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.