From BRYAN@wvnvm.wvnet.edu Fri Apr 14 17:08:12 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA27690; Fri, 14 Apr 95 17:08:12 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 3964; Fri, 14 Apr 95 17:06:54 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 0638; Fri, 14 Apr 1995 17:06:54 -0400 Message-Id: Date: Fri, 14 Apr 1995 17:06:48 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Repetitive Application of Elements of Q* Recall that Q* has been defined as the set of representatives Q* = {F*, R*, L*, B*, U*, D*, F'*, R'*, L'*, B'*, U'*, D'*} where * has been defined as a function selecting a representative element of an M-conjugacy class. I have done a little experimentation with cycles of the form X* ^ n. As long as the X* are directly in Q*, the sequences are quite short, and the final cycle is of length 2 in all cases. I found the latter surprising initially, but with the wisdom of hindsight, it was inevitable. Here is a table of lengths. Operation Length of Sequence F* 11 U* 10 L* 7 R* 3 D* 9 B* 2 F'* 7 U'* 10 L'* 4 R'* 6 D'* 7 B'* 2 A couple of points of clarification: 1. As an example, for F*, we take i(F*)^n (that is, apply F* to Start repeatedly). The sequence has 11 elements before it repeats, then the 10-th and 11-th element repeat over and over again. In all twelve cases, it is the last two elements of the sequence which repeat. 2. In order to replicate my results, you would have to define a representative element function exactly like mine. Every choice of representative element function can be expected to yield different results. To take a little more interesting case, I tried i(F*D'*) ^ n. In this case, there were 63 unique elements in the sequence, and then the 8-th through the 63-rd elements repeated. Hence, the final cycle had 56 elements rather than the 2 elements of the simpler cases. I suppose I could try quite a few other cases, but I have no idea how to predict how long the sequences or the terminal cycles might be. All I know to expect for sure is for things to be quite ill-behaved. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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