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Circles, spheres, and regular forms in further dimensions

chiisai littleone

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Posted: Thu Mar 15, 2007 3:45 pm

The circumference and area of a 2D circle can be calculated.
The circumference, surface area and volume of a 3D sphere can be calculated.
Does anybody know how you would go about working out the formulas for corresponding dimensions and things in a 4D space or 5D space and so on...equivalent of the circle and sphere?
Out of pure curiosity. It's mind boggling just trying to imagine another space dimension in my piddly 3D mind.

Posted: Thu Mar 15, 2007 5:43 pm

Check it out: Wikipedia knows all.

It's actually not that bad to calculate what the next volume is going to be. You just integrate over an additional parameter:

Given a n-sphere with "surface area" B_n and "volume" V_n, the area for an (n+1)-sphere is

B_{n+1} = B_n*(int_{x=-1}^1 (1-t^2)^{(n-1)/2}dt)

And the volume is

V_{n+1} = V_n*(int_{x=-1}^1(1-t^2)^{n/2}dt)

Getting closed forms out of these isn't too bad if you pay close attention to the indices.

Layra-chan
Crew

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Mathematics

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