From cube-lovers-errors@mc.lcs.mit.edu Fri Apr 3 18:45:49 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA16932; Fri, 3 Apr 1998 18:45:48 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Mar 29 18:56:43 1998 Date: Sun, 29 Mar 1998 18:57:08 -0400 (EDT) From: Jerry Bryan Subject: Re: All the Partial Isoglyphs In-Reply-To: To: Cube-Lovers Message-Id: On Sun, 29 Mar 1998, Jerry Bryan wrote: > Here is a breakdown of how the solid faces can be arranged. > > 97 - two solid faces, opposite to each other > 11 - two solid faces, adjacent to each other > 25 - one solid face > 1 - three solid faces, mutually adjacent to each other > 2 - three solid faces, not mutually adjacent to each other > 3 - four solid faces, other two opposite to each other > 1 - four solid faces, other two adjacent to each other > --- > 140 > As this table shows, the vast majority of partial isoglyphs involve two solid faces opposite to each other. The basic reason for this is the corners. If the corners are not fixed, then the only partial isoglyphs which are possible have two solid faces opposite to each other. Conversely, the 43 partial isoglyphs which do not have two solid faces opposite to each other do fix the corners. In fact, 67 of the partial isoglyphs derive from just 5 of the glyphs, namely those which fix the corners. If the corners of the partial isoglyph are fixed, you can think of the edges as consisting of a set of strongly constrained edge flips and swaps. (Be careful -- if the corners are fixed, then *any* resultant position can be thought of as just a bunch of edge flips and swaps. But for partial isoglyphs, the possible edge flips and swaps are strongly constrained.) The glyph which yields the most partial isoglyphs is the one my charts call BF, whick looks like the following. X0X XXX XXX With this glyph, each face of a partial isoglyph can have at most one edge cubie which is swapped or flipped, but on a cube-wide basis there are quite a few different ways to arrange for this to happen. Another interesting glyph which fixes the corners is called BD on my charts, and which appears as follows. X0X XXX X0X As an isoglyph, this glyph yields five different patterns on the 6-H theme. As a partial isoglyph, this glyph yields a number of pretty 2-H, 3-H, 4-H, and 5-H patterns. You may also think of the H patterns as complicated edge swappers/flippers, with exactly zero or two edges swapped/flipped on each face, and with the coloring requirements for partial isoglyphs being maintained. The following two glyphs (A7 and AF in my charts) are in the same spirit as the H, except that the configuration of the edges on each face which are swapped/flipped is slightly different than for the H. X0X X0X 0X0 0XX XXX XXX Finally, for completeness in the list of glyphs which fix the corners, the glyph called A5 on my charts appears as follows. X0X 0X0 X0X However, this glyph only yields two partial isoglyphs. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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