From @mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU Tue Jan 4 11:12:37 1994 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 AA23149; Tue, 4 Jan 94 11:12:37 EST Message-Id: <9401041612.AA23149@life.ai.mit.edu> Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 1951; Tue, 04 Jan 94 11:12:39 EST Received: from WVNVM.WVNET.EDU (NJE origin MAILER@WVNVM) by MITVMA.MIT.EDU (LMail V1.1d/1.7f) with BSMTP id 0432; Tue, 4 Jan 1994 11:12:39 -0500 Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 0016; Tue, 4 Jan 1994 11:10:03 -0500 X-Acknowledge-To: Date: Tue, 4 Jan 1994 11:10:02 EST From: "Jerry Bryan" To: "Cube Lovers List" Subject: Which is the Real Start? The net is so wonderful about answering questions, here are a few more: 1. Take a standard 3x3x3 Rubik's cube, and remove the corner and center labels to make an Edges Cube. (I am assuming that the underlying plastic is black. If the underlying plastic is white and one of the colors on the labels is also white, the Edges Cube is not so pretty). Scramble the cube. Give it to a cubemeister to solve. How will the cubemeister know if the cube is solved? In other words, how will the cubemeister distinguish Start from Pons Asinorum? One answer is that the cubemeister cannot. Unless the cubemeister saw the cube before it was scrambled, or unless the cubemeister was told which reflection of the colors was Start, there would be no way to tell. Another answer is that either one is Start -- that there are two Starts. However, if you like this answer, and if you identify the identity with Start, you are in the disquieting situation of having a group with two distinct identities (grin!). It is obvious that this problem does not arise if the labels are left on the centers. Almost as obvious is the fact that the problem does not arise if the labels are left on the corners, even if the labels are removed from the centers. The corner group cannot be turned inside out by a reflection as can be the edge group. 2. As silly as my second answer is, it leads to a second question. Just what is the 2x2x2 cube? Or more correctly, how do you know when it is solved? With any size of cube, if you restrict yourself to quarter-turns, by definition you cannot rotate the cube in space as a single operation. Yet, a simple quarter-turn sequence such as RL' does rotate the 2x2x2 cube because it is faceless. Is Start of the 2x2x2 operated on by RL' solved? If so, you can argue that the 2x2x2 has 24 Starts. Most people would not. They would argue that there is only one Start, and that 2x2x2 cubes that differed only by a rotation are equivalent. 3. Combining #1 and #2, I *think* that most people would argue that Start and Pons Asinorum on the Edge Cube are not equivalent, but that simple rotations of the 2x2x2 are equivalent. If I am correct about "most people", why? Is a rotation symmetry intrinsically a stronger or weaker symmetry than a reflection symmetry? 4. When I was first posting my results about the Edge Group, and particularly when it first began to sink in what the four equivalence classes with only 24 elements really were, I had a moment of panic. Since Start and Pons Asinorum differ only by a simple reflection, why had not my version of M-conjugation declared them to be equivalent? (I speak of "my version of M-conjugation", but the question is no different if you look at Dan Hoey's original M-conjugation). I think I know the answer, but I will leave the problem as an exercise for the student. Furthermore, I think the answer to #4 is really the same as the answer to #3. 5. What is a reflection, really? Here is an exercise to illustrate the question. Take two identically colored and oriented 3x3x3 cubes. On one, perform F and on the other perform F'. Examine the two cubes, plus their images in a mirror. Why are there four distinct cubes rather than only two? At one level of abstraction, the answer is simple. Of the four, one is not reflected, one is pre-reflected, one is post-reflected, and one is both pre- and post-reflected. Is this a sufficient answer, or is there something deeper? At this point, I can't help but note Martin Gardner's famous mirror question in Scientific American many years ago: why does a mirror reverse left and right but not up and down? 6. I found Dan Hoey's postings about the four special states of the Edge Group to be delightful. Some of the results were based on a computer search of the group, for example the fact that f(I)=(0,9,12,15) could only reasonably be determined from a computer search. However, the thought occurred to me that most of Dan's results were independent of the computer search, and I was curious precisely which results would stand without the search? For example, if we identified the group as being rectangular, would we be led to saying which of the four special states were diagonally opposed without the computer search? Without the search, I might be tempted to say that Start and Pons Asinorum were diagonally opposed. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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?