This is a bizarre feature of geometery that I forgot about: the volume of a the unit n-dimensional ball tends to zero as n increases past 5. A unit n-ball (or hypersphere) is the set of points that can be reached by going 1 unit in any direction. I think this is really counter-intuitive.
Image from Wolfram on ball.
Wikipedia on the curse of dimensionality says:
Thus, in some sense, nearly all of the high-dimensional space is “far away” from the centre, or, to put it another way, the high-dimensional unit space can be said to consist almost entirely of the “corners” of the hypercube, with almost no “middle”. (This is an important intuition for understanding the chi-squared distribution.)