From @mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU Fri Dec 17 00:56:31 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 AA11504; Fri, 17 Dec 93 00:56:31 EST Message-Id: <9312170556.AA11504@life.ai.mit.edu> Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 0779; Fri, 17 Dec 93 00:56:36 EST Received: from WVNVM.WVNET.EDU (NJE origin MAILER@WVNVM) by MITVMA.MIT.EDU (LMail V1.1d/1.7f) with BSMTP id 9437; Fri, 17 Dec 1993 00:56:36 -0500 Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 3544; Fri, 17 Dec 1993 00:54:02 -0500 X-Acknowledge-To: Date: Fri, 17 Dec 1993 00:54:00 EST From: "Jerry Bryan" To: "Cube Lovers List" Subject: Some Additional Distances in the Edge Group It is now known that using the qturn metric, Start has a unique antipode in the edge group, namely Mirror-Image- of-Edges-Flipped. The antipode is 15 qturns from Start. Also, I have a complete data base of equivalence classes in the edge group documenting the distance from Start for each configuration of the edges. It seems to me that given these two facts, some additional distances can be determined. For example, it is possible to determine the distance from any configuration to Mirror-Image-of-Edges-Flipped. Let Z be a sequence of operators that converts Start to Mirror-Image-of-Edges-Flipped, and let A be any configuration of the edges. Then apply Z' to A, look up the result in the data base of distances from Start, and that will be the distance from A to Mirror-Image-of-Edges-Flipped. The reason is quite simple. Let P be a sequence which takes Z'(A) to Start. Then, Z'PZ takes A to Mirror-Image-of-Edges-Flipped. This is a very nice use of conjugates. Another consequence of this result is the following: suppose you began with Mirror-Image-of-Edges-Flipped and performed a breadth-first exhaustive search. Start would be antipodal, and the number of nodes at each level of the tree would be identical to the existing tree which begins at Start. In addition, all of the above applies to Mirror-Image-of-Start and Edges-Flipped with respect to each other. They are mutually antipodal, and are 15 qturns apart. A tree built with either at the root would have exactly the same number of nodes at each level as the existing tree with Start at the root. Finally, the distance of any configuration from Mirror-Image-of-Start or Edges-Flipped can be determined. Let Y be a sequence of operators which converts Start to Mirror-Image-of-Start, and let X be a sequence of operators that converts Start to Edges-Flipped. Let A be any cube. Then, the distance of A from Mirror-Image-of-Start is the same as the distance of Y'(A) from Start, and the distance of A from Edges-Flipped is the same as the distance of X'(A) from Start. I have the sensation in describing this that the Edge group is square, with Start and Mirror-Image-of-Edges-Flipped 180 degrees apart, and Mirror-Image-of-Start and Edges-Flipped at the other two corners of the square. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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?