Volume of the Unit Hypersphere
Intro
"It is not easy to become an educated person." - Richard Hamming
This post is about n-dimensional space and how our naïve intuition often fails us when thinking about more than 3 dimensions. The content of this post has been almost entirely lifted from chapter 9 of Richard Hamming's excellent book The Art of Doing Science and Engineering. The main reason I wanted to write this up is that I got stuck on a particular step and wanted to make it crystal clear (a good way to practice my technical writing). Moreover I did not know this and I thought it was awesome and wanted to share it! A caveat for those with some more mathematical training, I will be doing the following in classical Euclidean space using the Pythagorean distance.
Recap
For those unfamiliar with n-dimensional space a quick recap: We geometrically represent one dimensional space as a line. You can only move forward or backwards on the line. Algebraically we can represent a point in one dimensional space with just one coordinate \((x_1)\). We think of this of where you are on the line relative to the point 0. Two dimensions we can represent with a plane, you can move up, down, and side to side. We represent points in two dimensional space with two coordinates \((x_1, x_2)\). Three [spatial] dimensions is where we live. It can be represented using three coordinate axes, an \(x, y,\) and \(z\). A point in three dimensional space looks like this \((x_1, x_2, x_3)\). More than three dimensions is basically impossible for us mortals to visualise. We can only really represent points and objects in n-dimensions algebraically. A point in n-dimensions looks like this \((x_1, x_2, ..., x_{n-1}, x_n)\)
What is a [Hyper]sphere?
We commonly think of a sphere as an object that only lives in three dimensions. If we extend ourselves a little and think of a sphere as a more general object then we can make it applicable over any number of dimensions. If we take the definition of a sphere to be "the collection of all points equidistant to a single particular point", then this defines a line segment in one dimension, a circle in two dimensions, and a sphere in three dimensions. Mathematicians use this definition for hyperspheres in \(n> 3\) dimensions. In this post we want to consider only the unit hyperspheres, that is, those hyperspheres where the radius (the equidistant length) is 1.
Volume of a unit Hypersphere
Hopefully you'll believe me that the volume (the amount of space enclosed by a shape) of a hypercube with side length \(x\) is \(x^n\). If we were to think about it, the volume of a hypersphere must have the form \(V = C_nr^n\). It must be linearly related to the volume of the hypercube with side length \(r\). One way to calculate volume is to think about volume as a sum of infinitesimally small shells (... remember volumes by revolution and volumes by shells?). Each shell would have an area and be multiplied by an small increment relative to the radius \(dr\). When we talk about the area of a shell what we are talking about turns out to be a surface area. Surface area is very easy to calculate for a sphere. For spheres it turns out to just be the derivative of volume \(\frac{dV_n(r)}{dr} = nC_nr^{n-1}\). The next few sections will be a little technical while we try and calculate \(C_n\).
Gamma Function
I am going to pull a magic function out of the air. Partly because I think you should read the chapter in full, and partly because it was an expression involving this function that I got stuck on. We will use this function to calculate \(C_n\) Here is the gamma function: \begin{align*} \Gamma (m) = \int_{-\infty}^\infty x ^ {m-1} e^{-x} dx\\ \end{align*} It is a generalization of the factorial (!) function. It has the property \(\Gamma(m) = (m-1)\Gamma(m-1)\) with \(\Gamma(1) = 1\). Surprisingly we also get \(\Gamma(\frac{1}{2}) = \sqrt\pi\). Now let's raise it to the \(n\)th power \begin{align*} \Gamma^n(m) &= (\int_{-\infty}^\infty x ^ {m-1} e^{-x} dx)^n\\ &= \Pi_{i=1}^n\int_{-\infty}^\infty {x_i} ^ {m-1} e^{-x_i} dx\\ \end{align*} and now let's set \(m = \frac{1}{2}\) \begin{align*} \Gamma^n(\frac{1}{2}) &= \Pi_{i=1}^n\int_0^\infty {x_i} ^ {-\frac{1}{2}} e^{-x_i} dx \end{align*} let \(x_i = {t_i}^2\) then \(d{x_i} = 2{t_i}d{t_i}\) which gives us \begin{align*} \Gamma^n(\frac{1}{2}) &= \Pi_{i=1}^n\int_0^\infty {t_i} ^ {-1} e^{{-t_i}^2} 2{t_i}d{t_i}\\ &= \Pi_{i=1}^n 2 \int_0^\infty e^{{-t_i}^2} d{t_i} \\ &= \Pi_{i=1}^n \int_{-\infty}^\infty e^{{-t_i}^2} d{t_i} \\ & = \int_{-\infty}^\infty e^{-({t_1}^2 + {t_2}^2 + ... + {t_n}^2)} d{t_1}d{t_2} ... d{t_n} \end{align*} To evaluate this integral we will use a common technique. We will change coordinates to hyperspherical coordinates. \(r^2 = {t_1}^2 + {t_2}^2 + ... + {t_n}^2\) and \(d{t_1}d{t_2} ... d{t_n} = r^{n-1}drd\Omega\) where \(d\Omega = d\theta_1 ... d\theta_{n-1}\) which gives us \begin{align*} \Gamma^n(\frac{1}{2}) &= \int_{\Omega}\int_0^\infty e^{-{r}^2} r^{n-1} drd\Omega\\ &= \int_0^\infty e^{-{r}^2} A r^{n-1} dr \end{align*} Now this \(Ar^{n-1}\) looks la lot like we had before, our \(nC_nr^{n-1}\) to be exact. We can easily change our arbitrary variable \(A\) to be \(nC_n\) to get \begin{align*} \Gamma^n(\frac{1}{2}) &= nC_n\int_0^\infty e^{-{r}^2} r^{n-1} dr\\ \end{align*} ... Home stretch now. Let \(t = r^2\) then \(dt = 2rdr\) and we get \begin{align*} \Gamma^n(\frac{1}{2}) &= nC_n\int_0^\infty e^{-{r}^2} r^{n-1} dr\\ & = \frac{nC_n}{2} \int_o^\infty e ^ {-t} t^{\frac{n}{2} - 1} dt\\ & = \frac{nC_n}{2} \Gamma(\frac{1}{2})\\ & = C_n \Gamma(\frac{n}{2} + 1) \end{align*} from which we get \begin{align*} C_n = \frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2} + 1)} = \frac{2 \pi}{n} C_{n-2} \end{align*}
... Finally: the volume of a unit hypersphere
\begin{align*} V &= C_nR^n\\ C_n &= \frac{2 \pi}{n} C_{n-2} \end{align*} Given this recursive relationship we can figure out the volume coefficient for any hypersphere. Starting with \(C_1 = 2\), and \(C_2 = \pi\) (why?) we get \(C_3 = \frac{4\pi}{3}\) It can be shown that for \(n = 2k\) we have \(C_n = \frac{\pi}{k!}\) which goes to zero as k increases. It can also be shown that the \(C_n\) is a smooth function so that it is fine for us to only consider the evens. What this says is that the volume of a unit hypersphere decreases to zero as the number of dimensions increase. HOW WEIRD IS THAT!?!