstein (talk) 00:16, 28 November 2021 (UTC)

The weakest structure that gives a ball of the right shape is a positive definite real quadratic space (a real vector space with a positive definite quadratic form Q). The set of all x such that Q(x) ≤ 1 is a unit ball with respect to Q. Comparing unit balls on different quadratic spaces requires fixing a linear transformation from one space to the other. The result, however you interpret it, is tautological: If you assume nothing about the transformation, then the relative sizes of the balls are of course arbitrary; if you assume that the transformation is an isometry onto its image, then you've given the two spaces consistent units, so you get the usual comparison of volumes of different balls. Ozob (talk) 00:44, 28 November 2021 (UTC)

In regard to your previous reply: Yes, we both believe this. It boils down to what "the volume" means – I just think that the normalization with the unit n-cube is implicit with almost every mention of "the volume" on Wikipedia. A1E6 (talk) 00:50, 28 November 2021 (UTC)

@David Eppstein: "being well defined mathematically is not enough to create a purpose for a concept." – If you dispute the notability: the sources had already been provided. A1E6 (talk) 01:21, 28 November 2021 (UTC)

Notability is a third thing, different from both rigor and interestingness. —David Eppstein (talk) 07:03, 28 November 2021 (UTC)

Please specify where exactly is the result/proof not rigorous. And interestingness is subjective. A1E6 (talk) 11:45, 28 November 2021 (UTC)

I think David meant his remark the other way (but he can correct me if I'm wrong): Even though the result is rigorous, it may not meet WP:N. Ozob (talk) 16:47, 28 November 2021 (UTC)

So, can we eventually keep the picture [3] in this article? Personally I find it interesting. Guswen (talk) 18:10, 28 November 2021 (UTC)

It's too dense for my taste. Ozob (talk) 21:21, 28 November 2021 (UTC)

I think the picture is quite good, but it is necessary to fix it: $V_{15}$ should be ${\tfrac {256\pi ^{7}}{2027025}}$ instead of ${\tfrac {15\pi ^{7}}{118771}}$. A1E6 (talk) 22:43, 28 November 2021 (UTC)

Perhaps I could make it lighter to fit Ozob taste? How about this one [4]. Of course I would first crop out the unsourced negative dimensions not to upset Ozob. Guswen (talk) 22:48, 28 November 2021 (UTC)

@Guswen: Regarding your picture: the "gamma-expression curve" with significant points (at the natural numbers) alone is better than mixing it all with linear interpolation. A1E6 (talk) 23:04, 28 November 2021 (UTC)

True. This spline interpolation was a bad idea. It's better to use known V_(n-2) continuous recurrence relation than spline over integer values. I'll remove it (as soon as it's accepted). Guswen (talk) 23:18, 28 November 2021 (UTC)

@Guswen: I would prefer: Volumes in one picture and surface areas in another; values indicated with dots; dots not connected by lines; starting at n = 0 or n = 1; and no dimension singled out, i.e., do not indicate that n = 5 is where the volume is maximized or that n = 7 is where the surface area is maximized. I'm not sure if there's enough space to include exact expressions for the volumes and surface areas while keeping the picture good; I'd have to see it. Ozob (talk) 23:49, 28 November 2021 (UTC)

Would you please send me a sketch of your pictures (e.g. via e-mail). I'll do my best to include your suggestions. Guswen (talk) 21:08, 29 November 2021 (UTC)

Gentlemen, I believe that we have reached some form of agreement concerning the pictures? Guswen (talk) 21:08, 29 November 2021 (UTC)

Perhaps the linear interpolation is OK. It doesn't create the illusion that we defined the n-sphere for non-integer n (at least not so much). But it is understandable that someone prefers a smooth graph (with significant points marked with dots) over a jaggy graph (with significant points indicated by corners). On the other hand, Ozob prefers a discrete plot (if I understand correctly). A1E6 (talk) 01:57, 30 November 2021 (UTC)

Yes, I would prefer plots like the ones I just made, displayed at right. Ozob (talk) 04:50, 30 November 2021 (UTC)

Surface areas of unit hyperspheres in dimensions 0 through 25

Volumes of unit balls in dimensions 0 through 25

How about now? Guswen (talk) 11:21, 30 November 2021 (UTC)

I still do not like connecting the dots. It suggests that there are balls of fractional dimension, and while I personally don't object to that idea, on Wikipedia that needs to be backed by a citation.

I also still do not like putting both the surface area and volume in the same picture. The vertical scales used by the two are completely different.

Finally, I don't see any reason to have two pictures for different radii. If showing what happens at different radii is of interest, then it would be better to put that in a single picture.

I tried updating the pictures I made before. I'm pretty happy with them; I like them better than what's in the article now. Ozob (talk) 05:26, 1 December 2021 (UTC)

I think there are balls of fractional dimension. $V_{n}(R)={\frac {2\pi }{n}}R^{2}V_{n-2}(R)$ recurrence relation admits fractional dimensions (real n). But indeed, that needs to be backed by a citation.

Indeed. The scales are different, so perhaps that is not a good idea to mix volume with surface on the same picture.

I think unit radius and unit diameter n-balls are of particular interest. But indeed, that idea requires further research.

I updated the article with your pictures. Guswen (talk) 10:07, 1 December 2021 (UTC)

A comparison of V_n to S_n for a given value of n is something that is meaningful. For example, one could meaningfully ask what is the ratio of the surface area of the northern hemisphere to its projection down onto the circle that slices the ball at its equator? These updated charts eliminate having V_n to S_n on the same graphic. IMHO, these updated charts eliminate the aspect that the originals had that is most plausibly of use. —Quantling (talk | contribs) 17:33, 2 December 2021 (UTC)

There is some point in this argument. Guswen (talk) 18:12, 2 December 2021 (UTC)

After a careful consideration I changed my mind: perhaps that is not a good idea to mix volume with surface on the same picture. Guswen (talk) 19:18, 2 December 2021 (UTC)

Using the same y axis values for all plotted values says to me that there is some circumstance under which it is practical to compare V_n(1) to, say, V_n+1(1) or V_n+2(1.1). Can you find a citation that indicates a context where such a comparison is practical? If not practical, can you find one that indicates that it is interesting in that it leads to something else of interest? I am looking for a reasonable citation to any argument that goes beyond "the comparing of volumes from different dimensions is interesting in and of itself". It is my fear that such comparisons are fringe; not sufficiently encyclopaedic. —Quantling (talk | contribs) 18:39, 2 December 2021 (UTC)

@Quantling: We've already had this discussion. It doesn't have to be "practical", it's pure math. If you dispute the notability: several sources have been provided. A1E6 (talk) 13:23, 3 December 2021 (UTC)

We are having a discussion and I thank you for being a part of it. The material does have to be notable and that can be achieved through a number of means. The only sources I see mentioned are the following. (If I missed some, please let me know!)

1. https://www2.karlin.mff.cuni.cz/~slavik/papers/p-balls.pdf appears to have been published in Archiv der Mathematik. "Archiv der Mathematik (AdM) publishes short high quality research papers in every area of mathematics which are addressed to a broad readership and not overly technical in nature." IMHO, that's not so strong as a primary source. Do we have evidence that this article is used as a secondary source?
2. D. J. Smith, M. K. Vamanamurthy: How small is a unit ball?, Math. Mag. 62(1989), 101–107 -- "Mathematics Magazine offers lively, readable, and appealing exposition on a wide range of mathematical topics." Again, Wikipedia isn't for presenting toy problems, but for summarizing accepted mathematics. IMHO, this is no stronger than the citation in Archiv der Mathematik.
3. https://mathworld.wolfram.com/Hypersphere.html -- this is a tertiary source that has substantial overlap with Wikipedia. I hesitate to classify it as the secondary source that it should be, and I hesitate to assume that it is independent of the Wikipedia pages that it is quite similar to.

Surely if this material is notable there is an actual primary source in the literature of mathematics research, or a notable secondary source / textbook. If we have to scrape the bottom of the barrel to find citations ... I fear that that is telling us that we are reaching too far. —Quantling (talk | contribs) 15:10, 3 December 2021 (UTC)

The articles on this topic often refer to each other. "Wikipedia isn't for presenting toy problems" I don't see why you consider that a toy problem, whatever that means. "...independent of the Wikipedia pages..." What do you mean?

I've been very confused by some of your writing lately, as it seems inconsistent... Is deleting Ozob's picture your intention/wish? A1E6 (talk) 18:31, 3 December 2021 (UTC)

Me too... On 17:33, 2 December 2021 you (Quantling) argue that a comparison of volume to surface for a given dimension is something that is meaningful and that updated charts eliminate that aspect that the originals had that is most plausibly of use (so you implicitly argue I favor of the previous charts), only to argue later (18:39) that any such a comparison is not practical... It does not make sense to me. Guswen (talk) 18:40, 3 December 2021 (UTC)

I am looking for a research article that compares V_n or S_n to V_m or S_m for m ≠ n. Best if it is in a decently high-impact journal intended to talk to actual mathematics researchers rather than a money-grubbing conference or something intended for amateurs. Surely there's got to be one somewhere. If we can find one, it would make me a lot more confident that we haven't gone off the deep end. —Quantling (talk | contribs) 21:32, 3 December 2021 (UTC)

Money-grubbing conference? Intended for amateurs? What are you even talking about? The given sources establish the notability of the topic. A1E6 (talk) 01:27, 4 December 2021 (UTC)

I apologize for the distraction that my "money-grubbing conference or something intended for amateurs" causes here. I am trying to say that it is my hope that we will find an article in a decently high-impact research journal that compares V_n or S_n to V_m or S_m for m ≠ n. —Quantling (talk | contribs) 03:59, 4 December 2021 (UTC)

The Mathematics Magazine article is a reliable secondary source—precisely what Wikipedia is supposed to cite—which discusses the behavior of V_n as n varies. (It also provides an exposition of several of the proofs of the volume formula.) Is that what you were asking for? Ozob (talk) 05:04, 4 December 2021 (UTC)

We haven't gone off the deep end. And see the Unit sphere article for example, where the similar pictures are present. Curiously this article also discusses fractional dimensions. Guswen (talk) 15:13, 4 December 2021 (UTC)

What is more curious, however is that you (Quantling) have created these pictures on January 13, 2009 and then inserted them in the new section concerning n-sphere fractional dimensions (?). Guswen (talk) 15:19, 4 December 2021 (UTC)

Thank you @Ozob: for your reply. Yes, secondary sources are key. However, if this is in fact accepted mathematics, I'd think that there would be a decent primary source somewhere. Lacking that, I still have doubts.

