I Higher Dimensional Spheres viz. Cubes

[Hornbein]

This is about Euclidian geometry in n dimensions. On this subject it is often noted that the volume of the unit sphere goes to zero as n increases. However this has more to do with the unit cube getting larger than the sphere growing smaller. As n increases the diagonal of the unit cube grows without bound while the diameter of the unit sphere remains constant, so the result is no surprise.

How about comparing what happens with spheres and cubes both with volume one? Then the sphere can also grow. Things cancel out nicely so using Stirling's approximation we get that with large n

(diagonal of cube)/(diameter of sphere) = (pi*e/2)^1/2 = approx. 2.066.

That is, the "diameter" of the cube is a bit more than twice that of the sphere of the same volume.
Taking it a step further, we can show that for large n a unit cube of dimension n/4 (rounded down) or less can be entirely enclosed by an n dimensional unit volume sphere.

 

[Hornbein]

Suppose we have a unit volume n-ball and n-cube that share the same center. How large is their intersection?

The volume of their intersection goes to zero at a rate of roughly 1/sqrt(n). Both figures grow at the same rate but in different directions.