Planetary and atomic systems in N dimensions

[black_squirrel]

Homework Statement

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?

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?

 

[robphy]

Check out:
https://www.physicsforums.com/showthread.php?p=809880#post809880
and
(if you have JSTOR access)
http://links.jstor.org/sici?sici=0080-4614(19831220)310%3A1512%3C337%3AD%3E2.0.CO%3B2-2

 

[Bill Foster]

1 dimention: a line $$x$$
2 dimensions: a square $$x^2$$
3 dimensions: a cube $$x^3$$
...
...
N dimentions: ? $$x^N$$

 

[black_squirrel]

Check out:
https://www.physicsforums.com/showthread.php?p=809880#post809880
and
(if you have JSTOR access)
http://links.jstor.org/sici?sici=0080-4614(19831220)310%3A1512%3C337%3AD%3E2.0.CO%3B2-2

wow those are great papers but I'm having a bit of trouble following the math/physics of the second paper and the first one just kind of states Ehrenfest's findings and doesn't explain how he came at those conclusions. I'd appreciate it if you could explain it in a bit.