From BRYAN@wvnvm.wvnet.edu Tue May 9 09:12:52 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA12086; Tue, 9 May 95 09:12:52 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 3544; Tue, 09 May 95 08:48:24 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 1219; Tue, 9 May 1995 08:48:24 -0400 Message-Id: Date: Tue, 9 May 1995 08:48:23 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: more on the slice group In-Reply-To: Message of 04/14/95 at 16:03:31 from mreid@ptc.com On 04/14/95 at 16:03:31 mreid@ptc.com said: >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.) Results for the slice group under M-conjugacy: Level Number of Number of Positions Local Maxima 0 1 0 1 1 0 2 2 0 3 6 0 4 16 0 5 15 1 6 9 9 Mike's conjecture that all 24 positions which are a local maxima at level 5 are equivalent under M-conjugation is correct. I don't yet understand why Mike's position is a local maximum in the full cube group. But assuming it is, it is not only the shortest local maximum, it is the first local maximum which is not Q-transitive (i.e, we have |{m'Xm}|=24, hence we have |Symm(X)|=2, and the size of the symmetry groups for Q-transitive positions must be divisible by 12.). = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU