What's bigger, the unit circle or the unit sphere?

This is a trick question, because and are incompatible units, so the fact that the area of one is less than the volume of the other (i.e. < ) doesn't tell you much. So let's ask a different question: if we inscribe an *n*-sphere inside an *n*-cube, is a unit circle bigger or smaller, relative to its bounding square, than the unit sphere relative to *its* bounding cube? And in general, what will be the ratio of their volumes as a function of ?

Suppose we fix the radius of the *n*-sphere to be 1. The edge length of the *n*-cube, then, is 2, and its volume is 2^{n}. So, that means that the ratio of the volumes is (1/2)^{n}, times some prefactors. Right?

Right. But the form of the prefactor here is quite fascinating. Here's what the volume of the *n*-sphere is:

See that Gamma function in the denominator? It grows like , meaning that our original estimate of the hypersphere/hypercube volume ratio as (1/2)^{n} is quite a few orders of magnitude off for even moderate *n*. As *n* grows, the volume of a unit *n*-sphere goes to zero *super*-exponentially.

This has an important implication when you want to cluster high-dimensional data. Intuitively, any clustering algorithm (e.g. *k*-means) involves drawing a boundary around a set of points that lie within a certain radius of their mean. But what we just found is that if we draw a sphere of even a relatively large radius around points in *n*-dimensions, for *n* larger than about 30, the volume of that region enclosed by the spherical boundary is approximately zero, meaning that it's highly unlikely to have any points! In that sense, *everything* is far apart in high dimensions. This is known as *the curse of dimensionality*.

Volumes of *n*-spheres are useful in other contexts (for instance, in statistical mechanics). In fact, back in my physics days I learned a very cool and easy derivation of the *n*-sphere volume formula above. I like it so much that I made not one but *two* videos about it. Below you can see me doing the derivation in 3 minutes flat. There is also the longer version where I do the same steps, but a little more slowly and methodically. Enjoy!

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