My friend Colin has been trying to give me a better intuition for higher dimensions. His first fun fact was that the volume of an n-dimensional hypersphere peaks somewhere around n=6 or so, I forget, and then approaches 0 as n approaches infinity. Damn! Meanwhile, of course the volume of a hypercube just diverges to infinity, as you'd expect. So if you inscribe a hypersphere in a hypercube, as dimension increases, more and more of the volume is in "the corners" — ultimately, almost all of it.
That's not what I'm trying to show here, it's just cool. This is somewhat different.
Colin also pointed out that, in a high-dimensional multivariate random normal distribution (with identity covariance matrix), all the mass ends up coming to be found in a sort of donut at some distance from the middle. There's very little mass in the middle. Of course the origin is still the mean/median/mode. The problem is that the middle is just so dang small, and there's SO MUCH SPACE as you go a little further out. So if you're jus