From @mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU Sat Dec 18 17:08:38 1993 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 AA00538; Sat, 18 Dec 93 17:08:38 EST Message-Id: <9312182208.AA00538@life.ai.mit.edu> Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 5020; Sat, 18 Dec 93 14:04:37 EST Received: from WVNVM.WVNET.EDU (NJE origin MAILER@WVNVM) by MITVMA.MIT.EDU (LMail V1.1d/1.7f) with BSMTP id 8380; Sat, 18 Dec 1993 14:04:37 -0500 Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 0957; Sat, 18 Dec 1993 14:02:02 -0500 X-Acknowledge-To: Date: Sat, 18 Dec 1993 14:02:01 EST From: "Jerry Bryan" To: "Cube Lovers List" Subject: Second Addendum - Size of Cube Group under M I feel like I am pestering the list to death with corrections. I still believe that the figure that I proposed for the size of the cube group under M is correct. The first post included a "correct" but I think unsatisfactory explanation. The second post improved upon one point that was unsatisfactory in the first post. Now, let's see if I can get it completely correct. The size of the corner group under (my version of) M is known. The size of the edge group under (my version of) M is known as well. Let C be the size of the corner group, and E be the size of the edge group. Remember, the elements of the groups are equivalence classes induced by (my version of) M. Here is an incorrect formula for G, the size of the entire cube group under (my version of) M. G = (C*24) * (E*24) / 2 The division by 2 is introduced to account for parity between the corner group and the edge group. But the value for G produced by this formula is only half as big as it should be. The problem is that M induces equivalence classes based on both rotations and reflections, not just base on rotations. Hence, we are led to the following (still incorrect) formula: G = (C*24*2) * (E*24*2) / 2 As before, the division by 2 takes care of parity between the corner group and the edge group. In addition, the multiplication by 2 takes care of reflecting each group. But the value for G produced by this formula is twice as big as it should be. The problem is that while any corner rotation can occur with any edge rotation (subject to parity), you must either reflect both groups, or else reflect neither group. Thus, we have the following (correct) formula: G = ((C*24) * (E*24) / 2) * 2 The division by 2 takes care of parity between the groups, and the multiplication by 2 takes care of reflection of the two groups as a unit. If we wish, we can cancel the multiplication and the division to yield G = (C*24) * (E*24) This is the same formula I originally posted, and I did say in the original post that the division by 2 cancelled out. However, I think that this post provides a better explanation of the cancellation than did the original post. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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?