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Planetary and atomic systems in N dimensions

1. The problem statement, all variables and given/known data

Suppose that a human head is a sphere with the bone thickness 10% of the radius of the
head. Find the fraction of the volume of the head occupied by the brain. Make a similar estimate (forget about factors of pi) in the N-dimensional space (d = N) in the limit N -> infinity.

For the existence of life of our type we must have atomic and planetary systems. Do stable atomic and planetary systems exist for d > 3? We also need to have free particles like electrons moving far away from atoms (for conductivity, and to have ions). Can we have them for d < 3?

3. The attempt at a solution

finding fraction of volume is simple but I don't know how you'd go about making an estimate in the N-dimensional space. would it just be r^N instead of r^3 in the volume?

For the second part, i'm not even sure what the question is trying to get at. I was thinking maybe Gauss law could be used to say that the field of charge is inversely proportional to the area of N-1 dimensional sphere surrounding it. Still that doesn't take me anywhere...any suggestions?
1 dimention: a line [tex]x[/tex]
2 dimensions: a square [tex]x^2[/tex]
3 dimensions: a cube [tex]x^3[/tex]
N dimentions: ? [tex]x^N[/tex]
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