Volume of Balls

I like the following question:

“Which dimension has the ‘biggest’ unit ball?”

To define a few words, a “ball” of radius r about a point x in a metric space with metric d (in particular, in R^n with standard metric) is defined by B_X(x,r) = \{ y \in X: d(x,y) \leq r \}.  So the unit ball in Euclidean space is the collection of points of distance less or equal to one from, say, the origin.  If you think I am being overly careful with this definition, you’re right!

1-, 2-, and 3-dimensional balls.

By “biggest”, I will mean the n-dimensional volume.  So if we define V(n) to be the volume of the unit ball in R^n, then V(1) = 2 (the 1-dimensional ball is a line), V(2) = \pi r^2 \approx 3.14, and V(3) = \frac{4}{3} \pi r^3 \approx 4.19.  So the volumes are getting bigger, and maybe we have a good question on our hands.  But now what’s the volume of a four dimensional sphere?

It turns out that one can derive the formula

V(n) = \frac{\pi^{n/2}}{\Gamma(n/2+1)},

where \Gamma is the gamma function, a generalization of the factorial (!) function!  In particular, \Gamma(n+1) = n! for integers n.  Then, for even dimensions,

V(2n) = \frac{\pi^{n}}{n!}.

Now so long as we are only worried about integer dimensions, we can use the half-integer identity for the gamma function:
\Gamma(n+1/2) = \frac{(2n)!}{4^nn!}\sqrt{\pi},
to get a similar formula for odd dimensions:
V(2n-1) = \frac{4^n\pi^{n-1}n!}{(2n)!}.

Then a quick calculation gives:

V(1) = 2.00
V(2) = 3.14
V(3) = 4.19
V(4) = 4.93
V(5) = 5.26
V(6) = 5.17
V(7) = 4.72
V(8) = 4.06
V(9) = 3.30.

We note that the denominator for both odd and even dimensions grows much faster than the numerator to conclude (in academia we could only nod our head suggestively so far at this conclusion, but we’re in industry now!) that the 5-dimensional ball has the most volume.

As an aside, and possibly a subject for a later post, if we allow for fractional dimensions (everything in V(n) is defined for n a real number, not just an integer), then the maximum value of V is at about 5.2569, where the unit ball has volume around 5.2778.

The function V(n), plotted for real numbers n, with max around (5.26, 5.28).


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