What is "n-sphere"

In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension. It is an n-dimensional manifold that can be embedded in Euclidean -space.

For any natural number n, an n-sphere of radius r may be defined in terms of an embedding in -dimensional Euclidean space as the set of points that are at distance r from a central point, where the radius r may be any positive real number. Thus, the n-sphere would be defined by:

S = {x ∈ R : ∥x∥ = r}.

In particular:

the pair of points at the ends of a (one-dimensional) line segment is 0-sphere, the circle, which is the one-dimensional circumference of a (two-dimensional) disk in the plane is a 1-sphere, the two-dimensional surface of a (three-dimensional) ball in three-dimensional space is a 2-sphere, often simply called a sphere, the three-dimensional boundary of a (four-dimensional) 4-ball in four-dimensional Euclidean is a 3-sphere, also known as a glome.

An n-sphere embedded in an -dimensional Euclidean space is called a hypersphere. The n-sphere of unit radius is called the unit n-sphere, denoted S. The unit n-sphere is often referred to as the n-sphere.

When embedded as described, an n-sphere is the surface or boundary of an -dimensional ball. For , the n-spheres are the simply connected n-dimensional manifolds of constant, positive curvature. The n-spheres admit several other topological descriptions: for example, they can be constructed by gluing two n-dimensional Euclidean spaces together, by identifying the boundary of an n-cube with a point, or (inductively) by forming the suspension of an -sphere.