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.
It is an n-dimensional manifold in Euclidean (n + 1)-space.
2-sphere wireframe as an orthogonal projection.
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.
Video: Hypersphere.
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.
An n-sphere is the surface or boundary of an (n + 1)-dimensional ball, and is an n-dimensional manifold. 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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