And although he made many excellent discoveries, he is said to have asked his kinsmen and friends to place over the grave where he should be buried a cylinder enclosing a sphere, with an inscription giving the proportion by which the containing solid exceeds the contained.
The proportion is , as Archimedes discovered. More generally, let be the ratio of the volumes of -dimensional unit ball and of the circumscribed cylinder . This sequence begins with , , , , , . Note that beginning with dimensions, the ball occupies less than half of the cylinder. It is not hard to see that the sequence monotonically decreases to zero. Indeed, the ratio of -dimensional cross-sections by the hyperplane , for , is . Hence where . The integrand monotonically converges to a.e. as , and therefore does the same.
The Archimedean ratio is rational when is odd, and is a rational multiple of when is even. Indeed, integrating by parts, we get
hence . In terms of , we have . Since and are known (the former being trivial and the latter being one of definitions of ), the claim follows.
Combining the monotonicity with the recurrence relation, we find that . Hence , which yields the Wallis product formula for . Indeed, the recurrence shows that for even, is a particular rational multiple of .
The above is essentially the inductive proof of Wallis’s formula I remember from my calculus (/real analysis) class. Unfortunately, between integration by parts and the squeeze lemma we lose geometry. In geometric terms, Wallis’s formula says that the volumes of are are asymptotically equivalent: their ratio tends to . Is there a way to see that these solids have a large overlap? Or, to begin with, that has the smaller volume? I don’t see this even with vs . By the way, the latter is the bidisc from complex analysis.
Aside: Wallis’s formula for statistics students by Byron Schmuland presents a nice 2-page proof (which uses the relation between and the Gaussian).
Added: Another way to generalize the Archimedean volume ratio to higher dimensions is to divide the volume of by the volume of . The quotient is always rational, namely . The sequence begins with and obviously tends to zero. It does not appear to be related to .