3-sphere

frame Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Due to conformal property of Stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect have infinite radius (= straight line).

In [mathematics] , a 3-sphere is a higher-dimensional analogue of a [sphere] . It consists of the set of points equidistant from a fixed central point in 4-dimensional [Euclidean space] . Just as an ordinary sphere (or 2-sphere) is a two dimensional [surface] that forms the boundary of a [ball] in three dimensions, a 3-sphere is an object with three [dimension] s that forms the boundary of a ball in four dimensions. A 3-sphere is an example of a [3-manifold] .

A 3-sphere is also called a hypersphere , although the term hypersphere can in general describe any [''n''-sphere] for n ≥ 3.

Definition
In [coordinates] , a 3-sphere with center ( C 0, C 1, C 2, C 3) and radius r is the set of all points ( x 0, x 1, x 2, x 3) in real, [4-dimensional space] ( R 4) such that
:\sum_{i=0}^3(x_i - C_i)^2 = ( x_0 - C_0 )^2 ( x_1 - C_1 )^2 ( x_2 - C_2 )^2 ( x_3 - C_3 )^2 = r^2.
The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted S 3:

:S^3 = \left\{(x_0,x_1,x_2,x_3)\in\mathbb{R}^4 : x_0^2 x_1^2 x_2^2 x_3^2 = 1\right\}.