Thank you @Guswen: for your reply. If you think it would bring in more opinions from more editors, we could note the existence of this discussion on other relevant Wikipedia pages. Also, if you would, please clarify your remarks about my having created pictures. —Quantling (talk | contribs) 20:11, 4 December 2021 (UTC)

I have no idea what is going on here. These ([5] and [6]) pictures were created on 13 January 2009 by user Quantling. This account is now deleted (marked as red) though I believe it wasn't on 15:19 when I discovered these pictures for the first time. The usernames are the same. Now on 15:01 20 August 2009, for example, you (existing user Quantling) changed the unit sphere article from "The formulae for A_n and V_n can be computed for any complex number with non-negative real part..." to "he formulae for A_n and V_n can be computed for any real number n ≥ 0...", which means that you not only support comparing V_n or S_n to V_m or S_m for m ≠ n but you also support n-balls in fractional and non-negative dimensions...

Also on 18:31, 13 January 2009 you added ([7]) a section concerning fractional dimensions to the unit sphere article... This is weird and confusing to me. Perhaps you are group that is coordinating your edits (?). Guswen (talk) 22:36, 4 December 2021 (UTC)

Thank you for clarifying. I have no recollection of making those graphics nearly 13 years ago, but the evidence sure looks like I did! The graphics that I apparently found appropriate then are something that I now have serious doubts about.

I suspect that that wikimedia account is not deleted despite the redlink.

FWIW, that volume for balls with positive real-valued dimensions can make sense is not the same as saying that different dimensions can be compared. For the former, you may be interested in, e.g., the use of ω_α for normalizing Hausdorff measure in https://encyclopediaofmath.org/wiki/Hausdorff_measure. —Quantling (talk | contribs) 02:48, 5 December 2021 (UTC)

So, if you now have serious doubts about, go to the unit sphere article and delete those graphics of yours, possibly with the whole wacky section that you added. I will gladly restore these graphics in their more sourced version. As a bonus you'll get a comparison of $V_{n}$ to $S_{n}$ for a given value of $n$ as something missing that you recently considered meaningful and demanded to restore. Guswen (talk) 14:34, 5 December 2021 (UTC)

As long as we have a productive discussion going here, I personally would be hesitant to make the unilateral WP:BRD edits you suggest that I make for unit sphere. I am thinking that one article at a time can work. —Quantling (talk | contribs) 18:00, 5 December 2021 (UTC)

Discussion here seems to be closed. But let me ask you a question: you have serious doubts about the (discrete) graphs of n-ball volumes w/r/t n in the Volume of an n-ball article, while at the same time you don't have any doubts about similar (though even more wacky, as extended to n-balls in positive fractional dimensions) graphs (of yours) in the Unit_sphere article? Right? Guswen (talk) 21:09, 5 December 2021 (UTC)

Based upon your description I would have serious doubts about the graphic in the unit sphere article as well. I base my response upon your description because I am not sure what graphic you are referring to in the unit sphere article. I don't think you mean the ones from 13 years ago that you already pointed out because I already state explicitly that I have serious doubts about them. If there is another graphic that is mine that I haven't opined about; your help in locating it would be much appreciated! —Quantling (talk | contribs) 15:16, 6 December 2021 (UTC)

I do apologize but I don't have time to locate your forgotten graphics or other stuff. Guswen (talk) 07:21, 8 December 2021 (UTC)

Thank you for deleting graphics from the unit sphere article on December 6. —Quantling (talk | contribs) 18:03, 8 December 2021 (UTC)

You're welcome. Wow, I see that you found a citation supporting your (previously unsourced) notion of n-balls in non-negative, fractional dimensions! Great job!

Would you do me favor and find a citation supporting my (currently unsourced) notion of n-balls in negative dimensions? Then we may eventually go on with this notion of n-balls in negative and fractional dimensions, ponder on the fact why there are no n-balls in negative and even dimensions, etc. I find it particularly interesting in the context of black holes.

