Date: Mon, 13 Dec 93 22:31:31 -0500 (EST)
From: Dan Hoey <hoey@aic.nrl.navy.mil >
~~~ Subject: Symmetry

I suppose it's time for a few observations on symmetry. After all,
tomorrow is the thirteenth anniversary of "Symmetry and Local Maxima."

As Jerry Bryan notes, we can perform the "R" turn by rotating the cube
to put the R face in front, performing "F", and undoing the rotation.
But we can also perform "R'" by reflecting the cube in a left-to-right
mirror, performing "L", and undoing the reflection. Thus conjugation
can be extended to use the 48-element group of rotations and
reflections, which we call M.

In the absence of face centers, there is another kind of reduction
that takes account of the 24 possible positions of the resulting
collection of edges in space. So two positions X and Y are considered
equivalent if
X = m' Y m c
where m is a rotation or reflection in M, and c is a rotation.

My understanding of Jerry Bryan's method is that he combines "m c"
into a single rotation or reflection, and factors out the reflection
on both sides. It seems to me that what he calls a a "color rotation"
is premultiplication, while a "cube rotation" is postmultiplication.
[I am somewhat uncertain of this, because it doesn't explain how there
can be a 1252-element symmetry group when face centers are present, so
perhaps I'm missing something crucial.]

But I think we are at least conceptually better off dealing with M
when dealing with conjugation, because it takes account of the
essential similarity between clockwise and anticlockwise turns. Alan
Bawden mentioned back in 1980 that certain positions with sufficient
symmetry were local maxima (in terms of distance from start), on the
grounds that any clockwise or anticlockwise turn gives us essentially
the same position. Jim Saxe and I formalized the notion in a paper
entitled "Symmetry and Local Maxima" that we posted on 14 December
1980. [You can find it in /pub/cube-lovers/cube-mail-1.Z on
FTP.AI.MIT.Edu].

We had some hope that some of these local maxima might turn out to be
global maxima. My hopes for that have been somewhat low in recent
years. That is perhaps my best excuse for not noticing immediately
that the single global maximum for the edge group turns out to be one
of these symmetric local maxima. In fact, all four of the positions
with 24-element equivalence classes appear in the list of M-symmetric
positions.

The paper on Symmetry and Local Maxima also catalogues the positions
that have 48-element equivalence classes and 72-element equivalence
classes. The The former are the H-symmetric positions, "Six-H" and
"Six-H with all edges flipped". The latter are the twelve T-symmetric
positions. For T-symmetry, the set of flipped edges may be any of
{none, girdle-edges, off-girdle-edges, or all}; the set of edges
exchanged with their antipodes may be any of the four as well. But if
we choose "none" or "all" for all both choices we get one of the four
M-symmetric positions with 24-element equivalence classes, so only
twelve of the sixteen possibilities have 72-element equivalence
classes.

With regard to the edge cube, I should mention that no one has
mentioned a 9 QT process for the all-flip nor a 15 QT process for the
pons-asinorum-all-flip. Of course, the latter would be somewhat more
interesting, being the longest optimal sequence.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil