From @mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU Sat Jan 8 10:21:12 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 AA26746; Sat, 8 Jan 94 10:21:12 EST Message-Id: <9401081521.AA26746@life.ai.mit.edu> Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 4208; Sat, 08 Jan 94 08:48:56 EST Received: from WVNVM.WVNET.EDU (NJE origin MAILER@WVNVM) by MITVMA.MIT.EDU (LMail V1.1d/1.7f) with BSMTP id 0001; Sat, 8 Jan 1994 08:48:56 -0500 Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 7240; Sat, 8 Jan 1994 08:46:21 -0500 X-Acknowledge-To: Date: Sat, 8 Jan 1994 08:46:20 EST From: "Jerry Bryan" To: "Cube Lovers List" Subject: Some Terminology Concerning B I have started to use "B" to indicate various aspects of the conjugacy class generated by m'Xmc. The choice of B is sort of an accident. I used "B" in the program fragment which I posted to the list, and Dan Hoey analyzed the program fragment. I have called it "B" in my mind ever since. However, I have used B in several inconsistent ways. This is a proposal to rectify that inconsistency. Let X be any cube. Then the set of B-conjugacy classes of X is the set of all m'Xmc for all m in M and all c in C. We denote this set as BClass(X). B is the function B(X)=min(BClass(X)). Note that we could have defined BClass(X) equivalently as the set of all mXm'c, or as the set of all cm'Xm, or as the set of all cmXm'. It is in general not the case that m'Xmc = mXm'c = cm'Xm = cmXm' for any fixed value of m and c. (Quite the contrary!). However, when we say "the set of all...", the four ways of generating BClass(X) become equivalent. This is the justification for the assertion in a previous note that Gx\B = (Gx\M)\C = (Gx\C)\M. Two cubes X and Y are B-equivalent if BClass(X) = BClass(Y). Equivalently, two cubes X and Y are B-equivalent if B(X) = B(Y). |X| is the length of X (the distance of X from Start). We have |B(X)| = |X| for centerless cubes, but it is generally not the case that |B(X)| = |X| for cubes with centers. In fact, let X and Y be cubes with centers such that B(X)=B(Y). It is not necessarily the case that |X| = |Y|. For example, consider the set GC of cubes with corners with centers without edges. We have B(RL')=B(I), but |RL'|=2 and |I|=0. |BClass(X)| is the number of elements in BClass(X). If |BClass(X)| = N, then X is said to have order-N symmetry. (I sincerely regret ever using this terminology. As has been noted on the list, it seems "backwards" somehow. But given that this usage exists, the value 1152/N is generally more useful than the value N.) We note the following: 1. B(X) is a cube. 2. BClass(X) is a set of cubes. 3. B(B(X)) = B(X) 4. BClass(B(X)) = BClass(X). 5. Both X and B(X) are in BClass(X). = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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?