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Date: Tue, 09 Sep 1997 11:02:32 -0400 (EDT)
From: Jerry Bryan
Subject: Re: Open and Closed Subgroups of G
In-Reply-To: <199709060107.VAA04503@sun30.aic.nrl.navy.mil>
To: Cube-Lovers
Cc: lvt-cfc@servtech.com
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On Fri, 5 Sep 1997, Dan Hoey wrote:
> Chris Chiesa , among other things, writes
>
> > If I now make the single turn
>
> > B'
>
> > I no longer find it so easy to characterize the corner-twist
> > parity state of the Cube, because (all of) the corner-cubies
> > affected by this particular Cube-state-change have left their
> > previous positions, leaving me to wonder, "RELATIVE TO WHAT" their
> > twist is to be assessed.
>
> At the risk of being repetitious, the answer is, "relative to the home
> orientation of the position they find themselves in". You choose a
> special facelet for each corner cubie. When the cubie is in its home
> position, its twist is the position of its special facelet relative to
> the home of the special facelet. When cubie X is in cubie Y's home
> position, the twist of cubie X is the position of X's special facelet
> relative to the home of Y's special facelet. The edges are done the
> same way, except mod 2.
Dan's response (plus his references in the Cube-Lovers archives) pretty
well covers it. I would just like to add a couple of points.
1. There is a reference in the archives to a way of demonstrating
conservation of twist without first establishing a frame of
reference, but I can't find the reference. The best I can
recall, the same technique did not work for edges. But I prefer
the frame of reference technique anyway because it is closely
tied to some of the more usual ways of representing the cube in a
computer.
2. For example, number the corner facelets from 1 to 24. Each
facelet has two companion facelets which are bound to it on
the same cubie. By knowing where one of the three facelets
of a cubie is in a computer program, you automatically know
where the other two facelets are, so you only have to store
one of the three facelets. The one that you store can be the
"special" facelet that Dan described for the purposes of
determining conservation of twist.
The collection of eight "special" facelets for the corners
have been described in the archives as constituting a
supplement for the group, but I have yet to find a discussion
group supplements in any group theory book.
As Dan says, your choice of "special" facelet is totally
arbitrary for each cubie, but most typically you choose
the Front and Back facelets, or the Right and Left facelets,
or something equally well organized.
3. For another example, number the corner cubies from 1 to 8, and
for each of the cubies describe the twist with a number from 0
to 2. This is essentially a wreath product representation of
the cube. The numbers from 0 to 2 which describe the twist
can be used to describe whether a cubie is twisted when it is
not home, and can therefore be used to prove conservation of
twist.
Without knowing any more than I do about supplements, it seems
very likely that it should be easy to represent any group
which can be representated as a supplement as a wreath product
and vice versa. The isomorphism seems obvious. I wonder if
anybody out there can shed any light on this issue?
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 539-7198
10915 Hardin Valley Road (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990