Editing 3-sphere (section)

==Definition==
In [[coordinates]], a 3-sphere with center {{math|(''C''<sub>0</sub>, ''C''<sub>1</sub>, ''C''<sub>2</sub>, ''C''<sub>3</sub>)}} and radius {{mvar|r}} is the set of all points {{math|(''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}} in real, [[Four-dimensional space|4-dimensional space]] ({{math|'''R'''<sup>4</sup>}}) such that
:<math>\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.</math>
The 3-sphere centered at the origin with radius 1 is called the '''unit 3-sphere''' and is usually denoted {{math|''S''<sup>3</sup>}}:

:<math>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\}.</math>

It is often convenient to regard {{math|'''R'''<sup>4</sup>}} as the space with 2 [[complex numbers|complex dimensions]] ({{math|'''C'''<sup>2</sup>}}) or the [[quaternion]]s ({{math|'''H'''}}). The unit 3-sphere is then given by

:<math>S^3 = \left\{(z_1,z_2)\in\mathbb{C}^2 : |z_1|^2 + |z_2|^2 = 1\right\}</math>
or
:<math>S^3 = \left\{q\in\mathbb{H} : \|q\| = 1\right\}.</math>

This description as the [[quaternion]]s of [[Quaternion#Conjugation, the norm, and reciprocal|norm]] one identifies the 3-sphere with the [[versor]]s in the quaternion [[division ring]]. Just as the [[unit circle]] is important for planar [[polar coordinates#Complex numbers|polar coordinates]], so the 3-sphere is important in the polar view of 4-space involved in quaternion multiplication. See [[polar decomposition#Quaternion polar decomposition|polar decomposition of a quaternion]] for details of this development of the three-sphere.
This view of the 3-sphere is the basis for the study of [[elliptic geometry#Elliptic space|elliptic space]] as developed by [[Georges Lemaître]].<ref>{{Cite journal |last=Lemaître |first=Georges |author-link=Georges Lemaître |date=1948 |title=Quaternions et espace elliptique |journal=Acta |publisher=[[Pontifical Academy of Sciences]] |volume=12 |pages=57–78}}</ref>