I recall a related fact from simple statistical mechanics: the volume of high-dimensional sphere is concentrated almost exclusively near its surface. This fact allowed some nasty calculation of volume in state space to be approximated by the area of the hypersphere.

IIRC, the basic notion is that volume is like r^n and area like n * r^(n-1). Quite different when n = 2, 3, ... and not so much at n = 10^23. I never thought of them as spikey but I get the picture now.