here are the mathmatical formulas used to prove different dimentions of such that our physical eyes cannot see. if you have learned anything in math up to trig, you should atleast understand up to the third dimension. maybe this is a little insentive for some of you to further your math education, remember, it is after all the universal language.
Formulas for N-Dimensional Spheres
The sequence is
r^n * pi^(n/2) / Gamma(n/2)
Here pi is the familiar ratio of circumference to diameter of a
circle. Gamma(x) is a function with the property that if k is a
positive integer, then
Gamma(k) = (k-1)!
Gamma(x) is the smooth continuous version of the discrete factorial
function.
When n is even, say n = 2*k, this becomes
r^n * pi^(n/2) / (n/2-1)!, n even.
You can see that this works for n = 2, when you get
r^2 * pi / 0! = pi * r^2
When n is odd, say n = 2*k - 1, you need the additional fact that
Gamma(k-1/2) = sqrt(pi)*(2*k)!/(k!*4^k)
In this case, the expression you get is
r^n * pi^([n-1]/2) * ([n+1]/2)! * 2^(n+1)/(n+1)!, n odd.
You can see that this works for n = 3, when you get
r^3 * pi * 2!*2^4 / 4!
= (4/3) * Pi * r^3
For n = 4, the expression is
r^4 * pi^2 / 1!
= Pi^2 * r^4
For n = 5, the expression is
r^5 * pi^2 * 6!/(3!*4^3)
= (15/8)* Pi^2 * r^5
Observe that the exponent of r increases by 1 each time, but the
exponent of pi increases by 1 every second time, and the coefficients
obey the peculiar sequence given above. A larger part of the sequence
is:
n F(n)
0 0
1 2*r
2 Pi*r^2
3 (4/3)*Pi*r^3
4 Pi^2*r^4
5 (15/8)*Pi^2*r^5
6 (1/2)*Pi^3*r^6
7 (16/105)*Pi^3*r^7
: :
For more, see "Formula for the Surface Area of a Sphere in Euclidean
N-Space" from the Sci.Math FAQ (editor Alex Lopez-Ortiz) at:
http://db.uwaterloo.ca/~alopez-o/math-faq/node75.html
- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
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