I also think that having your citation on board we might replace the graphics in this article into continuous versions (@Ozob:). Guswen (talk) 19:57, 9 December 2021 (UTC)

@Guswen: If you want to include fractional dimensions, the proofs in the article here need to be edited. They rely on the fact that n is an integer. For example, consider what happens when n is not an integer in

$\prod _{i=1}^{n}\left(\int _{-\infty }^{\infty }\exp \left(-{\tfrac {1}{2}}x_{i}^{2}\right)\,dx_{i}\right)=(2\pi )^{n/2}.$ A1E6 (talk) 20:30, 9 December 2021 (UTC)

Too bad :( I took this citation for granted not checking its content. It seems therefore that Quantling wacky section in the unit sphere article remains unsourced, and thus must be removed. I kinda felt serious doubts about that... Guswen (talk) 21:12, 9 December 2021 (UTC)

Seems like, in order to move forward, we need to consider the notion of fractional multiplication... Guswen (talk) 21:29, 9 December 2021 (UTC)

@Guswen: For the record, I was referring to the proofs in the Volume of an n-ball Wikipedia article, not the article mentioned by Quantling. A1E6 (talk) 22:52, 9 December 2021 (UTC)

@A1E6: thank you for the clarification. Because the section's source is not being challenged, I have restored that section. —Quantling (talk | contribs) 01:44, 10 December 2021 (UTC)

Well, I'm going to nitpick and note that the source given in n-sphere for fractional dimensions does not actually say that there is such a thing as a fractional dimensional ball or sphere. It defines normalized Hausdorff measure using a normalizing constant $\sigma (s)$ and then says, "When $s=n$, we have $\sigma (n)={\mathcal {L}}^{n}(B)/2^{n}$ where $B$ is the unit ball in $\mathbb {R} ^{n}$." (p. 18) Saying that this is a useful normalizing constant for fractional dimensional Hausdorff measure is not really the same as saying that fractional dimensional spheres and balls really exist. The n-sphere article is carefully worded to follow the source, so it's fine. I think it is fine if this article includes similarly careful wording as long as we don't create a WP:CONTENTFORK. Ozob (talk) 02:02, 10 December 2021 (UTC)

@Ozob: Did you mean the unit sphere article? In that case, I don't think it makes sense to say "The formulae for $A_{n}$ and $V_{n}$ can be computed for any real number $n\geq 0$." when fractional dimensional spheres were not defined in the first place. A1E6 (talk) 04:06, 10 December 2021 (UTC)

Yes, sorry, I did mean the unit sphere article. The only citeable fact that I think might be relevant to this article is that the interpolation of the volume and surface area formulas to fractional positive real numbers do in fact get used. Ozob (talk) 04:27, 10 December 2021 (UTC)

I deleted the stuff in the unit sphere article because a drastic rewording would be needed to make it "fine". A1E6 (talk) 05:49, 10 December 2021 (UTC)

I think there is agreement here that, with the right wording, one or both of these Wikipedia articles would benefit from a mention that the volume formula can be used with non-negative real numbers in the context of Hausdorff measure. I believe the currently removed wording is mine. I can try again, but we might converge more quickly if another interested editor makes the next edit. (If no one does, I will give it a try.) —Quantling (talk | contribs) 19:01, 10 December 2021 (UTC)

Actually, I will give a try for acceptable wording. If you have objections, your corrective edits rather than a wholesale revert would be appreciated. —Quantling (talk | contribs) 21:57, 10 December 2021 (UTC)

"The evaluation of the volume formula at non-negative real values of n is sometimes used for normalization of Hausdorff measure."? You got to be kidding. It can hardly be even called a try :) Try harder. Guswen (talk) 23:25, 10 December 2021 (UTC)

Thank you for providing feedback. If I may beg more ... mention of things to aim for is more helpful than mention of things to avoid. Given the goal of improving the article, what would you aim for here? —Quantling (talk | contribs) 00:59, 11 December 2021 (UTC)

@Quantling: It's better now, but I think the normalization should be described explicitly. A1E6 (talk) 05:32, 11 December 2021 (UTC)

@Quantling: A1E6 "is trying to teach you something. I advise listening". [8]

Just kidding :) Go on and expand this section (by describing the normalization explicitly). Guswen (talk) 23:19, 11 December 2021 (UTC)

Please do it despite having serious doubts about. Guswen (talk) 00:29, 12 December 2021 (UTC)

@Quantling: This source might be useful in expanding your section [9] (cf. 1.1.ii). Guswen (talk) 22:30, 16 December 2021 (UTC)

References

Historical source

A primary source for the material in this article (including the comparison of different dimensions) is likely to be something historical; it is likely to have been written by Newton, or Euler, or one of the Bernoullis, or Gauss, or Lagrange, or Laplace, or someone like that. I'm only now realizing that I know nothing about the history of this calculation, so I can only guess who was first. (If I had to guess, I'd guess Euler, but that's a good rule for most things in math.) So I have no evidence that the Mathematics Magazine article is not a primary source for the asymptotic behavior of V_n, though I'd be shocked if it were. Ozob (talk) 06:16, 5 December 2021 (UTC)

You're far too early. Higher dimensional norms and distances were first formulated by Cauchy, in the early 19th century. See Ratcliffe, Foundations of Hyperbolic Manifolds, p. 32. —David Eppstein (talk) 07:09, 5 December 2021 (UTC)

