Parameterizing the 3-Sphere: 3 Angles?

[pcr]

How would one parameterize the three-sphere? Would three angles work: latitude, longitude and something?

 

[jedishrfu]

Wikipedia has an article on it:

https://en.m.wikipedia.org/wiki/3-sphere

You need a 4th coordinate as to whether its an angle or some fixed axis is based on the coordinate system chosen.

A 3D analogue would be spherical vs cylindrical coordinates. One has two angles and the other just one but both have coordinate values to describe a point in 3D space.

https://plus.maths.org/content/richard-elwes

 

[stevendaryl]

How would one parameterize the three-sphere? Would three angles work: latitude, longitude and something?

Well, you can describe it using four coordinates $X$, $Y$, $Z$ and $W$ with the constraint $X^2 + Y^2 + Z^2 + W^2 = R^2$. You can parametrize it in terms of angles like this:

$W = R cos(\psi)$
$Z = R sin(\psi)cos(\theta)$
$X = R sin(\psi)sin(\theta)cos(\phi)$
$Y = R sin(\psi)sin(\theta)sin(\phi)$

 

[jedishrfu]

Humorously you could call the 4th coordinate:

Fortitude

Something you might need when studying higher dimensioned geometries.

 

[fresh_42]

The $3-$sphere (with radius $1$) is a three dimensional analytic manifold. Two possible ways of looking at it are
$$\mathbb{S}^3 \simeq U(1,\mathbb{H}) \cong SU(2,\mathbb{C})$$
There are others, but I find these convenient. This means you can look at points of $\mathbb{S}^3$ as unit quaternions or as unitary complex matrices with determinant one. Of course you can always use polar coordinates $(\varphi , \theta , \phi)$ or Cartesian coordinates
$$
x_1=\cos \varphi \sin \theta \sin \phi \\
x_2=\sin \varphi \sin \theta \sin \phi \\
x_3=\cos \theta \sin \phi \\
x_4=\cos \phi
$$
If you want to look at it as an isometry group of a four dimensional Euclidean space, then $\mathbb{S}^3\simeq SO(4,\mathbb{R})/SO(3,\mathbb{R})$ would be useful.

 

[pcr]

Wikipedia has an article on it:

https://en.m.wikipedia.org/wiki/3-sphere

You need a 4th coordinate as to whether its an angle or some fixed axis is based on the coordinate system chosen.

A 3D analogue would be spherical vs cylindrical coordinates. One has two angles and the other just one but both have coordinate values to describe a point in 3D space.

https://plus.maths.org/content/richard-elwes

Thank you this was very helpful!

Humorously you could call the 4th coordinate:

Fortitude

Something you might need when studying higher dimensioned geometries.

The $3-$sphere (with radius $1$) is a three dimensional analytic manifold. Two possible ways of looking at it are
$$\mathbb{S}^3 \simeq U(1,\mathbb{H}) \cong SU(2,\mathbb{C})$$
There are others, but I find these convenient. This means you can look at points of $\mathbb{S}^3$ as unit quaternions or as unitary complex matrices with determinant one. Of course you can always use polar coordinates $(\varphi , \theta , \phi)$ or Cartesian coordinates
$$
x_1=\cos \varphi \sin \theta \sin \phi \\
x_2=\sin \varphi \sin \theta \sin \phi \\
x_3=\cos \theta \sin \phi \\
x_4=\cos \phi
$$
If you want to look at it as an isometry group of a four dimensional Euclidean space, then $\mathbb{S}^3\simeq SO(4,\mathbb{R})/SO(3,\mathbb{R})$ would be useful.

 

[pcr]

Well, you can describe it using four coordinates $X$, $Y$, $Z$ and $W$ with the constraint $X^2 + Y^2 + Z^2 + W^2 = R^2$. You can parametrize it in terms of angles like this:

$W = R cos(\psi)$
$Z = R sin(\psi)cos(\theta)$
$X = R sin(\psi)sin(\theta)cos(\phi)$
$Y = R sin(\psi)sin(\theta)sin(\phi)$

Thank you, that was helpful

 

[pcr]

The $3-$sphere (with radius $1$) is a three dimensional analytic manifold. Two possible ways of looking at it are
$$\mathbb{S}^3 \simeq U(1,\mathbb{H}) \cong SU(2,\mathbb{C})$$
There are others, but I find these convenient. This means you can look at points of $\mathbb{S}^3$ as unit quaternions or as unitary complex matrices with determinant one. Of course you can always use polar coordinates $(\varphi , \theta , \phi)$ or Cartesian coordinates
$$
x_1=\cos \varphi \sin \theta \sin \phi \\
x_2=\sin \varphi \sin \theta \sin \phi \\
x_3=\cos \theta \sin \phi \\
x_4=\cos \phi
$$
If you want to look at it as an isometry group of a four dimensional Euclidean space, then $\mathbb{S}^3\simeq SO(4,\mathbb{R})/SO(3,\mathbb{R})$ would be useful.

Thank you, that was helpful!

 

[pcr]

Thank you this was very helpful!

Thank you, that was helpful!

 

[jedishrfu]

We do have a like button that is even more helpful and will save some time too.

 

[pcr]

Hello again. Today I would like to ask how many coordinate patches is takes to cover S3? And how many poles are there on a three sphere, if that makes sense?

 

[fresh_42]

Hello again. Today I would like to ask how many coordinate patches is takes to cover S3? And how many poles are there on a three sphere, if that makes sense?

The stereographic projection still works, so two and one pole which maps to the origin and the other one to infinity.

 

[mathwonk]

just as a 2 sphere is obtained by revolving a semi circle around the axis joining its endpoints,, it seems you obtain a 3 sphere by revolving a hemi (2-)sphere around the plane of its equator (revolve in 4-space). so maybe you need one angle in the plane perpendicular to that of the equator, for the angle of revolution, and then you need 2 angular spherical coordinates for the point on the hemi sphere? i.e. two angles give the point of the hemi sphere and then one more angle tells us how far you revolved it. just a suggestion for how to picture it. (I was able to use this picture to imagine how to give a deduction of the volume formula for a 3 ball that archimedes could have done, so it should work.)

this also seems to allow you to see how cover it by coordinate charts, since revolving a semi circle around all but one point of circle sweeps out a coordinate chart on the 2 sphere omitting exactly one semi circle. so just as you can see that it is possible to choose two semi circles to omit from the 2 sphere, hence you can cover the 2 sphere by two of these charts, so you may try to visualize how to choose two disjoint hemi (2-(spheres in the 3 sphere, hence yielding a cover by two of the analogous charts. of course stereographic projection works, but this is a suggestion to do it with the kind of spherical coordinates being used here.

Actually I am having troubling picturing this and wonder now if it works, maybe not since a 1 sphere, unlike a zero sphere, is connected.

 

[zwierz]

is there a classical mechanics system that would have 3-sphere as a configuration manifold?

 

[pcr]

Perhaps roll, pitch and Yaw for an airplane, but I don't really know. Perhaps someone else does.

 

[Ryan Rankin]

"is there a classical mechanics system that would have 3-sphere as a configuration manifold? "

The Einstein universe, the first proposed non-euclidean model of the universe was spatially just a 3-Sphere. Of the three shapes of the universe supporting isotropy and homogony (the Freidmann-Robertson-Walker metrics) the three sphere is again spatially one of those (though it's radius will change in time).