N-dimensional Sphere Volumes

This is solid & trig but in an arbitrarily large dimensional Euclidean Space. It's from American Scientist November-December 2011, not my pea brain, so direct opinions that it isn't true to that magazine [NOT the Scientific American one or me]. The conclusion is so surprising and counter intuitive I just had to share it. Honestly I'd never considered any sphere beyond the 3D one we all know since I can't imagine a 4D one. But mathematically they can be dealt with. For simplicity we choose the radius equal to one so r cubed is still one and r to higher powers doesn't increase with dimension. Here's the formula given in American Scientist:

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????

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66-70, M
1 Response Oct 20, 2011

Why does your intuition tell you that the volume of an n-ball is proportional to r rather than to r^d? (Where d is the complex dimension, including the degenerate cases when the real part is a positive integer and the complex part is zero, i.e., 0, 1, 2, 3, 4, . . . , n dimensions.)<br />
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My intuition tells me that an n-ball fits more & more efficiently into higher-dimensional spaces. And of course it converges to 0 -- that's a perfect "fit".

The introduction of complex numbers/dimensions/volumes is inappropriate - this article addresses real N-spheres, not complex ones. The author, presumably more educated in math than either of us, was surprised at the result, why wouldn't I be?
You also are slightly off as to fitting the hyperspheres. They are not placed in successively higher dimension SPACES, they are placed against the faces of successively higher dimension hypercubes - the N-spheres TOUCH the hypercube faces, a perfect fit in EVERY dimension, not better or worse. The circle inscribed in a square is a perfect fit, having non-zero area (2-volume) within and without as is the easily imaginable ball in a cube.

I think a key insight here is to consider how the cardinality of the number of points in an n-ball changes as the dimension increases.