From JBRYAN@pstcc.cc.tn.us Tue Dec 19 17:41:31 1995 Return-Path: Received: from PSTCC4.PSTCC.CC.TN.US by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16455; Tue, 19 Dec 95 17:41:31 EST Resent-From: JBRYAN@pstcc.cc.tn.us Resent-Message-Id: <9512192241.AA16455@life.ai.mit.edu> Received: from pstcc.cc.tn.us by pstcc.cc.tn.us (PMDF V5.0-3 #11457) id <01HZ00GZU6XS8WY0RV@pstcc.cc.tn.us> for cube-lovers@ai.mit.edu; Tue, 19 Dec 1995 17:43:05 -0400 (EDT) Resent-Date: Tue, 19 Dec 1995 17:43:05 -0400 (EDT) Date: Tue, 19 Dec 1995 17:43:03 -0400 (EDT) From: Jerry Bryan Subject: Physical Cubes and Models Thereof Sender: JBRYAN@pstcc.cc.tn.us To: Cube-Lovers Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Content-Transfer-Encoding: 7BIT The general subject of physical cubes and mathematical models thereof has been discussed many times before, but I have never been totally satisfied with all of the conclusions. I'm going to take one more crack at it. Let's start with the question of what constitutes a single move and the argument between the quarter-turners and the half-turners. There are good and valid arguments on both sides of the question, and there is no one "right" answer. However, the strongest and most succinct argument in favor of quarter turns is that they are conjugate. In the case of the standard 3x3x3 cube, the set Q of twelve quarter turns is M-conjugate, where M is the set of 48 rotations and reflections of the cube. A quarter-turner would normally generate G as G=. But given that Q is M-conjugate, we could say equivalently that G=<{m'Xm | m in M}> for any X in Q. Question: for the 3x3x3, are there any elements X in G other than those X in Q itself where we can generate G as G=<{m'Xm | m in M}>? Remember that in most cases we would have 48 generators available. Clearly, there are X in G such that <{m'Xm | m in M}> does not generate G. For example, the M-conjugates of F2 do not generate G. But I have a feeling that any group that is generated by <{m'Xm | m in M}> is an "M-symmetric group" (using the term "M-symmetric" very loosely and informally) and is therefore a somewhat interesting group. For the 4x4x4, I will use upper case letters for outer slab moves (face moves) and lower case letters for inner slab moves. For example, L'l'rR would rotate the entire cube away from you by 90 degrees, but the cube would otherwise look unchanged. If we denote the set of outer slab moves as Q and the set of inner slap moves as q, then we can generate a group as G4=. I am hesitant to say that G4 is "the" cube group for the 4x4x4, because it is so hard to agree on what "the" cube group is for higher order cubes. But in any case, Q and q are not M-conjugate with each other. There is in fact no way to have M-conjugate generators for the 4x4x4 and higher physical cubes. For a mathematical model, conjugacy can be repaired. For example, there is an operation called Evisceration where inner slabs and adjacent parallel outer slabs are exchanged. There is also an operation called Inflection where inner slabs are exchanged with their parallel inner slabs, and Exflection where outer slabs are exchanged with their parallel outer slabs. We can use Rotations, Reflections, Evisceration, Inflections, and Exflections to generate a 192 element symmetry group for the 4x4x4 called M4. We can then show that Q and q are M4-conjugate, and conjugacy is repaired. That is, we can generate G4 as G4=<{m'Xm | m in M4}> for any X in q or Q. (See Dan Hoey's article "Eccentric Slabism, Qubic, and S&LM" dated 1 June 1983.) In the previous paragraph, I used the term "symmetry group" quite deliberately, although some of you may not agree with the way I used it. I am still struggling to understand how narrowly or loosely we should really construe the preservation of a geometric property before we declare a permutation to be a symmetry. In the case of M4 above, I think the designation of "symmetry" is warranted, although it is a looser interpretation than is typical. But my purpose is to model physical cubes. Evisceration is not possible on physical cubes. Conjugate quarter turn generators are not possible for physical cubes larger than the 3x3x3 without Evisceration (or its generalization to the NxNxN case). Therefore, we abandon M-conjugation and its generalizations as a criterion for modeling physical cubes. Dan's Eccentric Slabism article talked about slab moves (a single plane of cubies turning together) and cut moves (all the cubies on each respective side of a plane cut of the cube turning together). Evisceration convinced Dan to convert from a Cutist view of the cube to a Slabist view of the cube. But Dan fully endorsed the Slabist view only for even-sided cubes. His phrase "Eccentric Slabism" refers to the fact that he still refused to make slab moves for the center slabs of odd-sided cubes. The problem is that center slab moves break M-conjugacy and its generalizations. But I've already given up M-conjugacy and its generalizations. Given that, it seems unnatural to leave out the center slab moves, so we leave them in. We next confront the issue that physical cubes are rotated in space with abandon. Different rotations of physical cubes are considered to be equivalent, and/or rotations of physical cubes are considered to be zero cost operations. But we desire a mathematical model of a physical cube to be a group. My preferred non-computing model of this situation is to treat the various configurations of the cube as cosets of C, the set of 24 rotations of the cube. However, this model is awkward for computing. For something like the 2x2x2, we more typically do something like fixing a corner. We hereby adopt "fixing a corner" as the solution for the general NxNxN case. See below for more details of how we propose to do so in the general case. We can note several things about the "fixing the corner" model: 1. It breaks M-conjugation. But we gave up M-conjugation anyway. Consider the 2x2x2 as a good example. If we insist on treating different rotations as equivalent, then the 2x2x2 really isn't M-conjugate. I am simply suggesting that the NxNxN physical cube really isn't M-conjugate, no matter the value of N, if we treat different rotations as equivalent. 