This is an incomplete blog post. Maybe you can help finish it.

One of the formulas I’ve looked up the most is the volume of a ball in *n* dimensions. I needed it often enough to be aware of it, but not often enough to remember it. Here’s the formula:

The factor of *r*^{n} is no surprise: of course the volume as a function of radius has to be proportional to *r*^{n}. So we can make the formula a little simpler by just remembering the formula for the volume of a *unit* ball.

Next, we can make the formula a simpler still by using factorials instead of the gamma function. If *n* is a non-negative integer, *n*! = Γ(*n*+1). We can use that to define factorial for non-integers. Then the volume of a unit ball is

That’s easier to remember.

It’s also curious. The *n*th term in the series for *e*^{x} is *x*^{n}/*n*!, so the volumes of unit balls look like series for *e*^{π} except compressed, with each index *n* cut in half. The volumes are not the coefficients in the series for *e*^{x}, but could they be the coefficients in the series for another familiar function? To find out, let’s stick back in the factor of *r*^{n} and sum.

This is the sum of the volumes of balls of radius *r* in all dimensions. That doesn’t make sense by itself, but you could also think of this as the generating function for the volumes of *unit* balls. So can we find a closed-form expression for the generating function? Yes:

If you work with probability, you probably find Φ more familiar than the error function (see notes relating these) and find exp(*x*^{2}/2) more familiar than exp(*x*^{2}). So you could rewrite the generating function as *f*(√(2π)r) where

That looks familiar, but I don’t know what to do with it.

I warned you this would an incomplete post. I feel like there’s an interesting connection to be made, but I’m not quite there. Any suggestions?

Minor observation:

exp(pi) – 1 = volume sum of all unit balls in even dimensions

The formula I can always remember is $int_{-infty}^infty e^{-pi x^2} dx = 1$, from which by Jacobian determinant and symmetry $int_mathbb{S^n} int_0^infty r^n e^{-pi r^2} dr dOmega = 1 $. I don’t know if that helps you much, but it does the job.

Is Coxeter’s Introduction to Geometry the best resource for learning more about n-dimensional solids like this?

$$f(x) = phi(x)/(frac{dphi(x)}{dx}/(sqrt{2}x))$$

the value of something as a fraction of the ratio of its derivative to a linear function…

See Hamming’s 9th lecture, http://www.youtube.com/playlist?list=PL2FF649D0C4407B30, on n-Dimensional space.

Michael Hartl posted some comments on the formula for the volume of an n-sphere in a discussion about his Tau Manifesto [1]. He also mentions this at the end of the the talk [2,3].

If you know the values for the 1-sphere (length of interval [-1, 1]) and the 2-sphere (area of unit circle) then the following recursion is easy to remember:

`V[1] = 1, V[2] = pi, V[n] = V[n-2] * (2*pi)/n`

There is also a very nice recursion that links the volume

`V[n]`

and the surface area`S[n]`

[4].[1] http://forums.xkcd.com/viewtopic.php?t=61958#p2225028

[2] http://tauday.com/tau-manifesto

[3] http://youtu.be/H69YH5TnNXI

[4] http://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_area

I think the reason why it looks familiar is that it’s almost the derivative of e^(-x^2/2) Phi[x]. In fact, I believe that f(x) = A d/dx [ e^(-x^2/2) Phi[x] ] + B for some constants A and B.

Hopcroft’s up and coming book http://www.cs.cmu.edu/~venkatg/teaching/CStheory-infoage/hopcroft-kannan-feb2012.pdf has a fantastic chapter on n-dimensional spheres/cubes

I’ve seen this before… I had to reach back to remember this one. Eventually I found it:

American Mathematical Monthly, Vol. 107, No. 3, Mar., 2000, pp. 256-258

“Generating the measures of n-Balls”

He calls the generating function b(r) and makes a nice point about b'(r) being the generator for surface areas of n-spheres. Unfortunately this doesn’t really help you complete the post – the article stops at more or less the same point as the incomplete post does, I think.

One other comment besides the reference… the form of the generating function is suggestive of some connection between n-spheres and the ordinary 1-dimensional normal distribution. I am aware of another connection of this sort: If one seeks a joint distribution of n i.i.d. real random variables that is also isotropic (spherical symmetry) then that distribution must be the multivariate normal formed by the products of identical 1-d normal distributions.

This is suggestive to me of successive constructions of n-spheres and their volumes (or surface areas), but I’m not sure where to go with it to connect close the loop. Hope this is helpful and not a red-herring.

Thank you for this thought-provoking post, I enjoy your blog & take a look now & again.

Can I ask how you arrived at your closed-form for the generating function? I think there may have been some typo and in terms of x it should be 2*exp(x^2/2)*Phi(x). Or maybe better sqrt(2/pi)*Phi(x)/phi(x), cf the Mills ratio.

Anyway thanks again for the post which was even better for being open-ended.

In the same period you wrote this post, I inserted a similar one in my blog: http://zibalsc.blogspot.fr/

We had quite the same conclusion, but we found closed-form expression for the generating function with a little difference: http://zibalsc.blogspot.fr/2013/02/115-somma-di-ipersfere.html

The blog is in Italian language, sorry, but you may easy translate it with google translator or simply to look at the formula. Let me know your opinion.