But what about the n-ball? As I have already noted, my early education failed to equip me with the necessary formula, and so I turned to the Web. What a marvel it is! (And it gets better all the time.) In two or three clicks I had before me a Wikipedia page titled “Deriving the volume of an n-ball.” Near the top of that page was the formula I sought:
Later in this column I’ll say a few words about where this formula came from, both mathematically and historically, but for now I merely note that the only part of the formula that ventures beyond routine arithmetic is the gamma function, Γ, which is an elaboration on the idea of a factorial. For positive integers, Γ(n+1)=n!=1×2×3×...×n. But the gamma function, unlike the factorial, is also defined for numbers other than integers. For example, Γ(½) is equal to the square root of π.
I (Brian Hayes, the author of the article) 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.
The volume of a unit ball in n dimensions reveals an intriguing spectrum of variations. Up to dimension 5, the ball’s volume increases with each increment to n; then the volume starts diminishing again, and ultimately goes to zero as n goes to infinity. If dimension is considered a continuous variable, the peak volume comes at n=5.2569464 (green dot).
Illustration by Brian Hayes.
Incredibly, the volume of what you'd think becomes larger with extra dimensions actually becomes "vanishingly small"!!!! Whodathunkit????