From @wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU Tue Jul 19 10:55:43 1994 Return-Path: <@wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU> Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA00780; Tue, 19 Jul 94 10:55:43 EDT Message-Id: <9407191455.AA00780@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 9928; Tue, 19 Jul 94 08:56:30 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 8616; Tue, 19 Jul 1994 08:56:30 -0400 X-Acknowledge-To: Date: Tue, 19 Jul 1994 08:56:28 EDT From: "Jerry Bryan" To: "Cube Lovers List" Subject: More on Centerless Cubes On 13 Feb 1994, I proposed a model for centerless cubes which I claimed met two criteria: 1) it was a group, and 2) it maintained the symmetrical nature of the problem. On 23 May 1994, I retracted the claim that the proposed model was a group. I am now of the opinion that it is impossible to satisfy both criteria simultaneously. I can make a very small modification to the proposed model to make it a group, but the small modification costs the model its cubic symmetry. G is the full cube group, GC is the corners only cube group, and GE is the edges only cube group. The proposed model for centerless cubes consisted of partitioning any of G, GC, or GE into sets of the form {Xc} for all c in C, where C is the set of twenty-four rotations of the cube and X is a cube. The sets are the elements of the proposed group. The sets are called cosets and can also be denoted as xC. The partitions are denoted as G/C, GC/C, and GE/C, respectively. Originally, the proposed group operator was {Xc} * {Yc} = {XYc}. This operator fails to maintain closure, and hence fails to define a group. In order to illustrate the slight modification which will define a group, we will start by restricting ourselves to GC. An operator which works to define GC/C as a group is {Xc} * {Yc} = {VWc}, where V is the unique element of {Xc} such that the urf cubie is properly positioned in the urf cubicle, and W is the unique element of {Yc} such that the urf cubie is properly positioned in the urf cubicle. Any other corner could have been used instead of urf, but once you choose a corner the problem loses its symmetric nature. Well, I guess it still has symmetry, but it is not the uniform symmetry of the cube any more, because there is a preferred orientation. I have found only limited discussion in the archives, but previous investigators have modeled a corners only, centerless cube by leaving one corner fixed. Such a model is clearly a group. For example, if we leave the urf corner fixed, we can generate the group JC as JC=, where we omit all twists of the U, R, and F faces from the set of generators. It is easy to find an isomorphism between GC/C and JC. I would express it as something like {Xc} = {Wc} <--> W, where W is defined as before. W is an element of JC, and as well is an element of {Xc} = {Wc}. {Xc} = {Wc} is an element of GC/C. But W is a particular element of {Xc} = {Wc}, whereas X is an arbitrary element. Also, X is in GC, but X is not in JC unless X = W. The mapping {Wc} <--> W is clearly one-to-one and onto in both directions. For the edges GE, we need to keep one edge cubie fixed, so the centerless cube could be generated by something like JE=, where we keep the uf cubie fixed by omitting all twists of the U and F faces from the set of generators. The isomorphism between GE/C and JE is expressed as {Xc} = {Wc} <--> W, where X is an arbitrary element of GE, and W is the unique element of {Xc} such that the uf cube is properly placed in the uf cubicle. As before, any edge cube would do as well, but once chosen, it is no longer arbitrary. For the whole cube G, at first blush it appears we could model centerless cubes either by keeping a corner cubie fixed, or by keeping an edge cubie fixed. But if we keep a corner cubie fixed, the three immediately adjacent edge cubies are never moved by any Q-turns. We could solve the difficulty by admitting slice turns. But slice quarter-turns are odd on edges and even on corners, so we have to restrict ourselves to slice half-turns. I find this ugly, plus I would prefer to generate G with Q-turns only. Hence, I would prefer to model a centerless full cube as J=, where it is an edge cubie which is held fixed rather than a corner cubie. I said at the beginning that I thought it was impossible for a model of centerless cubes both to be a group and also to maintain cubic symmetry. The reason is as follows: it seems to me that for any model which is a group, it should be possible to find an isomorphism between the model and J (or JC or JE, as appropriate). But J and JC and JE do not have cubic symmetry because there is a preferred orientation. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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?