## There was this mensa IQ question I saw back when I was 16. “Fill in the blank” it asked,

It’s one of the earliest problems that got me to jump down the math rabbit-hole, so I’m happy to share the solution here with you today, its intuition, and perplexing ramifications.

First: the pattern. It’s the surface area, and then volume of an n-dimensional sphere with radius r, starting from 0 dimensions (first 2 terms), 1 dimension (second 2 terms) and 2 dimensions (last 2 terms).

What about the blank? This would be the surface area of a 3-dimensional sphere with radius r. Turns out with a bit of multivariable calculus, one can show that in general (check out the video at the end for the derivation!), the surface area of a sphere in (n-1)-dimensions with radius r is:

Where Gamma is the Gamma function. So, for n=4 (remember we’re looking for the surface area of a 3-sphere with radius r), our missing blank must be,

Let’s not just stop there. What’s so-interesting about the general result for surface area is that we can integrate it in r, to obtain the volume of the (n-1)-dimensional sphere of radius r:

Let’s not just stop there either. That is, what if we look at large values of n? That is, what happens to the volume of these (n-1) dimensional spheres in high dimensions? Great question!

I plotted the volume of the unit sphere (r=1) in n-dimensions as a function of n below (of course, the trend would hold for any fixed r as well). This makes sense if we look at the asymptotic form of V for large n using Stirling’s Approximation for the Gamma Function,

So for good measure, I plotted both:

Where’d the volume go?? Great question. You gotta check out the video below to find out (skip to the last screen if you want to skip the math-proofs)…View the Video

Prerequisites/concurrent learning: multivariable calculus