From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 9 14:56:02 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA17811; Tue, 9 Sep 1997 14:56:01 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Tue Sep 9 11:06:36 1997 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 Message-Id: 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