# Is there an intuitive explanation of $n$-dimensional sphere's surface and volume being maximal in dimensions 5 and 6? [duplicate]

I was wondering if there is an intuitive explanation why the surface of an $n$-dimensional sphere is maximal at $n=6$ and for the volume $n=5$. I know that for the surface you have: $$A_n=\frac{2\pi^\frac{(n+1)}{2}}{\Gamma(\frac{n+1}{2})}$$

And for the volume
$$V_n=\frac{A_{n-1}}{n}$$

Finally you get
$$A_n(R)=A_nR^n$$ $$V_n(R)=V_nR^n$$
Where $R$ is the radius of the ball.
Plugging in the dimensions it's easy to show that $V_n$ is max for $n=5$ and the surface for $n=6$
Is there an intuitive way of explaining this?

## marked as duplicate by J. M. is a poor mathematician, dantopa, carmichael561, The Dead Legend, Especially LimeMay 19 '17 at 7:53

Here is a naive intuitive explanation for the growth in volume of the unit $n$-sphere as $n$ increases.
Embed the $n$-sphere in an $n$-cube which has $2^n$ corners, $2n$ facets and volume 1. The space in the cube not occupied by the sphere is concentrated near the corners and the space in the cube occupied by the sphere is concentrated near the middle of the facets. For $n \leq 4$, the latter effect overrides the former. But for $n > 4$, the corners win.