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Consider a 3-dimensional body $B$. The space of all orientations of $B$ in 3-space is the rotation group $SO(3)$, and is often represented in computer science applications as the quaternions with $q$ and $-q$ identified. However, if $B$ is a cube, for example, not all its orientations are distinguishable. From my cursory understanding of group theory, it seems that the space of all distinguishable orientations of $B$ is the quotient group of $SO(3)$ modulo the symmetry group of $B$, in this example the (oriented?) octahedral symmetry group. Is there a more specific name for such a group? And can this kind of group be represented in a nice way as a particular subset of the quaternions?

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up vote 4 down vote accepted

Your question is based on a misconception. For the quotient $SO(3)/\mathrm{Sym}(B)$ to be a group, the group of symmetries $\mathrm{Sym}(B)$ would have to be normal in $SO(3)$. But this is rarely (never?) the case.

To take the example of $B$ a cube, if $\mathrm{Sym}(B)$ were normal then it would follow that every symmetry of $B$ is also a symmetry of any rotated copy of $B$. But that is clearly not the case.

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You're right; I realized that this morning, but did not get around to editing or deleting the question. I'm now left wondering whether the space of distinguishable orientations of $B$ does have some structure, if not that of a quotient group, but I'll think harder about that myself first before I ask. –  Rahul Mar 23 '11 at 19:46
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