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.]

There are many "dimensionless" ratios: choosing $R$ as the linear measurement is arbitrary.
For instance, the ratio of the volume of the sphere to the volume of the circumscribed cube
has a maximum at $n=1$.  The ratio to the volume of the inscribed cube never attains a maximum.  There are intermediate geometrically-related "midscribed" cubes, where all faces of some dimension are tangent to the unit sphere.  Here is the graph for the ratio of
volumes when the codimension 2 faces of a cube are tangent. It attains the maximum for $n = 12$ (just barely more than for $n=11$).  There are many other reasonable dimensionless comparisons, for instance comparing to a simplex, etc. etc.  Since the Gamma function
grows super-exponentially, these simple geometric variations tend to shift the maximum --- there's nothing special about 5 or 7. 

       (source: Wayback Machine)