Aha! Following the reference in the book you cited led me to Rosenfeld, "A History of Non-Euclidean Geometry", [10]. It appears (p. 248) that the first calculation of the surface area of a unit hypersphere is due to Jacobi in an 1834 paper entitled, De binis quibuslibet functionibus homogeneis secundi ordinis per substitutiones lineares in alias binas transformandis, quae solis quadratis variabilium constant; una cum variis theorematis de transformatione integralium, or in English, "On the transformation of two arbitrary homogeneous functions of the second order by means of linear substitutions into two others containing only squares of the variables; together with many theorems on the transformation of multiple integrals." (Works, vol. 3, pp. 191–268.) Jacobi actually calculates the surface area of just the positive orthant of the sphere, not the surface area of the whole sphere itself (Works, vol. 3, p. 267), and apparently he worked purely algebraically with no reference to higher-dimensional geometry. It might be interesting to know what Jacobi actually did, but I can't do it myself since I don't speak Latin. Ozob (talk) 22:35, 5 December 2021 (UTC)

Nice! Perhaps some of this can be worked into the article, maybe as part of the history of this calculation. (If this is something that we need to discuss further, we might want to break it into its own talk section because it seemingly has nothing to do with maximizing volume or comparing volume across multiple dimensions.) —Quantling (talk | contribs) 15:16, 6 December 2021 (UTC)

If anyone speaks Latin, the original is here. Ozob (talk) 03:46, 7 December 2021 (UTC)

Unfortunately I don't. Perhaps User:Quantling does? He's active in this area despite serious doubts about. Guswen (talk) 08:34, 7 December 2021 (UTC)

I read some Latin but can't bring myself to grind through the calculations prior to page 267, which is clear enough as stating the area formula for even ("par") and odd ("impar") dimensions separately. I think that and the Rosenfeld source are sufficient to add this history to our article. —David Eppstein (talk) 08:18, 8 December 2021 (UTC)

Contributors to this article

@Guswen: @Parejkoj: @Ozob: @A1E6: Are you the same person (or are you a group that is coordinating your edits)? —Quantling (talk | contribs) 23:40, 28 November 2021 (UTC)

Did you actually read this thread? Ozob (talk) 23:42, 28 November 2021 (UTC)

@Quantling: Alright, I really didn't expect this :) A1E6 (talk) 00:23, 29 November 2021 (UTC)

@A1E6:Welcome to my world :) User:Quantling seems to be just another conspiracy theory aficionado. Guswen (talk) 08:24, 29 November 2021 (UTC)

Content fork

In the previous thread, Guswen wrote Gentlemen, Let's just delete this whole speculative, wacky article and eventually close this futile discussion. n-sphere article seems sufficient.

Comparing the two articles, it appears that this article is a WP:REDUNDANTFORK of n-sphere#Volume and surface area. This fork has been created in 2009 by a user who has been indefinitely blocked soon after.

Consequently, I will redirect this article to that section. Apparently, nothing useful will be lost by this edit. In any case, the history of the article will remain accessible for restoring, some useful deleted content, if any, in the new target. D.Lazard (talk) 10:18, 26 November 2021 (UTC)

Makes sense to me. XOR'easter (talk) 19:08, 26 November 2021 (UTC)

I don't feel strongly that this is an independently notable subtopic or needs to be its own separate article. —David Eppstein (talk) 19:49, 26 November 2021 (UTC)

This article may have started as a redundant fork, but it isn't anymore. There are two topics covered by this article that are not in the n-sphere article. One is proofs, and the other is generalizations. Proofs are usually not of encyclopedic interest, but in this case my feeling is that they are. The generalizations to L^p balls are of interest in some applications but would be out of place in the n-sphere article.

There is some duplicated material, but I think that's better handled by applying Wikipedia:Summary style to n-sphere.

I'm worried that the redirection may be a reaction to the behavior of some editors, not to the state of the article. Consequently I have undone it for the time being. Ozob (talk) 01:54, 27 November 2021 (UTC)

The redirection is not a reaction ot anybody's behavior. It is simply that, for making my opinion on the above Guswen's quotation, I looked on N-sphere, where I found essentially the same content. That, in Wikipedia terminology, a content fork. I have chosen a redirect rather than a merge, because the content of this article that is not already in N-sphere seems WP:OR and does not deserve to be kept for the moment (if it is needed in the future, it remains accessible on the history of the article).

Also, the relationship between spheres and balls is fundamental here, and appears very late in this article, while, in n-sphere#Volume and surface area, it appears from the beginning (and even in the heading). This makes the target section much better, even if it needs to be strongly cleaned up.

About proofs: In n-sphere#Volume and surface area there is a proof that is simple and natural. Here there are several unsourced proofs that are less illuminating than the proof given there. I see no reason for having them in an encyclopedia. If they have been reliably published, a citation of the relevant article is sufficient.

About the generalization to $L^{p}$ norms: it is not established that this is not WP:OR. If it is not WP:OR, I cannot imagine how people interested in metric properties of spheres and balls in a metric space can find this section. So, if it deserve to be kept in Wikipedia, this is not in this article.

