I saw a piccy recently of the Riemann sphere on the projective plane; the sphere was tesselated with an octahedral/cubic graph (these of course, are duals). ed: Please Register or Log in to view the hidden image! So the space of fractional linear transformations of the modular group (aka the Mobius group), obviously acts on this dual graph on the sphere. What I want to know is if I can rotate half the sphere about the octahedral (blue) lines, thus the projective plane will have one half transformed, the other fixed. Fix the other half and rotate the initially fixed half, and that's equivalent to a rotation of the 'whole' sphere (you don't say?). What I need is a formula for a Mobius transform that rotates the sphere, say 180°, one hemisphere at a time. Or more generally, one that rotates the sphere any angle, but one hemisphere at a time. Anyhoo, I've already worked out that the cube group has a representation (a partition as equivalence classes), which is a tensor product space, the primary 'object' is the direct product of \( \mathbb Z_2 \) with \( \mathbb Z_3 \). The tensor product is very much like what happens in the Temperley Lieb diagram monoids, the graph of the cube group (embedded) in \( \mathbb Z × \mathbb Z \) is just a composition of (some finite number of) \( \mathbb Z_2 \oplus \mathbb Z_3 \). I could show some of my working (i.e. reasoning), but I'm not sure if the king is really setting that kind of test here.