Thanks for this nice link. It's a robust phenomenon that kicks in pretty well even in modestly large dimensions. One name for this is concentration of measure http://en.wikipedia.org/wiki/Concentration_of_measure
See figure 6 in the paper linked in the above comment for a nice illustration of how this works.
If you are interested in these things, this survey is amazing: http://www.math.ucdavis.edu/~deloera/MISC/BIBLIOTECA/trunk/B...