From cube-lovers-errors@mc.lcs.mit.edu Sat Aug 16 22:34:17 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA23066; Sat, 16 Aug 1997 22:34:17 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Sat Aug 16 22:35:27 1997 Date: Sat, 16 Aug 1997 22:35:13 -0400 Message-Id: <199708170235.WAA07984@sun30.aic.nrl.navy.mil> From: Dan Hoey To: reid@math.brown.edu Cc: cube-lovers@ai.mit.edu In-Reply-To: <199708152344.TAA05912@life.ai.mit.edu> (message from michael reid on Fri, 15 Aug 1997 19:49:54 -0400) Subject: Re: isoglyphs > perhaps i am missing something, but doesn't > D2 R2 U2 D2 R2 U D (12q, 7f) > produce an isoglyph of type 4D ? Oops, you're right. I goofed because I half-remembered a different result, that there are no 6-bar patterns of the nice symmetric sort, with three mutually perpendicular pairs of parallel bars. > ... > i hadn't even considered chiral versus achiral isoglyphs. indeed, > all the "continuous" isoglyphs given by herbert are chiral. > achiral isoglyphs certainly exist, for example > D2 R2 U' B' L B U B L F2 R D' L2 U2 B2 D (22q, 16f) > of type 11; pattern > *.. > *** > *** > and others can be derived from this. i suspect that there is no > chiral form of this isoglyph, but i'm not absolutely certain. Modulo some oversight, I think this is true, and not hard to demonstrate. Recall that a "ground" facet is one that is not on its home face. First note that a corner cubie will have 0, 2, or 3 ground facets. So on any isoglyph of corner type 1, there are a total of 6 ground corner facets, and these ground facets must appear on two corner cubies (three ground facets each) or three corner cubies (two ground facets each). If two corner cubies, those cubies must be antipodes, and they are either rotated (forming a meson, FTR+ BLD- or equivalent) or exchanged ((FTR,BLD) or equivalent, implying odd edge permutation parity). If ground facets appear on three corner cubies, the cubies must be a three-cycle of cubies on nonadjacent corners ((FTR,FDL,BTL) or equivalent). I've done some analysis by facets on these three cases, which is too messy to describe, but which leads me to the conclusion that the above position is the only isoglyph of its pattern, implying the conclusion that there is no chiral form. Dan Hoey Hoey@AIC.NRL.Navy.Mil