An Adventure in the Nth Dimension
On the mystery of a ball that fills a box, but vanishes in the vastness of higher dimensions
Slicing the Onion
The volume formulas I learned as a child were incantations to be memorized rather than understood. I would like to do better now. Although I cannot give a full derivation of the n-ball formula—for lack of both space and mathematical acumen—perhaps the following remarks will shed some light.
The key idea is that an n-ball has within it an infinity of (n–1)-balls. For example, a series of parallel slices through the body of an onion turns a 3-ball into a stack of 2-balls. Another set of cuts, perpendicular to the first series, reduces each disklike slice to a collection of 1-balls—linear ribbons of onion. If you go on to dice the ribbons, you have a heap of 0-balls. (With real onions and knives these operations only approximate the forms of true n-balls, but the methods work perfectly in the mathematical kitchen.)
This decomposition suggests a recursive algorithm for computing the volume of an n-ball: Slice it into many (n–1)-balls and sum up the volumes of the slices. How do you compute the volumes of the slices? Apply the same method, cutting the (n–1)-balls into (n–2)-balls. Eventually the recursion bottoms out at n=1 or n=0, where the answers are known. (The volume of a 1-ball is 2r; the 0-ball is assigned a volume of 1.) Letting the thickness of the slices go to zero turns the sum into an integral and leads to an exact result.
In practice, it’s convenient to use a slightly different recursion with a step size of 2. That is, the volume of an n-ball is computed from that of an (n–2)-ball. The specific rule is: Given the volume of an (n–2)-ball, multiply by 2πr2/n to get the volume of the corresponding n-ball. (Showing why the multiplicative factor takes this particular form is the hard part of the derivation, which I am going to gingerly avoid; it requires an exercise in multivariable calculus that lies beyond my abilities.)
The procedure is easy to express in the form of a computer program:
For even n, the sequence of operations carried out by this program amounts to:
For odd n, the result is instead the product of these terms:
For all integer values of n the program yields the same output as the formula based on the gamma function.