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3-sphere

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\}.

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