From BRYAN@wvnvm.wvnet.edu Sat Jun 17 00:35:23 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23622; Sat, 17 Jun 95 00:35:23 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 9876; Sat, 17 Jun 95 00:35:32 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 4531; Sat, 17 Jun 1995 00:35:32 -0400 Message-Id: Date: Sat, 17 Jun 1995 00:35:31 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: 10q Local Maxima Positions 1. (UU) (F'B')(RL)(RL)(FB) 2. (UD') (F'B')(RL)(RL)(FB) 3. (UD) (F'B')(RL)(RL)(FB) 4. (UD')(FB')(LR')(FB')(FB') #4 is the one Mike Reid already found in the slice group. #3 is the one I already found in the antislice group. #1, #2, and #3 are obviously closely related. #1 and #2 appear not to be in either slice or antislice, but I have been fooled before by alternative sequences which yield the same position. #1, #2, and #3 all have the property that |{m'Xm}|=6 and |Symm(X)|=8. As has already been discussed, #4 has the property that |{m'Xm}|=24 and |Symm(X)|=2. The symmetry groups for #1, #2, and #3 are of a type Dan Hoey's taxonomy calls class P, class S, and class AX, respectively. These particular classes are hard to describe succinctly without introducing a lot of notation. But in all three cases, the symmetry groups (subgroups of M such that X=m'Xm} consist of four rotations and four reflections, and have as an axis of symmetry one of the three major axes of the cube (U-D, F-B, or R-L). There are three groups P1, P2, P3 with axis of symmetry U-D, F-B, and R-L, respectively, and similarly for S1, S2, and S3, and for AX1, AX2, and AX3. For #4, we have Symm(X)=HV in Dan's taxonomy, where HV={i,v}, and where i is the identity in M and v is the central inversion in M. If proper typography were available, the i and the v would be upper case script letters to follow Frey and Singmaster. There are relatively few positions in all of cube space for which Symm(X)=Pi or Symm(X)=Si or Symm(X)=AXi (i in 1..3). There are only 10 P positions through level 10 in the search tree (of which just one is a local maximum). There is only one S position through level 10, and only one AX position through level 10, both of which are of course local maxima. The positions are not Q-transitive, but the positions look "symmetric", and they fulfill the (incorrect) intuition that "symmetric" positions must be local maxima. We have no reason to say that other P or S or AX positions further down the search will be local maxima. I find position #4 extremely intriguing. In general, HV is not very strong symmetry, and there are relatively speaking, quite a few HV positions in cube space. We could create an HV position as follows. Put any edge cubie anywhere (say UF in RD). Put the "opposite" cubie in the "opposite" cubicle (DB in LU in this case). Continue for the remaining edge cubies, and then do the same thing for the corners, remembering only to make sure the edges and corners have the same parity. You can easily make an HV position that looks quite "random" to the casual glance, and in fact most HV positions don't look very "symmetric". But Mike's position looks very "symmetric" at a casual glance, as if its symmetry must be much stronger than HV. I certainly would not have expected to find an HV position as a local maximum close to Start. I think the "look" of Mike's position as "symmetric", and the fact that it is a local maximum close to Start are related. Without getting too long winded, I think the reasons are two-fold. First, the corners and edges have much stronger symmetry separately than they do collectively. Second, the symmetry looks much stronger if you ignore the centers (i.e., if you ignore the rotational positioning of the cubies), perhaps in the sense of Dan's CSymm function. For example, the corners are properly positioned with respect to each other, even though they are not properly positioned with respect to the fixed face centers. In the next few days, I intend to calculate Symm(X) for the corners and edges separately for Mike's position, as well as calculating CSymm(X) for the corners and edges separately and combined. I think the results will be enlightening. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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