So, my opinion is that the redirect must be restored, and followed by a clean-up of the target section/article. D.Lazard (talk) 11:17, 27 November 2021 (UTC)

I agree with restoring the redirect. A1E6 (talk) 13:55, 27 November 2021 (UTC)

I agree that the present situation is a content fork and must be fixed. I'm not yet convinced that the right solution is to merge everything into n-sphere.

The content which I believe should be kept in either this article or the n-sphere article is: The statements of the volume and surface area in terms of the radius; the statement of the radius in terms of the volume (I know I've seen this used in a survey article on convex geometry); the statements of the recurrences; the table in low dimensions (but it should stop earlier, maybe at n = 10); the asymptotic behavior in high dimensions; and the relation of volume and surface area.

The proofs should only be kept if they are themselves notable. Possibly the two-dimension recursion formula is notable because it's the correct generalization of what Archimedes did. I have the impression that the proof via Gaussian integrals is important in measure concentration. I also think I read somewhere that the group-invariance idea used in that proof has other applications, but I would have to search for sources to confirm that. I'm not convinced that the other proofs are notable in the same way. The exposition is nice, though; maybe there's a place for these proofs somewhere on Wikibooks?

The generalizations are not OR; as the last section of the article says, $B_{p_{1},\dots ,p_{n}}(L)$ was known to Dirichlet. But it might be better to put this content at Lp space.

I'm worried that, even after removing some of the proofs and the generalizations, there might be too much content to merge into n-sphere. I think this content would either form a large section of n-sphere or a medium-sized independent article. Right now I'm in favor of keeping it as an independent article (and removing duplicate content from n-sphere). Ozob (talk) 20:15, 27 November 2021 (UTC)

Comparison of L^p to L^q ball

Does anyone have a solid source for comparisons of the volumes or L^p balls vs. L^q balls for p ≠ q? I am thinking of something along the lines of "In n-dimensional space, the ratio of the volume of an L^p ball to the volume of the smallest enclosing L^q ball is ...". —Quantling (talk | contribs) 16:04, 3 December 2021 (UTC)

Generalizations section in terms of the harmonic mean

I seek your feedback for whether this change should be allowed under WP:CALC or be rejected under WP:OR. Specifically, the Generalizations section uses the sum ${\tfrac {1}{p_{1}}}+\cdots +{\tfrac {1}{p_{n}}}$ in several places and could instead be rewritten more concisely in terms of the harmonic mean $p={\frac {n}{{\tfrac {1}{p_{1}}}+\cdots +{\tfrac {1}{p_{n}}}}}$ and the substitution $L=R^{p}$. If anyone has seen this version in print that would be the best way to accept this. Barring that, if anyone objects then that is evidence of the lack of consensus needed for WP:CALC. I am proposing:

=== Generalizations ===

The volume formula can be generalized even further. For positive real numbers p₁, …, p_n, define the (p₁, …, p_n) ball with radius R ≥ 0 to be

$B_{p_{1},\ldots ,p_{n}}(R)=\left\{x=(x_{1},\ldots ,x_{n})\in \mathbf {R} ^{n}:\vert x_{1}\vert ^{p_{1}}+\cdots +\vert x_{n}\vert ^{p_{n}}\leq R^{p}\right\}$

where p is the harmonic mean

$p={\frac {n}{{\tfrac {1}{p_{1}}}+\cdots +{\tfrac {1}{p_{n}}}}}\,.$

The volume of this ball has been known since the time of Dirichlet:

$V{\bigl (}B_{p_{1},\ldots ,p_{n}}(R){\bigr )}=2^{n}{\frac {\Gamma {\bigl (}{\tfrac {1}{p_{1}}}+1{\bigr )}\cdots \Gamma {\bigl (}{\tfrac {1}{p_{n}}}+1{\bigr )}}{\Gamma {\bigl (}{\tfrac {n}{p}}+1{\bigr )}}}R^{n}.$

In the case that p₁ = … = p_n = p, this reduces to the formula for the $L^{p}$ ball:

$V_{n}^{p}(R)={\frac {{\Bigl (}2\,\Gamma {\bigl (}{\tfrac {1}{p}}+1{\bigr )}R{\Bigr )}^{n}}{\Gamma {\bigl (}{\tfrac {n}{p}}+1{\bigr )}}}.$

Feedback, please. —Quantling (talk | contribs) 01:21, 10 December 2021 (UTC)

So, your proposed text is clearly correct, and the manipulation is simple enough that I think it's covered under WP:CALC. But I don't think that resolves the question. I mean, there are lots of correct ways of manipulating formulas, but we don't put them all into Wikipedia.

