Sphere in n-Dimensional Space
Here's the elegant result clearly explained.
Take a point in n-dimensional Euclidean space (x1,x2,x3,...,xn)
Then the surface of a sphere of radius r is the set of points with:
and the volume consists of all the points with:
x1^2+x2^2+x3^2+....+xn^2 less than or equal to r^2
Then the Theorem is simply this..
"In n-dimensional Euclidean space a sphere with surface area=volume has radius n"
The radius of the sphere is equal to the dimension of the space!!!! I think this is a neat result because it relates the dimension of the space to a certain class of spheres within it. It also has a "holographic" aspect because it equates surface area and volume.
If you're interested in the proof it goes like this..
The volume of the sphere is c(n)*(r^n) and its surface area is n*c(n)*r^(n-1)
So equating them gives r=n.
Note that the function c(n) cancels out so its value is not needed, but c(n)=(pi^(n/2))/gamma(1+n/2). Where gamma is the Euler Gamma Function.
Content written and posted by Ken Abbott email@example.com