What shape does SO(3)/A5 describe and how can it be visualized?

[nateHI]

I was watching this video on Abstract Algebra and the professor was discussing how at one point a few mathematicians conjectured the special orthogonal group in $\mathbb{R}^3$ mod the symmetries of an icosahedron described the shape of the universe (near the end of the video).

My question is, what shape does $SO(3)/A_5$ describe? Also, I just started a course in algebraic topology so forgive my ignorance; but, is it correct to say that a good way to try and picture this shape would be to imagine shrinking all the points in $SO(3)$ corresponding to a rotational symmetry of an icosahedron to a point? If I understand correctly this would produce something in $\mathbb{R^4}$.

 

[DrDu]

The elements of SO(3) can be parameterised by a unit vector n, describing the direction of the rotation axis, and an angle phi. All the unit vectors lie on a sphere surface with antipodal points identified. If you take the angle as radial coordinate, you get a ball. Then SO(3)/A5 is probably a pentagonal prism corresponding to the face of a dodecahedron or the like.