For comparison's sake: I feel like every formula about L^p spaces makes more sense if you think of it in terms of 1/p instead of p. Just take a formula and write, say, α = 1/p, and you pretty much always get something better. So in this case, if you define:

$B'_{\alpha _{1},\dots ,\alpha _{n}}(L)=\{x=(x_{1},\dots ,x_{n})\in \mathbf {R} ^{n}\mid |x_{1}|^{1/\alpha _{1}}+\dots +|x_{n}|^{1/\alpha _{n}}\leq L\},$

then

$V{\bigl (}B'_{\alpha _{1},\ldots ,\alpha _{n}}(L){\bigr )}=2^{n}{\frac {\Gamma {\bigl (}\alpha _{1}+1{\bigr )}\cdots \Gamma {\bigl (}\alpha _{n}+1{\bigr )}}{\Gamma {\bigl (}\alpha _{1}+\dots +\alpha _{n}+1{\bigr )}}}L^{\alpha _{1}+\dots +\alpha _{n}}.$

On the whole, I think this formula for V looks better than what's on the page right now; but I'd never put it there. The problem is that Wikipedia is supposed to conform to existing usage, and everybody writes their formulas using p, not α. So while I like the way the formula with the harmonic mean looks (and actually I like using the arithmetic mean of α₁, ..., α_n best of all), I'm not really convinced that it's what should go on the page. Is there any evidence that the harmonic mean shows up in the literature in ways similar to what you've proposed? Even if not in this precise formula? Ozob (talk) 04:21, 10 December 2021 (UTC)

Thank you @Ozob: for the feedback. I have no reference to cite that uses the harmonic mean explicitly in this kind of context (though I don't have any that do it the other way either!). I interpret your response as opposing this edit and I think that this edit needs more than just my support, so I will not be making it. (Unless I have somehow misunderstood your reply ... then please let me know. Alternatively, if you think it might be productive to make the edit under WP:BRD in order to solicit discussion from additional editors then I'd do that ... please let me know.) Thank you —Quantling (talk | contribs) 18:12, 13 December 2021 (UTC)

You're correct; I oppose this edit. But I would make it myself in a heartbeat if I thought I could justify it! Ozob (talk) 01:15, 14 December 2021 (UTC)

I think that we can mention this approach to the extent that it connects two topics that are notable. I'll give it a try to show you what I mean. —Quantling (talk | contribs) 02:26, 23 February 2022 (UTC)

References

whether to privilege recurrence relation or closed formula

Copying discussion here from user talk: jacobolus. It seems more relevant here and others can participate. –jacobolus (t) 00:24, 17 September 2022 (UTC)

I see you [jacobolus] reverted my change to the page _Volume of the n-sphere_, with a hint that the recurrence relation is somewhere below. The intention of the edit was to put the most revealing, most useful, most accessible representation at the start. Before, the closed form with the monstrosity that is the Gamma function was featured -- I'm not sure that's very useful.

What's your take?

(Disclosure: I have a degree in math.) Nschloe (talk) 18:36, 8 September 2022 (UTC)

I switched the order of the two sections after the closed formula. Does that cut it? –jacobolus (t) 19:33, 8 September 2022 (UTC)

Not really. The first recurrence relation should be on the top. It describes the nature of the ball volumes.

The closed formula is a derivation of it. It makes matters unnecessarily complicated, highlighted by the fact that one needs to explain what the Gamma function is, for example. Nschloe (talk) 19:57, 16 September 2022 (UTC)

Personally I think this is the "wrong" recurrence to care about, with the "wrong" definition of hyper-volume to properly relate quantities in different dimensions. See § perhaps worth being explicit about the definition of 'volume' above. But anyway, I don’t really have any strong preference between recurrences vs. closed formulas; neither one seems more or less fundamental (it really depends on your perspective and what other kinds of patterns you are comparing to), and either is trivially provable from the other. –jacobolus (t) 00:26, 17 September 2022 (UTC)

I would like the article to show the closed form first. The recurrence relations are an exploration of why that closed form is correct and they are key to understanding the closed form. But without the closed form first, I find the recurrence relations poorly motivated. —Quantling (talk | contribs) 01:16, 20 September 2022 (UTC)

This discussion died out without any consensus. I'm going to restart it with a bold edit. —Quantling (talk | contribs) 20:45, 26 July 2023 (UTC)

recurrence relations are an exploration of why that closed form is correct – This is entirely a matter of perspective. You could just as well say "the recurrence formulas are fundamental and can be proven e.g. by integration, and the closed formula is a bit of trivia, the result of algebraically manipulating the recurrence formula". There's not really an a priori reason to declare one more "motivated" than the other. –jacobolus (t) 21:04, 26 July 2023 (UTC)

My bold edit was made without consensus so you can undo it. As a general rule for me, closed form is the pinnacle of achievement that one aims for. Other forms have their merits (in proving the closed form, quicker computation, etc.) but nonetheless don't achieve the same status. I suppose that the gamma function could be interpreted as not closed form in that it doesn't make the short list of "addition, subtraction, multiplication, division, and exponentiation". In that sense it is a place-holder to cover up something complicated. However, given that most programming languages and many calculators make it as easy to compute Γ(x) as, say, to compute exp(x), I think of it as pretty darn close to that short list of closed-form operations. That's my 2¢. —Quantling (talk | contribs) 16:14, 27 July 2023 (UTC)

closed form is the pinnacle of achievement – this is a dogmatic take based on an idiosyncratic personal ideology. (A view which, to be clear, there's nothing inherently wrong with. Everyone has preferences.) –jacobolus (t) 15:49, 6 August 2023 (UTC)

