From @mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU Wed Dec 8 14:42:11 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 AA19673; Wed, 8 Dec 93 14:42:11 EST Message-Id: <9312081942.AA19673@life.ai.mit.edu> Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 5941; Wed, 08 Dec 93 14:11:24 EST Received: from WVNVM.WVNET.EDU (NJE origin MAILER@WVNVM) by MITVMA.MIT.EDU (LMail V1.1d/1.7f) with BSMTP id 1649; Wed, 8 Dec 1993 14:11:24 -0500 Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 9868; Wed, 8 Dec 1993 14:08:47 -0500 X-Acknowledge-To: Date: Wed, 8 Dec 1993 14:08:15 EST From: "Jerry Bryan" To: "Cube Lovers List" Subject: 1152-fold Symmetry and 24-fold Symmetry I guess it's time to try to explain what I mean by 1152-fold symmetry and 24-fold symmetry. Let me start with two or three very simple ideas. First, consider two equally colored and oriented cubes at Start. To one of them, apply F, and to the other one apply R. The obvious solution to the first one is then F' and the obvious solution to the second on is then R'. But take both cubes and toss them through the air to someone else, so that the spatial orientation is lost. Almost anyone would solve either cube by finding the one face that was twisted clockwise and simply twisting it counter-clockwise. No distinction between F and R would be made, and it would be "obvious" to any reasonable person that the cubes were equivalent. As a slightly more formal application of this idea, consider again Start to which R has been applied. We could rotate the whole cube in space using Singmaster's script-U operation. That is, grasp the Up (top) of the cube and turn the whole cube in space clockwise. Now, the cube looks like F has been applied rather than R, and the solution looks like F' rather than R'. If we applied F', the cube would be solved, but the colors would be oriented wrong. We could restore the colors by script-U'. Thus, (script-U F' script U') is exactly the same thing as R' (we are just using conjugates in a very simple way). Continuing in this vein, take any two equally colored and oriented cubes at Start. To one of them, apply some long sequence of permutations P. To the second one, apply (script-U P script-U'). At this point, the two cubes are definitely not "equal" in some sense -- you could clearly tell them apart. Yet, they are clearly "equivalent" in some sense, because if P' is a solution to the first cube, then (script-U P' script-U') is a solution to the second one. Furthermore, it should be obvious that it is not really necessary to use the (script-U script-U') conjugate on the second cube. Rather we can think of some rotation as having been performed on P to give Q, and then of Q as having been performed on Start, so that the same rotation that was applied to P could be applied to P' to give Q', and Q' is equivalent to (script-U P' script-U'). If I can wax sophomorically philosophical for a minute, I tend to think of there being two kinds of permutations in mathematics. The first is the "permutations and combinations" kind of thing you run into in probability and statistics. The second is the permutation operator kind of thing you run into in abstract algebra or group theory. With this kind of thinking, the cube itself represents the first kind of permutation, where the cube is an object being operated on, and the twists of the cube are the second kind of permutation, where the twists are permutation operators and are doing the operating. Well, at some deep level, the two kinds of permutations are very much the same thing, so it is very natural to think of operating on (rotating, in this particular case) a permutation P, where P itself is an operator. I need one more simple idea. Again, think of a cube in Start, and think of Singmaster's script-U operator. We can (informally) write script-U = (Front --> Left --> Back --> Right --> Front). But suppose the cube is colored as Font=Red, Left=White, Back=Orange, Right=Blue). We could just as well write script-U = (Red --> White --> Orange --> Blue --> Red). It looks as if for any fixed coloring, we can freely interchange Singmaster's notation with a notation based on colors. But we can't really. For example, colored as I described it above, script-F is equivalent to script-Red. Either is the same as grasping the front of the cube and rotating the whole cube clockwise. But first perform script-U. Now, script-F is the same as script-Blue). The Front/Back/Up/Down/Left/Right notation is fixed in space, but the color notation is not. Now, we try to put all this together. Once again, consider two equally colored and equally oriented cubes in space, and apply F to the first one and (R script-U) to the second one. Both cubes can now be described as "Start with the front twisted clockwise by 90 degrees), but the colors are not the same. They are clearly equivalent, but under my internal computer model for the cube, they are not equal. Furthermore, no amount of additional application of Singmaster's whole cube "script" operators will make them equal. The only thing that will make them equal will be to rotate the colors. The exact same thing applies to reflections. Consider two equally colored and oriented cubes in Start, and apply F to one and F' to the other. The cubes are equivalent but not equal. Hold up the cube to which F' has been applied to a mirror. The mirror-image now has F applied instead of F', but the colors are wrong (they have been reflected). To make the cubes equal, it is necessary to reflect the colors of the mirror-image. Hence, my program generates equivalence classes by applying a cube rotation, a color rotation, a cube reflection, and a color reflection. There are 24 cube rotations and 24 color rotations (one of each is the identity), and any cube rotation can occur with any color rotation. There are 2 cube reflections and 2 color reflections (one each is the identity), but the cube reflection identity must occur with the color reflection identity and vice versa. Thus, there are (in general) 24x24x2 elements in each equivalence class. I only store one element of each equivalence class in my data base. Some of the equivalence classes have fewer than 24x24x2 elements, namely those for which the cube configuration inherently has a high degree of symmetry. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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?