# 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????

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.

http://en.wikipedia.org/wiki/Hyperreal_numbers