firstname.lastname@example.org (Mark Longridge) writes:
Perhaps my description of the rotations was unclear...
...Perhaps it is better to use the form
old FACE A -> new FACE A
old FACE B -> new FACE B
Where the faces A & B are adjacent.
That will serve to uniquely identify a rotation, but it's somewhat
verbose. Worse, it does not suffice to uniquely identify a symmetry
from the group of rotations and reflections, M. I find it's far more
informative to identify a rotation or reflection as a permutation of
the faces, in cycle format. There are only ten kinds:
Even rotations: I=Identity (1), (FRT)(BLD)=120-degree rotation (8), (FB)(RL)=180-degree orthogonal rotation (3). Odd rotations: (FRBL)=90-degree rotation (6), (FB)(TR)(DL)=180-degree diagonal rotation (6). Even reflections: (FR)(BL)=diagonal reflection (6), (FRBL)(TD)=90-degree glide reflection (6), Odd reflections: (FB)=orthogonal reflection (3), (FRTBLD)=60-degree glide reflection (8), (FB)(RL)(TD)=central reflection (1).
In case it isn't clear, the cycle notation for (e.g.) a 120-degree
rotation (FTL)(BDR) means that the F, T, L, B, D, and R faces move to
the T, L, F, D, R, and B, locations, respectively. The only thing I'm
afraid of with this notation is that someone will think I'm describing
a magic-cube process rather than a whole-cube move.
So when you say Top->Down, Front->Left, I would say (TD)(FL)(BR) for
the 180-degree diagonal rotation, to distinguish it from (TD)(FLBR)
the 90-degree glide reflection.
....wait a second, I don't think faces A & B have to be
adjacent for the rotation to be unambiguous. Any 2 faces
No, you're back to your original bogosity. Knowing the destinations
of two opposite faces doesn't give you any more information than
knowing the destination of one (unless you go breaking the axles).