Sphere - Generalization To Other Dimensions

Generalization To Other Dimensions

Spheres can be generalized to spaces of any dimension. For any natural number n, an "n-sphere," often written as Sn, is the set of points in (n + 1)-dimensional Euclidean space which are at a fixed distance r from a central point of that space, where r is, as before, a positive real number. In particular:

• a 0-sphere is a pair of endpoints of an interval (−r, r) of the real line
• a 1-sphere is a circle of radius r
• a 2-sphere is an ordinary sphere
• a 3-sphere is a sphere in 4-dimensional Euclidean space.

Spheres for n > 2 are sometimes called hyperspheres.

The n-sphere of unit radius centered at the origin is denoted Sn and is often referred to as "the" n-sphere. Note that the ordinary sphere is a 2-sphere, because it is a 2-dimensional surface (which is embedded in 3-dimensional space).

The surface area of the (n − 1)-sphere of radius 1 is

where Γ(z) is Euler's Gamma function.

Another expression for the surface area is

and the volume is the surface area times or