TITLE: Parameterizing rotations of a cube
QUESTION [10 upvotes]: For $g\in\mathrm{SO}(3),S\subseteq \mathbb{R}^3,$ define $g\cdot S:=\{g\cdot p : p\in S\}.$  In words, if $g$ is a rotation of $\mathbb{R}^3$, $g\cdot S$ is the set of elements of $S$ rotated by $g$.  For two elements $g,g'\in\mathrm{SO}(3)$, define an equivalence relation $\sim$ via rotations of the axis-aligned cube:  $g\sim g'\iff (g\cdot [-1,1]^3 = g'\cdot [-1,1]^3)$.
What is the structure of $\mathrm{SO}(3)/\sim$?  Does this space admit a simple parameterization, measure, or metric?
More broadly, is this a well-studied space with a name?  Some interesting problems in discrete geometry seem to involve this structure.
Note:  $\mathrm{SO}(3)$ is a simple group, so I expect this to a quotient space rather than a quotient group.  This is a refined version of an earlier question.

REPLY [3 votes]: More useful than embedding it in some higher-dimensional space will be to give it as a manifold with identifications, I think. Topologically, $SO(3)/\!\sim$ is the same space as what you get by taking an octahedron and gluing the opposite faces with a $1/3$ turn. This is related to the 24-cell, one of the 6 different regular 4-dimensional polytopes (analogues of Platonic solids in 4 dimensions), using the fact that $SO(3)$ is topologically $RP^3$, the quotient of the 3-sphere $S^3$ by the antipodal map.
The space of course comes with a natural metric and measure (both inherited from the one on $SO(3)$. You can see this metric/measure on the octahedron by embedding it as a regular octahedron of an appropriate size subset of $SO(3)$. You can work out the size by looking at the volume: 12 of these octahedra tile $SO(3)$, or 24 tile the double-cover $S^3$. Alternatively, each corner of the octahedron in $SO(3)$ is a rotation by $\pi/4$ around one of the faces of the cube, and the center of each face of the octahedron is a rotation by $\pi/3$ around one of the vertices of the cube.