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I know that the full symmetry group of the tetrahedron and the rotation group of the cube are both isomorphic to $S_4$. But I want to show a direct, visual isomorphism. I tried looking at a tetrahedron inscribed in a cube, but odd rotations of the cube take that tetrahedron to the 2nd tetrahedron that can be inscribed in a cube. If I inscribe both tetrahedra in a cube, the rotation group of the resulting stella octangula is isomorphic to the cubic rotation group, and I guess reflections of one tetrahedron to itself (which would not preserve the cube), correspond to rotations of one of the tetrahedra into the other, but that does not seem to be a direct isomorphism.

So the main place I'm getting stuck is finding a correspondence between reflections of a tetrahedron into itself and odd rotations of the cube.

Nishant
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  • I believe there's a correspondence that takes edges of the tetrahedron to the faces of the cube; I'm not sure the cleanest representation of this visually, but I suspect that's the core of the isomorphism you're after... – Steven Stadnicki Dec 04 '13 at 18:25
  • consider spatial diagonals of the cube. – ccorn Dec 04 '13 at 18:46
  • Hmm, so a reflection of a tetrahedron fixes 2 edges and sends 2 pairs of edges to each other, which means that 2 faces of the cube are fixed while 2 pairs of faces are sent to each other, giving a 180 degree rotation of the cube about one of its face centers. But this seems to be an even permutation of the cube's 4 diagonals, corresponding to something like (1 3)(2 4) in $S_4$. But reflections should represent odd permutations, right? – Nishant Dec 04 '13 at 19:07

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