From hoey@aic.nrl.navy.mil  Tue Dec 28 18:42:22 1993
Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA04472; Tue, 28 Dec 93 18:42:22 EST
Received: from sun30.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0)
	id AA06105; Tue, 28 Dec 93 18:40:52 EST
Return-Path: <hoey@aic.nrl.navy.mil>
Received: by sun30.aic.nrl.navy.mil; Tue, 28 Dec 93 18:40:52 EST
Date: Tue, 28 Dec 93 18:40:52 EST
From: hoey@aic.nrl.navy.mil
Message-Id: <9312282340.AA25691@sun30.aic.nrl.navy.mil>
To: Cube-Lovers@ai.mit.edu
Subject: Re: Cube Rotations
Cc: CRSO.Cube@canrem.com

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
of two opposite faces doesn't give you any more information than
knowing the destination of one (unless you go breaking the axles).

Dan Hoey
Hoey@AIC.NRL.Navy.Mil