Masterpiece of Craftsmanship: Calculating the Volume of an $n$-dimensional Sphere

Standard Approach

Simply put, the volume of an n-dimensional sphere is the following n-fold integral: V_n(r) = \int_{x_1^2+x_2^2+\dots+x_n^2 \leq r^2} dx_1 dx_2 \dots dx_n Using a more geometric approach, we partition the sphere using a set of parallel planes ((n-1)-dimensional hyperplanes), decomposing the n-dimensional sphere into a series of approximate small cylinders. Thus, we can obtain a recurrence formula: V_n(r) = \int_{-r}^r V_{n-1} \left(\sqrt{r^2-t^2}\right) dt Let t = r \sin \theta_1, then we have: V_n(r) = r \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} V_{n-1} (r \cos \theta_1) \cos \theta_1 d\theta_1 Iterating once, we get: V_n(r) = r^2 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} V_{n-2} (r \cos \theta_1 \cos \theta_2) \cos \theta_1 \cos^2 \theta_2 d\theta_1 d\theta_2 Iterating n-1 times: \begin{aligned} V_n(r) = & r^{n-1} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \dots \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} V_1(r \cos \theta_1 \cos \theta_2 \dots \cos \theta_{n-1}) \times \\ & \cos \theta_1 \cos^2 \theta_2 \dots \cos^{n-1} \theta_{n-1} d\theta_1 d\theta_2 \dots d\theta_{n-1} \end{aligned} where V_1(r) = 2r, which is a line segment of length twice the radius. Thus: V_n(r) = 2r^n \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \dots \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos^2 \theta_1 \cos^3 \theta_2 \dots \cos^n \theta_{n-1} d\theta_1 d\theta_2 \dots d\theta_{n-1} Completing this integral finally yields the formula for the volume of an n-dimensional sphere. This integral can naturally be evaluated (it is simply the product of n-1 one-dimensional integrals). However, these steps are quite arduous, and much work remains to connect it to the Gamma function. Overall, this is a standard method that is neither easy to remember nor particularly elegant.

An Ingenious Approach

There is an ingenious trick using Gaussian integrals that allows us to directly connect the volume of a sphere to the Gamma function. The entire process is a masterpiece of craftsmanship and gives the impression of being truly unique and unparalleled. It is said that this technique is well-known among physics students; I saw it on Baidu Wenku, and the original source is Example 5.2, “Statistical Calculation of the Entropy of an Ideal Gas,” in Thermodynamics and Statistical Mechanics by Greiner (Germany).

This ingenious approach begins with calculating the Gaussian integral using two different methods: G(n) = \int_{-\infty}^{+\infty} \dots \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \exp(-x_1^2 - x_2^2 - \dots - x_n^2) dx_1 dx_2 \dots dx_n \tag{1} On one hand, treating (1) as an n-fold iterated integral, since we have already calculated (refer to here): \int_{-\infty}^{+\infty} \exp(-t^2) dt = \sqrt{\pi} and (1) is simply the product of n such integrals, therefore: G(n) = \pi^{n/2} \tag{2} On the other hand, treating (1) as an n-dimensional multiple integral, since the integrand depends only on the radial length r = \sqrt{x_1^2 + x_2^2 + \dots + x_n^2}, it is natural to think of spherical coordinates. In n-dimensional space, these are called “hyperspherical coordinates.” There is no need to write out the full hyperspherical coordinates; one only needs to notice that the integral over the sphere can be transformed into an integration over the “spherical shells” followed by an integration over the radius: G(n) = \int_{0}^{+\infty} dr \int_{S_n(r)} \exp(-r^2) dS_n \tag{3} Here S_n(r) denotes the surface of an n-dimensional sphere of radius r (as well as its surface area; where no confusion arises, no distinction is made here). However, noting that the integrand depends only on r, integrating over the sphere’s surface is equivalent to multiplying the original function by the sphere’s surface area. Thus, the result for G(n) is: G(n) = \int_{0}^{+\infty} dr \exp(-r^2) S_n(r) \tag{4} Although we do not know the formulas for the volume and surface area of an n-dimensional sphere, we can be certain that the volume must be proportional to r^n, i.e.: V_n(r) = V_n(1) r^n The surface area of the sphere is the first derivative of the sphere’s volume (considering the shell decomposition), so: S_n(r) = n V_n(1) r^{n-1} Substituting into (4), we get: \begin{aligned} G(n) &= n V_n(1) \int_{0}^{+\infty} r^{n-1} \exp(-r^2) dr \\ &= \frac{1}{2} n V_n(1) \int_{0}^{+\infty} (r^2)^{n/2-1} \exp(-r^2) d(r^2) \\ &= \frac{1}{2} n V_n(1) \int_{0}^{+\infty} z^{n/2-1} \exp(-z) dz \quad (z=r^2) \\ &= \frac{1}{2} n V_n(1) \Gamma\left(\frac{n}{2}\right) \end{aligned} \tag{5} Combining this with (2), we have: \pi^{n/2} = G(n) = \frac{1}{2} n V_n(1) \Gamma\left(\frac{n}{2}\right) Thus: V_n(1) = \frac{\pi^{n/2}}{\frac{1}{2} n \Gamma\left(\frac{n}{2}\right)} = \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2} + 1\right)} Finally: V_n(r) = \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2} + 1\right)} r^n And thus we have obtained the formula for the volume of an n-dimensional sphere!! Differentiating with respect to r gives the formula for the surface area of an n-dimensional sphere: S_n(r) = \frac{2\pi^{n/2}}{\Gamma\left(\frac{n}{2}\right)} r^{n-1} Combining the two methods, we obtain: \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2} + 1\right)} = 2 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \dots \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos^2 \theta_1 \cos^3 \theta_2 \dots \cos^n \theta_{n-1} d\theta_1 d\theta_2 \dots d\theta_{n-1}

Brief Commentary

This technique is quite beautiful and concise. Gaussian integrals and spherical coordinate transformations are very familiar to physics students. With just a simple clever twist, the result is obtained. This is clearly an ingenious idea that only a physics student could have conceived!

Even more remarkably, we find this approach so wonderful that we might want to apply it to more problems, but a little investigation leads to the conclusion: nothing more can be done! That is to say, the entire process seems to be custom-made solely for calculating the volume of an n-dimensional sphere! Truly “one of a kind”! Marvelous!

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