Brian Hayes has [very nice article (Wayback Machine)][1] about the volume of the n-ball in the current issue of *American Scientist*  (Nov 2011). In particular, he discusses the surprising fact that the maximum volume occurs at $n=5$.


  [1]: https://web.archive.org/web/20111118101953/http://www.americanscientist.org:80/issues/pub/an-adventure-in-the-nth-dimension