Best visualization of SO(3)

[Matterwave]

TL;DR

Trying to find a good visualization of SO(3)

I know a decent number of properties of SO(3) and obviously I can visualize the group action on three space as rotations. But a good visualization of the group itself eludes me.

SU(2) is somehow more intuitive to visualize since it simply is $S^3$. So I just think of a 2-d sphere and say "well it has one more dimension" (lol).

Sometimes, because SU(2) double covers SO(3) I see some descriptions of SO(3) as "half of $S^3$". Sometimes this is tempting but it has some undesirable properties. For one, imagining a sphere cut in half introduces an edge to the sphere. The cut is also entirely arbitrary.

Really, it's two antipodal points of SU(2) maps to one point on SO(3) but this doesn't help me "get a picture" in my head. At least not in a way where, for example, the non simply-connectedness of SO(3) becomes obvious to me.

Anyone know of some good visualizations? To help build intuition?

I suspect this will also help me understand non simply connected as more than "there's a hole".

 

[anuttarasammyak]

How about Bloch sphere for qubit? Perhaps you want more than that.

 

[wrobel]

To me, the best way to visualize $\text{SO}(3)$ is as a rigid body with a fixed point. In particular by examining its motion through Euler angles, it becomes clear that $\text{SO}(3)$ is a fiber bundle with base $\mathbb S^2$ and fiber $\mathbb S^1$.

 

[Paul Colby]

Visualize as a manifold seems to be all you consider. There’s the algebraic aspect of $SO(3)$ as well.

 

[lavinia]

To expand @wroblel's point in post #3: The action of SO(3) on the 2 sphere can be extended to an action on its tangent circle bundle by the mapping r:(x.v)-> (r(x),dr(v)) where r is a rotation of the sphere. This mapping defines a diffeomorphism of SO(3) onto the tangent circle bundle of the 2 sphere.

There are infinitely many different circle bundles over the 2 sphere. So one needs a way to distinguish them.

There are a few ways to do this. One is to notice that if one cuts any circle bundle along the equator of the sphere, it splits into two trivial circle bundles over the two hemishperes and since a hemisphere is homeomorphic to a closed disk, one gets two copies of D^2xS^1 , the Cartesian product of a closed disk with a circle and these are both topological solid tori. To see this,think of the Cartesian product as a circle of disks. So one sees that any circle bundle over the 2 sphere is two solid tori pasted together along their boundaries. Each is distinguised by the ways these pastings are done.

Interestingly, since the 3 sphere is the total space of the Hopf fibration which is itself a circle bundle over the 2 sphere , the sphere in four dimensions can be made from two solid tori that are pasted along their boundaries. To see this visually one one might try to see how this pasting happens through the stereographic projection of the 3 sphere into R^3 and then look at the way the images of the Clifford tori fit together.

There are other ways to do this which I am happy to describe.

 

[lavinia]

A mathematician showed me another way to see the 3 sphere.

Slice the 3 sphere along its equatorial 2 sphere. Its two hemispheres are 3 dimensional balls which can be seen by projecting them vertically into three dimensions. This is the same as slicing a regular sphere in three dimensions. If one imagines that the sphere is an inflated balloon, then the two hemispheres will naturally deflate until they become flat disks. Then they are two dimensional rather than three. So think of the three sphere as an inflated three dimensional balloon.

Next core out a solid tube from each of these balls . Make sure to take care that back in the 3 sphere, these tubes connect at the ends to form a solid torus. Put them together to make a solid torus. Now look at one of the cored solid balls. Imagine that it is very stretchable, and put your hands in the top of the hole an warp it into a wider circle and then pull it down until it is flush with the bottom of the hole and is concentric with it. In this process, warp the entire outside surface of the cored ball downward until it becomes flattened out. Note that the inner surface where the tube was removed will curve over during the stretching. This makes half of a solid torus, a half bagel. Do this with the other one and paste the two together to make a whole bagel. Now there are two solid tori.

One knows that the solid torus made from the two cores fits into the bagel through its hole but because of the strectching, in order to recover the spots where it came from, its boundary must be spread out over the entire boundary of the bagel. In other words the two solid tori are pasted together along their boundaries.

If I knew how to post pictures I could make some drawings but this visualization is a good one to struggle with one one's own.

