From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 24 15:32:46 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id PAA02566; Mon, 24 Aug 1998 15:32:45 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Sun, 23 Aug 1998 23:19:35 -0400 (EDT) From: Jerry Bryan Subject: "Basic" vs. Superflipped Positions To: Cube-Lovers Message-Id: I recently commented on the fact that half of Mike Reid's X-symmetric positions were "basic" and the other half were superflipped versions of the first half. I further commented that this was true for all positions associated with any symmetry group -- half are "basic", and the other half are superflipped versions of the first half. Well, I am not quite sure that this is true in general. Or more correctly, I am not sure you can always tell the "basic" version apart from the superflipped version. Consider any two positions x and xf, where f is the superflip. We would say that x is the "basic" position and xf is the superflipped position. But if we define y=xf, then y is the "basic" position and yf is the superflipped position, and it is also true that x=yf. So which is the "basic" position, x or y? It appears that there is no way to tell. Yet when you look at X-symmetric positions, it is trivial for the eye to see which ones are "basic", and which ones are superflipped. So what is going on here? I briefly (*very* briefly) hoped to find a unique subgroup H of G with index 2 which did not contain superflip. Then, it would have been natural to call H the "basic" positions and Hf the superflipped positions. But it is well known to Cube-Lovers that the only subgroup of G with index 2 is the subgroup consisting of those positions where are an even number of quarter turns from Start. And this subgroup does contain the superflip. Therefore, there seems to me to be little possibility of a general way to distinguish between "basic" positions and superflipped positions. Upon further reflection, it seems to me that there is a natural way to tell "basic" positions apart from superflipped positions for some symmetry groups but not for others. I have not examined all 98 symmetry groups (33 symmetry classes) of the cube in this regard, but I have looked at a few of them, and can give a few examples. Before looking at examples, we need to look at a subtle but important point. We may think of a position x as consisting of corners and edges separately, so that x=x[c]*x[e]. Similarly, we may look at the symmetry of the corners and the edges separately, as in Symm(x[c]) and Symm(x[e]). The equation that relates the symmetries is Symm(x)=Symm(x[c]*x[e])=Symm(x[c]) intersect Symm(x[e]). But because Superflip affects only the edges, we need consider only Symm(x[e]) when we compare "basic" positions to superflipped positions. Example 1. Suppose Symm(x[e])=M. Then it seems natural to view the position as "basic" if all four edge facelets on each face are the same color, and to view the position as superflipped otherwise. The eye sees this distintion very clearly. Example 2. Suppose Symm(x[e])=X1. X1 is the symmetry group in Dan Hoey's taxonomy which preserves the U-D axis. X2 and X3 are conjugate subgroups preserving the F-B and R-L axes, respectively. X is the symmetry class consisting of X1, X2, and X3. All of Mike Reid's X-symmetric positions are in particular X1-symmetric. For X1, it seems natural to view the position as "basic" if all four edge facelets on the U and D faces are the same color, and to view the position as superflipped otherwise. For X2, the same rule would apply to the F and B faces. FOr X3, the same rule would apply to the R and L faces. The eye sees this distinction very clearly. Example 3. Suppose Symm(x[e])={i,v}, where v is the central inversion. For such a position, any particular edge cubie could be placed anywhere, but each edge cubie would have to be placed diametrically opposite its diametrically opposed edge cubie. For example, if cubie uf were placed in the rd cubicle, then cubie db would have to be placed in the lu cubicle, etc. Also, for Symm(x[e]) to be {i,v} the edges could not have any additional symmetry. In this case, I don't think there is any natural way to distinguish between a "basic" position and a superflipped position. Example 4. Suppose Symm(x[e])=I={i}. In other words, the edges have no symmetry. In this case, I don't think there is any natural way to distinguish between a "basic" position and a superflipped position. Example 5. Suppose Symm(x[e])={i,c_u2}, where c_u2 is a 180 degree whole cube rotation around the U-D axis. In this case, the position would be "basic" if opposite edge facelets on the U face were the same color and if opposite edge facelets on the D face were the same color, and would be superflipped otherwise. The eye sees this distinction very clearly. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990