David Harden thinks he’s found a way! But I haven’t carefully checked that it works: it takes some calculations. Can you check it?

]]>The eigenstates of the three Pauli operators form a regular octahedron inscribed in the Bloch sphere. Transformations of the qubit state space that permute the vertices of this octahedron should correspond to elements of the binary octahedral group. Offhand, though, I can’t recall of facts about the binary octahedral group being used to prove things in quantum information. I think that when people mod out the phase freedom in the qubit Clifford group to get a discrete group, they go all the way and end up with the symmetries of a regular octahedron, which are just the symmetries of a cube.

]]>That sounds vaguely familiar, but it’s been a long time since I’ve read that book!

]]>Yes, that’s very nice! The room centers form the vertices of the cross-polytope dual to the hypercube; together they form the vertices of a 24-cell. The midpoints of the walls of the cube form the vertices of the dual 24-cell. Interestingly there doesn’t seem to be a *canonical* way to partition these vertices into those of a hypercube and a 24-cells, given the structure we have so far.

The rotations of these figures in 4D are of a type that doesn’t leave any point on the 3-sphere fixed, but moves them all with equal velocity; you get that kind of rotation by combining rotations by equal angles in two orthogonal planes.

The projection is just something generic that I chose by trial and error to look OK throughout the course of the rotation.

]]>Greg, do your pictures show a 2-D parallel projection of a 24-cell rotating in 4-space around a fixed axis (so basically you’re just dropping two coordinates before displaying)? If so, is there any special relationship between the (planar) rotation axis and the plane of projection, other than the fact that they’re not equal? ]]>

The colours in the previous image were a bit of a trick; I wanted to make two octahedral cells visible by distinguishing them, but I also wanted to exploit the symmetry of the figure in order to show only half a rotation and make the GIF smaller. So the hues aren’t strictly linked to any coordinate, but just depend on how much the figure has been rotated, in such a way that the two cells swap colours after half a rotation.

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