From @mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU Mon Jan 17 09:09:39 1994 Return-Path: <@mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU> Received: from mitvma.mit.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA27830; Mon, 17 Jan 94 09:09:39 EST Message-Id: <9401171409.AA27830@life.ai.mit.edu> Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 0725; Mon, 17 Jan 94 09:09:39 EST Received: from WVNVM.WVNET.EDU (NJE origin MAILER@WVNVM) by MITVMA.MIT.EDU (LMail V1.1d/1.7f) with BSMTP id 6845; Mon, 17 Jan 1994 09:09:37 -0500 Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 5415; Mon, 17 Jan 1994 09:06:59 -0500 X-Acknowledge-To: Date: Mon, 17 Jan 1994 09:06:59 EST From: "Jerry Bryan" To: "Cube Lovers List" Subject: Number of M-Conjugate Classes for GC\M On 4 December 1993, I posted results from a breadth-first exhaustive search of GC\M, the corners of the 3x3x3, reduced by M-conjugation. The posting included a summary of how many conjugate classes there were at each level of the search tree (i.e., distance from Start). It occurred to me that I had not also posted a summary of M-conjugates for the corners of the 3x3x3 by the size of the conjugate classes. I searched my records, only to discover that I had never calculated such sizes. If I had, I probably would have been forced to analyze properly the distinction between M-conjugation and B-conjugation, because B-conjugation makes no sense for the corners of the 3x3x3. B-conjugation *can* be performed for the corners of the 3x3x3, but you end up with the 2x2x2 instead because B-conjugation effectively removes the centers. Anyway, I have now calculated M-conjugate class sizes for GC\M via computer search, and here are the results. M-Class Number Size of Classes 1 1 2 1 3 3 4 1 6 34 8 33 12 301 16 104 24 9064 48 1832428 Total 1841970 Notice that with M-conjugation, the maximum class is size is 48, rather than 1152 as it is with B-conjugation. Hence, my posting of 4 December 1993 incorrectly identified the results as being for "1152 fold symmetry". The results are correct, but they should be labeled as being for "48 fold symmetry", i.e., for M-conjugation rather than for B-conjugation. In calculating M-conjugate class sizes for GC\M, I did not "start from scratch". Rather, I used the existing results for B-conjugate classes as a base. In the case of B-conjugate classes of order 1152, no calculations are required. Each such B-class can simply be partitioned into 24 M-classes of order 48. Hence, I had to perform calculations for less than 4% of the B-classes. Here is a summary matrix, showing for each B-class size the number of each M-class size which are derived. M-Class Size 1 2 3 4 6 8 12 16 24 48 Total 24 1 0 1 0 2 1 0 0 0 0 5 B-Class 48 0 1 0 0 1 2 2 0 0 0 6 Size 72 0 0 2 0 11 0 2 0 5 0 20 96 0 0 0 1 0 1 3 0 2 0 7 144 0 0 0 0 20 0 42 0 30 14 106 192 0 0 0 0 0 29 0 8 73 16 126 288 0 0 0 0 0 0 252 0 682 406 1340 384 0 0 0 0 0 0 0 96 0 224 320 576 0 0 0 0 0 0 0 0 8272 22360 30632 1152 0 0 0 0 0 0 0 0 0 1809408 1809408 Total 1 1 3 1 34 33 301 104 9064 1832428 1841970 The first row of the matrix exemplifies the process of calculating M-Class sizes from B-Class sizes. In the case of corners, there is only one B-class of order 24, namely Start. The 24 elements of the B-class are the 24 elements of the form Ic, where c is in C, the 24 rotations of the cube. Under B-conjugation, these 24 elements are equivalent (i.e., in a centerless cube such as the 2x2x2, the 24 rotations of I are indistinguishable). But in a cube with centers, such as the corners of the 3x3x3, the 24 elements are not equivalent. For example, the M-class of order 1 is {I}. One of the M-classes of order 6 is {FB', UD', RL', LR', BF', DU'}. The M-class of order 3 is {FFB'B', RRL'L', UUD'D'}. That's as many as I can do in my head, but I think the pattern is clear. M-classes are a partition of the B-classes. In the case of B-classes of order 1152, the partition is regular -- i.e., you get exactly 24 M-classes of order 48. However, all partitions are not regular. In the partition of the B-class of I which we just discussed, there is 1 M-class of order 1, 1 M-class of order 3, 2 M-classes of order 6, and 1 M-class of order 8, for a total of 24 M-classes. Many other partitions are not regular, as well. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 If you don't have time to do it right today, what makes you think you are going to have time to do it over again tomorrow?