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 (n + 1)-space.

The 0-sphere is a pair of points, the 1-sphere is a circle, and the 2-sphere is an ordinary sphere. Generally, when embedded in an (n + 1)-dimensional Euclidean space, an n-sphere is the surface or boundary of an (n + 1)-dimensional ball. That is, for any natural number n, an n-sphere of radius r may be defined in terms of an embedding in (n + 1)-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:

In particular:

An n-sphere embedded in an (n + 1)-dimensional Euclidean space is called a hypersphere. The n-sphere of unit radius is called the unit n-sphere, denoted Sⁿ, often referred to as the n-sphere.

When embedded as described, an n-sphere is the surface or boundary of an (n + 1)-dimensional ball. For n ≥ 2, 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 (n − 1)-sphere.

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