Very nice video. For those less able to follow the calculus, another way to get at least a little intuition about this result is to think about how far the far

Very nice video. For those less able to follow the calculus, another way to get at least a little intuition about this result is to think about how far the farthest points in a unit cube are from the origin - in 20 dimensions, the point (1,1,...1,1) is sqrt(20) ~= 4.47 units from the origin, so the 19-sphere only "reaches" 1/sqrt(20) ~= 0.22 of the way to the corners of the 20-cube.