From cube-lovers-errors@mc.lcs.mit.edu Fri Sep 5 21:03:58 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA04289; Fri, 5 Sep 1997 21:03:57 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Fri Sep 5 21:08:00 1997 Date: Fri, 5 Sep 1997 21:07:48 -0400 Message-Id: <199709060107.VAA04503@sun30.aic.nrl.navy.mil> From: Dan Hoey To: lvt-cfc@servtech.com Cc: cube-lovers@ai.mit.edu In-Reply-To: <199709031702.NAA20567@cyber1.servtech.com> (lvt-cfc@servtech.com) Subject: Re: Open and Closed Subgroups of G (fwd) 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. Cube-lovers can find this in Vanderschel's article (6 Aug 1980) and the extension by Saxe (3 September 1980). I mentioned (23 September 1982) that the choice of special facelets is arbitrary, and that a conservation of twist occurs for a set of pieces of any puzzle that 1. have an Abelian orientation group, and 2. are moved in untwisted cycles by the generators. This is true even if not all the cycles have the same length. For instance, we could have a Rubik's cube in which generators move corners in permutations like (FTR,FRD,FDL,FLT)(BRT,BTL,BLD), and twist would be preserved. The key is that for each piece, the minimum power of the generator that returns that piece to its home position must also return it to its home orientation. I'm quite uncertain about what orientation constraints can arise in puzzles with non-Abelian orientation groups. For instance, the hypercorners of a Rubik's tesseract have the symmetry group A4, and any orientation is achievable up to a constraint imposed by an Abelian quotient of A4 of type 3 (See 22 Oct 1982). Does every group have a unique maximal Abelian quotient? Is that the only orientation constraint that can occur? Dan Hoey Hoey@AIC.NRL.Navy.Mil [ Moderator's Note: Cube-lovers will be down Saturday and Sunday due to major electrical work at MIT. ]