Spherical volume....

[Chaotic42]

The area of a circle is ðr². The volume of a sphere is (4/3)ðr^3.

Is there an easy way to determine the formula for the hyper-volume of an x-dimensional
sphere?

 

[Ender]

Math makes me jolly with excitement.

 

[Heisenberg]

If you could integrate a volume element in n-dimension space I guess you could. It seems like it'd be pretty tough, but I've never really thought about it.

 

[Chaotic42]

Originally posted by: Heisenberg
If you could integrate a volume element in n-dimension space I guess you could. It seems like it'd be pretty tough, but I've never really thought about it.

Hmm....

Google isn't producing anything useful, it won't even let me shop for best prices on 9 dimensional hyper-spheres. I was in the market, too....

Edit: This site has some good info, but apparently an easy way of calculating the formulae is elusive. Sounds like a fun project.

 

[Heisenberg]

Originally posted by: Chaotic42

Originally posted by: Heisenberg
If you could integrate a volume element in n-dimension space I guess you could. It seems like it'd be pretty tough, but I've never really thought about it.

Hmm....

Google isn't producing anything useful, it won't even let me shop for best prices on 9 dimensional hyper-spheres. I was in the market, too....

Edit: This site has some good info, but apparently an easy way of calculating the formulae is elusive. Sounds like a fun project.

Yeah, they basically express the volume in n-dimensional space in terms of just the radial coordinate.

 

[Chaotic42]

Hmm, well, this math is on the edge of my abilities. I'm not entirely comfortable working with it. I couldn't be sure I wouldn't make a mistake.

Ah well, off to the books.

 

[Oscar1613]

yes... if you would've asked this sometime last semester i would've been able to give a complete answer, but it's all been lost in the "cool but not really practical so you can forget it" area....


but then again i could be thinking of something entirely different...

 

[KF]

Originally posted by: Chaotic42
The area of a circle is ðr². The volume of a sphere is (4/3)ðr^3.

Is there an easy way to determine the formula for the hyper-volume of an x-dimensional
sphere?

From that page you linked to, you can see that the formula for n dimensions is always 2 x pi x r^2 / n times the formula for n-2 dimensions.

So for n even the formula is, letting m = n/2 :


pi^m x r^n / 1 x 2 x 3 x... x m

for instance for n = 12:

pi^6 x r^12 / (2 x 3 x 4 x 5 x 6)

And for n odd, letting q = (n-1)/2 :

2 x (2 x pi)^q x r^n / 1 x 3 x 5 x ... x n

For instance for n = 11,

2 x (2 x pi)^5 x r^11 / (3 x 5 x 7 x 9 x 11)

These are elegant formulas, but it is a litttle surprising that the formulas for odd an even dimensions look different. There is probably a way to massage the formula so they look more alike.