Date: Tue, 28 Dec 93 18:40:52 -0500 (EST)
From: Dan Hoey <hoey@aic.nrl.navy.mil >
~~~ Subject: Re: Cube Rotations

mark.longridge@canrem.com (Mark Longridge) writes:

Perhaps my description of the rotations was unclear...

Yes.

...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
should do!

No, you're back to your original bogosity. Knowing the destinations