From BRYAN@wvnvm.wvnet.edu Sat Jan 7 11:09:27 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16758; Sat, 7 Jan 95 11:09:27 EST Message-Id: <9501071609.AA16758@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 3760; Sat, 07 Jan 95 10:15:53 EST Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 2792; Sat, 7 Jan 1995 10:15:53 -0500 X-Acknowledge-To: Date: Sat, 7 Jan 1995 10:15:52 EST From: "Jerry Bryan" To: Subject: Re: Cube with GAP In-Reply-To: Message of 01/06/95 at 23:51:00 from mark.longridge@canrem.com On 01/06/95 at 23:51:00 mark.longridge@canrem.com said: > As I understand it, the format Martin uses in GAP is to represent >the 3x3x3 cube by assigning each individual facelet an unique >number like so (by the way, the following part is all from the >GAP documentation). >---------------------------------------------------------------------- > +--------------+ > | 1 2 3 | > | 4 top 5 | > | 6 7 8 | > +--------------+--------------+--------------+--------------+ > | 9 10 11 | 17 18 19 | 25 26 27 | 33 34 35 | > | 12 left 13 | 20 front 21 | 28 right 29 | 36 rear 37 | > | 14 15 16 | 22 23 24 | 30 31 32 | 38 39 40 | > +--------------+--------------+--------------+--------------+ > | 41 42 43 | > | 44 bottom 45 | > | 46 47 48 | > +--------------+ 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. What I don't see is how to model the Supergroup in GAP. It looks like you would have to label each Face center with four numbers so you could see the rotations of the Face centers, but that seems like overkill. I call this kind of model a facelet model rather than a cubie model, and the twists and flips are implicit in a facelet model. I would think that the twists and flips would have to be made explicit in a cubie model. Dan Hoey reported to me once that he had an error wherein his corners turned themselves inside out. I can't totally picture how that happened, but it was related to the fact that he was using a cubie model with a little multiplication table for the twists. I have always used a facelet model, except that I number the corners from 1 to 24 and the edges from 1 to 24 for historical reasons. That is, I started with corners only or edges only, and have only lately put the two together. It really does not create any problems for me to use the same numbers for both edges and corners because the edges and corners are stored disjointly, as are the edge and corner permutations for quarter and half turns, and as are the edge and corner permutations for rotations and reflections. 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. > ... I don't know how a normal 4x4x4 could >be represented though. I fail to see the problem. Just number the facelets. The only problem would then lie in deciding what the generators are -- i.e., which kind of slice moves do you accept. You would also have to decide whether to model the invisible 2x2x2 inside, but again if you did, just number the invisible facelets and include their movements with your generators. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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