HYPERSPHERE / HIPERESFERA

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.

2-sphere wireframe as an orthogonal projection.

Just as a stereographic projection can project a sphere's surface to a plane, it can also project the surface of a 3-sphere into 3-space. This image shows three coordinate directions projected to 3-space: parallels (red), meridians (blue) and hypermeridians (green). Due to the conformal property of the stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All of the curves are circles: the curves that intersect <0> have an infinite radius (= straight line).

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.