TITLE: volume of a cube and a ball in n-dimensions?
QUESTION [12 upvotes]: Why does the volume of an $n$-dimensional cube approach $\infty$ as $n \to \infty$, but the volume of an $n$-dimensional sphere approach $0$ as $n \to \infty$? 
To be more precise,
$$\frac{V_n(\text{Sphere})}{V_n (\text{Cube})} \to 0, \,\text{as}\; n \to \infty$$
I understand how this is evident using the expression for the volume of an $n$-dimensional sphere and cube. 
Volume of an $n$-dim sphere (radius = $1$) $$V_n=\frac{\pi^{n/2}}{\Gamma(n/2+1)}$$
Volume of an $n$-dim cube (side length = $2$) $$V_n = 2^n$$
I am looking for a geometric explanation. In what way is a n-dim sphere different from a n-dim cube? Are there more general families which exhibit these properties?
Edit: For the sake of keeping the problem relatively simple, I am referring to a sphere of unit radius and a cube with side length equal to the diameter of the sphere.

REPLY [20 votes]: In very high dimensional space, the opposite corners of a cube are very far from one another. The unit cube is a huge object as compared to the unit sphere, where every point is 2 units or less from every other point.