From BRYAN@wvnvm.wvnet.edu Sat Dec 10 10:13:39 1994 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA10971; Sat, 10 Dec 94 10:13:39 EST Message-Id: <9412101513.AA10971@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 5452; Sat, 10 Dec 94 10:13:41 EST Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 1045; Sat, 10 Dec 1994 10:13:42 -0500 X-Acknowledge-To: Date: Sat, 10 Dec 1994 10:13:33 -0500 (EST) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Cubic Symmetry of the 2x2x2 (Again) The argument has been made that the 2x2x2 cube (or really any 2Nx2Nx2N) cube cannot have the "symmetry of the cube". In order for a real 2x2x2 cube to have the "symmetry of the cube", you would have to adopt unreasonable rules, such as no rotations (or if you use rotations they have a cost of at least 2) and the cube is only solved when the colors are oriented properly. But a 2x2x2 cube certainly feels like a real cube when you hold it in your hands. I offer the following interpretation that "sort of" gives the 2x2x2 cube the symmetry of the cube. Since I will only be talking about the 2x2x2, I will simplify the notation by talking about C, G, Q, etc. rather than C[C], G[C], Q[C], etc. As I have discussed several times before, my favorite model for the 2x2x2 is G/C (or CG/C, if you prefer; G=CG for the 2x2x2). The group operation is (xC)(yC)=(uv)C, where u and v are the elements of xC and yC, respectively, which fix a particular corner. (xC)(yC)=(xy)C doesn't work because C is not normal. There is an obvious isomorphism between G/C and , where the three q-turns are those which fix the same corner as the selection function for u and v. There are eight corners, and hence there are eight conjugate selection functions and eight conjugate subgroups G_m of the form which fix a particular corner. If you think of mapping G/C simultaneously and in parallel to the all the elements in the set {G_1, G_2, G_3, G_4, G_5, G_6, G_7, G_8}, then in a loose sense you have preserved the cubic nature of the problem. That is, none of the individual G_m's have the same symmetry as the cube, but in a loose sense the entire collection {G_m} does. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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