2. With the "cosets of C" model, we can make the cosets into a group by taking as a representative for each coset the unique element which fixes the same corner. There is then an easy isomorphism between the "cosets of C" model and the "fixed corner model". My only trouble with the "cosets of C" model is that I keep wanting to call it G/C, and you can't call it that. C is not a normal subgroup of G, and we cannot speak of G/C as a factor group of G. 3. We can have conjugation and we can have symmetry with a "fixed corner" model. It is just not M-conjugation. Rather, it is the symmetry that preserves the fixed corner, and conjugation within that symmetry group. The 4x4x4 is a good example of how we propose to "fix the corner" for the general NxNxN case. Consider our status after L'l'r. A physical cubist would say that you were only one move from Start, and would "solve" the cube simply with R. But R would yield L'l'rR, which would leave the cube rotated. This is fine for our physical cube, but not so fine for a mathematical model of a physical cube which seeks to fix a corner. Hence, we define R as R=Llr', and similarly for the other slab moves which would otherwise move the fixed corner. The generalization to cubes higher than 4x4x4 is obvious. Actually, I would prefer a slightly different but equivalent definition for those slab moves which fix a corner. Frey and Singmaster use script letters for whole cube moves (those moves in C). I would implement R as follows: perform R in the normal sense of the operation composed with Script-R' (and similarly for other slab moves that would move the otherwise fixed corner). So for the 4x4x4, let's suppose we fixed the TRF corner. Our generators would be, L',l',r,(R)(Script-R'), B',b',f,(F)(Script-F'), D',d',t,(T)(Script-T') and their inverses. Clearly, the same technique works not only for the 4x4x4 and above, but also for the 2x2x2 and for the 3x3x3. I am thinking of this in a Slabist interpretation. However, a case could be made that the (R)(Script-R') type of moves are really Cutist moves. I think all the other problems associated with a mathematical model of a physical cube can be unified under the heading "Invisible Moves of Facelets". The most obvious example is that the Supergroup is invisible on the 3x3x3 unless the orientations of the face centers are marked somehow or other. But with larger cubes (e.g., the face centers of the 4x4x4), it is not just changes in orientation that are invisible; there are also invisible changes in location. In all cases, I would propose initially modeling the "larger group" (call it L), where invisible changes in location and orientation are visible. Number all 16 facelets of each face on the 4x4x4, for example. You do have to decide how "large" you wish your larger group L to be. For example, to make invisible orientation changes visible, you have to give a facelet four numbers rather than just one. The set of all positions that are equivalent when the "invisible" changes are ignored is a subgroup K. Your final model is then the cosets of K in L. The "cosets of K in L" model will always work, but it may be difficult to deal with computationally. Ideally, you would be able to find a subgroup G of L for which you could find an easy isomorphism with the cosets of K. As an example, consider the Supergroup of the 3x3x3 and call it L. Within L, there is a subgroup K which fixes the corners and edges. K is just all the legal face center reorientations. Therefore, if we wish to ignore face center orientations our model can be the cosets of K in L. There is an easy isomorphism between the cosets of K in L, and our standard model for the 3x3x3 which we call G. In truth, we would never model the 3x3x3 in such a convoluted fashion. We would just use G and be done with it. But for the 4x4x4 and larger cubes, I am not sure there is any choice. As the cubes get larger, you would generally find that there was a nested sequence of subgroups -- K_0, K_1, etc. -- for which the cosets of K_n in the larger group L would produce a useful model. For example, on the 4x4x4 one of your K's might be the group of permutations that fixed everything but the positions of the center facelets within a face (keeping them the proper color, of course). But a more stringent K might be the group of permutations that fixed everything but the orientations of the center facelets within a face. I will end by pointing out that Goldilocks would really like the 3x3x3. Papa Bear's 4x4x4 is too large and Baby Bear's 2x2x2 is too small. But Mama Bear's 3x3x3 is just right. The physical 3x3x3 is the only physical NxNxN which can be modeled with M-conjugate generators (assuming we fix the face centers). And the 3x3x3 is the NxNxN (physical or mathematical) with the nicest isomorphism between the cosets of K in L and some reasonable group G. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us