#n-sphere

In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to a n-dimensional space. For any natural number n, an n-sphere of radius r is defined as the set of points in (n + 1)-dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real number. Thus, the n-sphere centred at the origin is defined by:

It is an n-dimensional manifold in Euclidean (n + 1)-space.

In particular::

a 0-sphere is the pair of points at the ends of a (one-dimensional) line segment,a 1-sphere is the circle, which is the one-dimensional circumference of a (two-dimensional) disk in the plane,a 2-sphere is the two-dimensional surface of a (three-dimensional) ball in three-dimensional space.

Spheres of dimension n > 2 are sometimes called hyperspheres, with 3-spheres sometimes known as glomes. The n-sphere of unit radius centered at the origin is called the unit n-sphere, denoted Sn. The unit n-sphere is often referred to as the n-sphere.