TITLE: Volumes of n-balls: what is so special about n=5?
QUESTION [42 upvotes]: I am reposting this question from math.stackexchange where it has not yet generated an answer I had been looking for. 

The volume of an $n$-dimensional ball of radius $R$ is given by the classical formula
$$V_n(R)=\frac{\pi^{n/2}R^n}{\Gamma(n/2+1)}.$$ 
It is not difficult to see that the "dimensionless" ratio $V_n(R)/R^n$ attains its maximal value when $n=5$.
The "dimensionless" ratio $S_n(R)/R^n$ where $S_n(R)$ is the $n$-dimensional volume of an $n$-sphere attains its maximum when $n=7$.

Question. Is there a purely geometric explanation of why the maximal values in each case are attained at these particular values of the dimension?
[EDIT. Thanks to all for the answers and comments.]

REPLY [7 votes]: Brian Hayes has very nice article (Wayback Machine) 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 $n=5$.