From BRYAN@wvnvm.wvnet.edu Sat Sep 9 09:52:21 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA24673; Sat, 9 Sep 95 09:52:21 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R3) with BSMTP id 1908; Sat, 09 Sep 95 09:51:57 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 1333; Sat, 9 Sep 1995 09:51:57 -0400 Message-Id: Date: Sat, 9 Sep 1995 09:51:56 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: A Proposal for a More General Definition of Symmetry Our normal definition of symmetry for Rubik's cube is based ultimately on the 48 symmetries of the standard math book wire model of a cube, and the 48 symmetries were discovered long before Rubik's cube was ever dreamed of. This note is based on the conviction that these 48 symmetries do not really capture all that we might think of as "symmetry" when we think of Rubik's cube. This note has the further purpose to propose a more general definition of symmetry for Rubik's cube. I want to start with a couple of really basic concepts. I think every reader of this list knows what a permutation is, namely it is a one-to-one onto function on a set. In the case of a finite set as we have with Rubik, a function on a set is one-to-one if and only if it onto, so we can sometimes get by with speaking only of one-to-one or by speaking only of onto. But what is a symmetry? A very standard definition is something like "the set of all rigid motions that transform a given geometric figure onto itself" (James and James Mathematics Dictionary). Another way to say it is that the transformation preserves the figure. Working with that definition, a symmetry almost inevitably may be interpreted as a permutation. With simple geometric figures, the permutation would usually be described as being a permutation on Euclidean n-space -- 2-space for a square or circle, etc., and 3-space for a cube or sphere, etc. Hence, we might think of a symmetry as being a special kind of permutation, namely one that preserves a geometric figure in Euclidean n-space. I have had a difficult time finding books that address the issue of symmetry vs. permutation to my satisfaction. It is very hard to think of a symmetry abstractly enough that it doesn't simply turn into a permutation right before your eyes. Paul Yale's "Geometry and Symmetry" doesn't really seem to define a symmetry (it sort of assumes you know what one is), but it does describe the relationship between a symmetry and a permutation. I would paraphrase as follows. Label your geometric figure in some fashion -- e.g., label the edges, label the axes, label the vertices, or label *something*. Then, there is a homomorphism between the set of symmetries and the corresponding set of permutations on the labels. But I repeat that it is hard for me to conceive of the set of symmetries in a sufficiently abstract fashion that the symmetries themselves aren't already permutations on *some* set or other. So it seems to me that Yale could just as well be talking about homomorphisms between one set of permutations and another set of permutations as talking about homomorphisms between symmetries and permutations. A couple of quick additional points, and then I will go on: 1) since we are talking about homomorphisms, it is obvious that both the set of symmetries and the set of permutations to which they map are groups, and 2) most homomorphisms between symmetries and permutations turn out in fact to be isomorphisms. This latter observation gives added weight to the notion that symmetries are just a special kind of permutation. Given all that has been said so far, we could informally say that the normal definition of a symmetry is that it is a permutation that preserves a geometric figure. Our more general definition will simply be that a symmetry is a permutation that preserves some property. If we were sufficiently liberal in our notion of "preserving some property", then most any permutation could be interpreted as a symmetry. We will not be quite that liberal by the time we are done, but we will be more liberal than would be permitted by the standard 48 math book symmetries of the cube. But what property of Rubik's cube should we try to preserve if we want a more general definition of symmetry than the normal one? I wish to motivate our definition of that property in several steps. The standard Rubik's cube definition of symmetry for a position X is Symm(X) is the set of all m in M such that X=m'Xm, or equivalently such that mX=Xm. M is the set of 48 permutations on the Rubik's cube corresponding to the 48 symmetries of a cube. Write a position Z as Z=XY, where X is the permutation on the corners and Y is the permutation on the edges. We have Symm(Z)=Symm(XY)=Symm(X) intersect Symm(Y). For example, we could have Symm(X)=M, Symm(Y)=I, and Symm(Z)=Symm(XY)=I. Such a position would look very "symmetrical" because the corners would be fixed (or "solved"), although the edges would be scrambled. Most people would consider such a position to be more "symmetrical" than one where both the corners and edges were scrambled, although we would have Symm(Z)=I in either case. A couple of points before proceeding: 1) From an information theory point of view, Symm(X) and Symm(Y) separately contain more information than does Symm(XY). There is an obvious loss of information when we calculate Symm(X) intersect Symm(Y). This is a strong indication that Symm(XY) does not tell us everything we might like to know about the symmetry of a position. 2) The set of positions Z=XY for which Symm(X)=M forms a group (as does the set of positions for which Symm(Y)=M). This anticipates where we are headed, namely that group membership is the property that we should seek to preserve in a more general definition of symmetry. A third (and equivalent) definition for Symm(X) is that Symm(X) is the set of all m in M such that X'm'Xm=I. Most readers will recognize X'm'Xm as a commutator. Per Dan Hoey, we can generalize and define CSymm(X) to be the set of all m in M such that X'm'Xm is in C, the set of 24 rotations of the cube. For example, if we have Z=XY as before, then CSymm(X)=M means that the corners are positioned properly with respect to each other, although they might be rotated with respect to the fixed face centers. Such a position would look fairly "symmetrical", even to a non-cubemeister, even though we might have Symm(Z)=I. Again, we have the set of all positions for which CSymm(X)=M forms a group. Similarly, the set of all positions for which CSymm(Y)=M forms a group, and the set of all positions for which CSymm(Z)=CSymm(XY)=M forms a group. Recall that there are 98 subgroups of M. For each subgroup K of M, there is a corresponding subgroup of G consisting of all the K-symmetric positions. So would could just as well define symmetry in terms of these 98 subgroups of G. But there are far more than 98 subgroups of G. (We don't know how many, and I doubt than even GAP could tell us). Why not simply define symmetry in G in terms of subgroup membership in G? The symmetry of a position X is then the set of all subgroups of H of G for which X is in H. And a symmetry operation (in the sense that a symmetry is a permutation that preserves something) is an operation that preserves subgroup membership. That pretty much completes my proposal, but I have a few closing remarks. 1) The proposed general definition of symmetry is analogous to the Thistlethwaite algorithm for solving the cube. Typical cube solutions gradually solve more and more of the cube. The "more and more of the cube" that gets solved can be characterized as a sequence of nested subgroups. Thistlethwaite reversed the process and created a sequence of nested subgroups which in turn solves more and more of the cube. Similarly, the standard definition of symmetry implies a set of 98 subgroups of G. We reverse the process and let all the subgroups of G define symmetry instead. 2) The proposed general definition of symmetry has the virtue that it includes the standard definition as a special case, since the 98 K-symmetric subgroups of G are in fact subgroups of G. 3) The proposed general definition of symmetry has the virtue that there is only one position that is "completely symmetric", namely Start itself (the identity permutation). The standard definition of symmetry has four positions which are "completely symmetric", which to me is an unsatisfactory state of affairs. (Recall that we have Symm(X)=M for Start, Pons Asinorum, Superflip, and the composition of Pons and Superflip. I am still bummed out that this is the case while at the same time only Start and Superflip are in the center of G. This suggests that Superflip is "more symmetric" than Pons. I wonder if such a suggestion would be supported by my proposed general definition 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