An Adventure in the Nth Dimension

On the mystery of a ball that fills a box, but vanishes in the vastness of higher dimensions

Brian Hayes

The Incredible Shrinking n-Ball

When I discovered the n-ball formula, I did not pause to investigate its provenance or derivation. I was impatient to plug in some numbers and see what would come out. So I wrote a hasty one-line program in Mathematica and began tabulating the volume of a unit ball in various dimensions. I had definite expectations about the outcome. I believed that the volume of the unit ball would increase steadily with n, though at a lower rate than the volume of the enclosing s=2 cube, thereby confirming Bellman’s curse of dimensionality. Here are the first few results returned by the program:

I noted immediately that the values for one, two and three dimensions agreed with the results I already knew. (This kind of confirmation is always reassuring when you run a program for the first time.) I also observed that the volume was slowly increasing with n, as I had expected.

But then I looked at the continuation of the table:

Beyond the fifth dimension, the volume of a unit n-ball decreases as n increases! I tried a few larger values of n, finding that V(20,1) is about 0.0258, and V(100,1) is in the neighborhood of 10^–40. Thus it looked very much like the n-ball dwindles away to nothing as n approaches infinity.