Editing 3-sphere (section)

===Hopf coordinates===
[[File:Hopf Fibration.png|right|250px|thumb|The Hopf fibration can be visualized using a [[stereographic projection]] of {{math|''S''<sup>3</sup>}} to {{math|'''R'''<sup>3</sup>}} and then compressing {{math|''R''<sup>3</sup>}} to a ball.  This image shows points on {{math|''S''<sup>2</sup>}} and their corresponding fibers with the same color.]]
For unit radius another choice of hyperspherical coordinates, {{math|(''η'', ''ξ''<sub>1</sub>, ''ξ''<sub>2</sub>)}}, makes use of the embedding of {{math|''S''<sup>3</sup>}} in {{math|'''C'''<sup>2</sup>}}. In complex coordinates {{math|(''z''<sub>1</sub>, ''z''<sub>2</sub>) ∈ '''C'''<sup>2</sup>}} we write
:<math>\begin{align}
z_1 &= e^{i\,\xi_1}\sin\eta \\
z_2 &= e^{i\,\xi_2}\cos\eta. \end{align}</math>

This could also be expressed in {{math|'''R'''<sup>4</sup>}} as
:<math>\begin{align}
x_0 &= \cos\xi_1\sin\eta \\
x_1 &= \sin\xi_1\sin\eta \\
x_2 &= \cos\xi_2\cos\eta \\
x_3 &= \sin\xi_2\cos\eta. \end{align}</math>
Here {{mvar|η}} runs over the range 0 to {{sfrac|{{pi}}|2}}, and {{math|''ξ''<sub>1</sub>}} and {{math|''ξ''<sub>2</sub>}} can take any values between 0 and 2{{pi}}. These coordinates are useful in the description of the 3-sphere as the [[Hopf bundle]]
:<math>S^1 \to S^3 \to S^2.\,</math>

[[File:Toroidal coord.png|thumb|A diagram depicting the poloidal ({{math|''ξ''<sub>1</sub>}}) direction, represented by the red arrow, and the toroidal ({{math|''ξ''<sub>2</sub>}}) direction, represented by the blue arrow, although the terms ''poloidal'' and ''toroidal'' are arbitrary in this ''[[Flat torus#Flat torus|flat torus]]'' case.]]
For any fixed value of {{mvar|η}} between 0 and {{sfrac|{{pi}}|2}}, the coordinates {{math|(''ξ''<sub>1</sub>, ''ξ''<sub>2</sub>)}} parameterize a 2-dimensional [[torus]]. Rings of constant {{math|''ξ''<sub>1</sub>}} and {{math|''ξ''<sub>2</sub>}} above form simple orthogonal grids on the tori. See image to right. In the degenerate cases, when {{mvar|η}} equals 0 or {{sfrac|{{pi}}|2}}, these coordinates describe a [[circle]].

The round metric on the 3-sphere in these coordinates is given by
:<math>ds^2 = d\eta^2 + \sin^2\eta\,d\xi_1^2 + \cos^2\eta\,d\xi_2^2</math>
and the volume form by
:<math>dV = \sin\eta\cos\eta\,d\eta\wedge d\xi_1\wedge d\xi_2.</math>

To get the interlocking circles of the [[Hopf fibration]], make a simple substitution in the equations above<ref>{{cite web|last1=Banchoff|first1=Thomas|title=The Flat Torus in the Three-Sphere|url=http://www.geom.uiuc.edu/~banchoff/script/b3d/hypertorus.html}}</ref>
:<math>\begin{align}
z_1 &= e^{i\,(\xi_1+\xi_2)}\sin\eta \\
z_2 &= e^{i\,(\xi_2-\xi_1)}\cos\eta. \end{align}</math>

In this case {{mvar|η}}, and {{math|''ξ''<sub>1</sub>}} specify which circle, and {{math|''ξ''<sub>2</sub>}} specifies the position along each circle. One round trip (0 to 2{{pi}}) of {{math|''ξ''<sub>1</sub>}} or {{math|''ξ''<sub>2</sub>}} equates to a round trip of the torus in the 2 respective directions.