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

$\begin{cases} \displaystyle \frac{(2\pi)^{n/2}\,r^{n-1}}{2 \cdot 4 \cdots (n-2)}, & \text{if } n \text{ is even}; \\ \\ \displaystyle \frac{2(2\pi)^{(n-1)/2}\,r^{n-1}}{1 \cdot 3 \cdots (n-2)}, & \text{if } n \text{ is odd}. \end{cases}$

and the volume is the surface area times or

$\begin{cases} \displaystyle \frac{(2\pi)^{n/2}\,r^n}{2 \cdot 4 \cdots n}, & \text{if } n \text{ is even}; \\ \\ \displaystyle \frac{2(2\pi)^{(n-1)/2}\,r^n}{1 \cdot 3 \cdots n}, & \text{if } n \text{ is odd}. \end{cases}$

Famous quotes containing the word dimensions:

“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
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