From BRYAN@wvnvm.wvnet.edu Wed Jan 11 14:01:19 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25989; Wed, 11 Jan 95 14:01:19 EST Message-Id: <9501111901.AA25989@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 2905; Wed, 11 Jan 95 13:53:57 EST Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 2283; Wed, 11 Jan 1995 13:53:31 -0500 X-Acknowledge-To: Date: Wed, 11 Jan 1995 13:53:30 EST From: "Jerry Bryan" To: Subject: Re: Re: Cube with GAP In-Reply-To: Message of 01/08/95 at 13:47:00 from , Martin.Schoenert@Math.RWTH-Aachen.DE On 01/08/95 at 13:47:00 Martin Schoenert said: >Jerry Bryan wrote in his e-mail message of 1995/01/07 > Note that this model does not include the face centers. That is, it > is G[C,E] rather than G[C,E,F]. 56 numbers would be required to > include the face centers. The distinction between 48 facelets and > 56 facelets bears on the nitpicky question of whether the set C of > rotations is a subgroup of G or not. >Absolutely right. This part of the GAP documentation was written years >ago. These days I represent MG, CG, etc. as permutation groups on 54 >points. I also changed the numbering, so that the [1..24] represent the >edges, [25..48] points represent the edges, and [49..54] represent the >centers. Martin was kind enough not to point out that 6*9 is 54 rather than 56. > When I write the model out to disk, I only write out 8 corner facelets > and 12 edge facelets. For example, I only write out the front and > back corner facelets. This saves space and converts the model from > a facelet model to a cubie model, with the twists implicitly encoded > rather than being explicitly encoded via multiplication tables. It > also automatically establishes a frame of reference by which a > proof of conservation of twist and flip can be accomplished. >In terms of computational group theory this sequence of 8 corner and >12 edgde facelets is called a *base* for the permutation group G. >That is, each element of the group is uniquely determined by the >images of those 20 facelets. Of course if you have already proved >that no single corner can be twisted and no single edge can be flipped, >you can reduce this to 7 corner and 11 edge facelets. With some of my models it can be reduced slightly more. Since representative elements of equivalence classes are what are being stored, the representative element can be chosen to fix a cubie of your choice. The cubie which is being fixed need not be stored. This trick does not work, of course, if all cube positions are being stored rather than just representative elements. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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