 

[Ibix]

If I knew how to post pictures I could make some drawings but this visualization is a good one to struggle with one one's own.

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[Hornbein]

Slice the 3 sphere along its equatorial 2 sphere.

The equation of the equator of a 3-sphere is w^2 + x^2 = y^2 + z^2 = 1/2. It looks like the equation for a 2-sphere but unlike a 2-sphere it can't be embedded in a 3D space. Instead it is a sort of 2D torus embedded in 4D.

The equator of a 3-sphere is the set of points equidistant from the two "poles." These "poles" are both circles, w^2 + x^2 = 1 and y^2 + z^2 = 1. Every point on each circle is the same distance from every point on the other circle.

Each circle may be embedded in a 2D plane. The intersection of the two planes is the point at the center of the 3-sphere.

 

[lavinia]

The Tangent Circle Bundle of the 2-Sphere

Here is a way to “visualize” the tangent circle bundle of the 2-sphere that connects to a lot of rich topology.

Start with a smooth vector field, V, that has a single isolated zero at the north pole of the sphere. This vector field has index 2 which means that is winds around the the zero twice.

Divide each vector except the zero by its length to get a unit-length vector field. That is a field of vectors that lies in the unit tangent circle bundle except at the north pole. Now remove a small open disk around the north pole and let the radius of the disk shrink to zero. The unit-length vectors at the boundary of this disk form a closed curve that converges onto the fiber at the zero and in the limit attaches to it. One can imagine this boundary curve winding around the zero point getting closer and closer until it finally lands on the fiber above the zero by winding around it twice. For simplicity, assume a choice of vector field where this attaching map is strictly 2-to-1.

The complement of the removed open disk is a topological closed disk, D. The normalized vector field over it is also a closed disk because it is the image of a smooth section of the bundle over D. As the radius of the open disk shrinks to zero, this disk of unit vectors expands over the entire sphere, and its boundary circle attaches to the fiber above the north pole by the 2 to 1 mapping on its boundary circle . One then gets a closed 2-disk attached along its boundary by a continuous 2 to 1 mapping to a circle. This is a surface homeomorphic to the real projective plane RP^2.

Now, imagine rotating this normalized vector field through an angle. This rotated vector field produces a second projective plane that is also attached to the fiber circle at the north pole by a 2-to-1 mapping. Sweeping through all possible rotations gives a set of vector fields that span the entire unit tangent circle bundle and fill it with projective planes that intersect only in a common circle. Topologically this configuration is a circle of closed disks that are bound to a single circle each by a 2-to-1 mapping on their boundaries.

Alternatively, look back at the bundle over the closed disk, D, before the limit is taken. The set of all these rotated vector fields fills out a solid torus, D × S¹. Letting the radius of the disk shrink to zero, this solid torus expands and its boundary, a regular torus, T, becomes glued to the fiber at the north pole. T is completely covered by the parallel trajectories of the rotated vector fields. When attached to the fiber circle, the 2-to-1 mapping collapses each of these parallel trajectories onto the fiber.

So the unit tangent circle bundle can be described as a solid torus whose boundary torus, T, is attached to a circle by a mapping that collapses a family of parallel closed curves covering T by a 2 to 1 mapping on each curve.


Notes

The method of visualizing a topological space by breaking it into simpler pieces and then describing how they are put back together is a foundational technique in geometric topology.

The Poincaré–Hopf Index Theorem: On a compact smooth manifold, the sum of the indices of a vector field with isolated zeros must equal its Euler characteristic. Since χ(S²) = 2, a single isolated zero must have index 2. It os also true that on a connected closed manifold a vector field with aan isolated zer can always be found no matter what its dimensions

Connections to Other Circle Bundles: Other circle bundles over the 2-sphere can be constructed using this exact geometric framework, altered only by changing the degree of the attaching map. In the case of the 3-sphere (corresponding to the Hopf fibration), the attaching map is 1-to-1. For other bundles , the mapping is n-to-1 with n > 2 and the individual "pages" are not manifolds.For the cases where the circle bundles are not tangent, the vector fields aren’t vector fields in the sense of flow fields on the sphere. They are not tangent to the sphere. Instad, hey are called sections of the vector bundle.