From mreid@ptc.com Fri Apr 14 17:30:05 1995 Return-Path: Received: from ptc.com (poster.ptc.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA29204; Fri, 14 Apr 95 17:30:05 EDT Received: from ducie by ptc.com (5.0/SMI-SVR4-NN) id AA11438; Fri, 14 Apr 95 17:22:59 EDT Received: by ducie (1.38.193.4/sendmail.28-May-87) id AA04391; Fri, 14 Apr 1995 16:03:31 -0400 Date: Fri, 14 Apr 1995 16:03:31 -0400 From: mreid@ptc.com (michael reid) Message-Id: <9504142003.AA04391@ducie> To: cube-lovers@ai.mit.edu, mark.longridge@canrem.com, CRSO.Cube@canrem.com Subject: more on the slice group Content-Length: 2394 mark's post got me thinking ... i made a quick hack for the slice group (which is easy to represent by fixing the corners). my figures concur with his. i wanted to see the number of local maxima. 90 degree number of number of slice turns positions local maxima 0 1 0 1 6 0 2 27 0 3 120 0 4 287 0 5 258 24 6 69 69 as i'd hoped, there are local maxima at distance 5. one such is: (FB') (RL') (U'D) (R2L2) = (R2L2) (F'B) (RL') (UD') = (R'L) (FB') (RL') (F'B) (U'D) = (U'D) (F'B) (RL') (U'D) (F'B) = (R'L) (UD') (F'B) (RL') (FB') (actually i think all are equivalent to this one under symmetries of the cube.) this is especially interesting because it is a local maximum in the full cube group (quarter turn metric) at distance 10q. according to jerry bryan's results, there are no local maxima within 9q of start, so this gives the closest local maximum. (there may well be others.) i also calculated for the other slice metric. in this metric, neighbors can have the same distance from start, so a "strong" local maximum is a position all of whose neighbors are strictly closer to start. a "weak" local maximum is a position none of whose neighbors are further from start. 90 or 180 degree number of number of strong number of weak slice turns positions local maxima local maxima 0 1 0 0 1 9 0 0 2 51 0 0 3 247 0 7 4 428 0 212 5 32 8 32 the strict local maxima are all equivalent under symmetries of the cube. they are the composition of pons asinorum with any of the eight positions called "six dots". in the same way, local maxima (within the antislice group) in the 90 degree antislice metric are local maxima in the full cube group (quarter turn metric). perhaps mark will tell us more about this. mike