I agree with Jacobulus, although I would have used a less rude formulation. Here, we have a numerical sequence (the volume of a unit n-ball) that is primarily defined by recurrence relations (I do not know any way to compute directly the needed integrals). Then, as for every sequence, one can use all techniques of combinatorics for studying the sequence, and, if possible finding a closed form. Here, the closed form is far from an achievement, since it does not say whether $V_{n}$ is an elementary function of n nor whether the sequence is a holonomic sequence. For both questions, the closed form is not useful, and the only way to answer them is to study the recurrence relation. Except for the aesthetic, which is a personal opinion, and the asymptotic study, the closed form is not useful for anything. D.Lazard (talk) 16:45, 6 August 2023 (UTC)

That's not quite fair. The closed form is useful for one thing: computing the volume of the ball for some particular n without needing to look up an n-specific formula or go through several steps. –jacobolus (t) 17:12, 6 August 2023 (UTC)

The closed form is useful for people who want to compute the volume on their calculator or in a single line of FORTRAN, etc. Folks who want more than a quick formula, but a deeper understanding, would then read more deeply into the article, wherein lies the mathematically beautiful 2-dimensions-at-a-time integration and resulting recursion.

Perhaps if we use the formulations with factorials instead of with the gamma function. Would that help?

Regardless, I don't want to be the lone holdout here; if you two agree then that is good enough for me. —Quantling (talk | contribs) 17:38, 6 August 2023 (UTC)

I don't personally care very strongly. You might try recruiting someone new to read two versions and see which one they think flows better / is easier to follow. –jacobolus (t) 17:44, 6 August 2023 (UTC)

Dimension that maximizes volume as a function of radius

@David Eppstein: you reverted my edit that specified which integer dimension will maximize the volume, with a comment about fractional dimension crankery ... so I am somewhat confused. If you would provide additional details here, I'd appreciate it. Thanks —Quantling (talk | contribs) 20:13, 25 July 2023 (UTC)

Maybe I misinterpreted. I was under the impression that you gave a formula that has a max at a fractional value, which we had problems a few years back related to a user pushing original research. If you are restricting to integers then at least it is relevant to the topic, but it is still not obvious to me that maximizing a sequence of numbers that have different units for different numbers in the sequence is in any way meaningful. —David Eppstein (talk) 20:39, 25 July 2023 (UTC)

Why do you want to maximize the ratio of the hypervolume of a diameter-$2R$ ball to the volume of a side-length-$1$ cube? It's an arbitrary (and pretty trivial) problem. –jacobolus (t) 02:37, 26 July 2023 (UTC)

My goal was to highlight that the result of n = 5 for R = 1, which I have been unable to remove from the article, is not particularly meaningful: there's the nonsense of comparing numbers with different units (or relative to unit hypercubes), and (perhaps equivalently) it is also the case that the value of n that maximizes is a function of R. I was able to achieve my aim with "Which value of n maximizes V_n(R) depends upon the value of R", so I am no longer as motivated to include the statement that was reverted. —Quantling (talk | contribs) 14:13, 26 July 2023 (UTC)

I would prefer to remove the n=5 R=1 material as well. —David Eppstein (talk) 16:36, 26 July 2023 (UTC)

Usefulness of formulas for computing radius from volume

User:Ozob re-inserts formulas for computing radius from a volume with the comment "The radius of a ball of unit volume is actually useful in convex geometry and concentration of measure, so I think this information is worth keeping". I see no mention of it in the convex geometry article. In the concentration of measure article there is use of a hyperball radius (represented by a variable R) but no apparent use of a formula for computing a radius from a volume. So, if you can give more detail for your edit comment, it would be appreciated. Thanks! —Quantling (talk | contribs) 16:37, 6 August 2023 (UTC)

See, for example, [11]. This normalization is dimension-independent and aids asymptotic comparisons with probability measures, specifically the standard Gaussian on $\mathbf {R} ^{n}$. Ozob (talk) 02:39, 8 August 2023 (UTC)

Thank you for that reference. Skimming through it, I see at the bottom of Page 5 that an asymptotic formula for the radius of a unit ball is given. I wasn't particularly impressed by the usefulness or beauty of the brief discussion there nor did I see that result used later (though I was just skimming). This isn't my specialty and I am unable to judge the significance of the referenced paper. So, although I am still hesitant, if you think these radius(volume) formulas are due the weight that we are giving them then I am prepared to yield to your call. Thank you for finding that reference for me. —Quantling (talk | contribs) 15:33, 8 August 2023 (UTC)

Please fix recurrence relation

For n=3 it gives 2 pi / 3 which should be 4 pi /3. Eijkhout (talk) 15:33, 20 October 2023 (UTC)

The recurrence given in the article expands to: $V_{3}=2\pi V_{1}/3={\tfrac {4}{3}}\pi ,$ since $V_{1}=2.$ –jacobolus (t) 17:32, 20 October 2023 (UTC)