Date: Sat, 17 Jun 95 00:35:31 -0400
From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~ ~~~ Subject: 10q Local Maxima Positions
```1. (UU)  (F'B')(RL)(RL)(FB)
2. (UD') (F'B')(RL)(RL)(FB)
3. (UD)  (F'B')(RL)(RL)(FB)

4. (UD')(FB')(LR')(FB')(FB')
```

#4 is the one Mike Reid already found in the slice group. #3 is the
one I already found in the antislice group. #1, #2, and #3 are
obviously closely related. #1 and #2 appear not to be in either
slice or antislice, but I have been fooled before by alternative
sequences which yield the same position.

```#1, #2, and #3 all have the property that |{m'Xm}|=6 and
|Symm(X)|=8.  As has already been discussed, #4 has the
property that |{m'Xm}|=24 and |Symm(X)|=2.
```

The symmetry groups for #1, #2, and #3 are of a type Dan Hoey's
taxonomy calls class P, class S, and class AX, respectively.
These particular classes are hard to describe succinctly without
introducing a lot of notation. But in all three cases, the symmetry
groups (subgroups of M such that X=m'Xm} consist of four rotations
and four reflections, and have as an axis of symmetry one of
the three major axes of the cube (U-D, F-B, or R-L). There
are three groups P1, P2, P3 with axis of symmetry U-D, F-B,
and R-L, respectively, and similarly for S1, S2, and S3, and
for AX1, AX2, and AX3.

For #4, we have Symm(X)=HV in Dan's taxonomy, where HV={i,v}, and
where i is the identity in M and v is the central inversion in M.
If proper typography were available, the i and the v would be
upper case script letters to follow Frey and Singmaster.

There are relatively few positions in all of cube space
for which Symm(X)=Pi or Symm(X)=Si or Symm(X)=AXi (i in 1..3).
There are only 10 P positions through level 10 in the search tree (of which
just one is a local maximum). There is only one S position through
level 10, and only one AX position through level 10, both of
which are of course local maxima. The positions
are not Q-transitive, but the positions look "symmetric", and they
fulfill the (incorrect) intuition that "symmetric" positions must
be local maxima. We have no reason to say that other P or S or AX
positions further down the search will be local maxima.

I find position #4 extremely intriguing. In general, HV is not
very strong symmetry, and there are relatively speaking, quite
a few HV positions in cube space.

We could create an HV position as follows.
Put any edge cubie anywhere (say UF in RD). Put the "opposite"
cubie in the "opposite" cubicle (DB in LU in this case). Continue
for the remaining edge cubies, and then do the same thing for the
corners, remembering only to make sure the edges and corners have
the same parity. You can easily make an HV position that looks
quite "random" to the casual glance, and in fact most HV positions
don't look very "symmetric".

But Mike's position looks very "symmetric" at a casual glance, as
if its symmetry must be much stronger than HV. I certainly would
not have expected to find an HV position as a local maximum close
to Start. I think the "look" of Mike's position as "symmetric",
and the fact that it is a local maximum close to Start are
related. Without getting too long winded, I think the reasons
are two-fold. First, the corners and edges have much stronger
symmetry separately than they do collectively. Second, the
symmetry looks much stronger if you ignore the centers (i.e.,
if you ignore the rotational positioning of the cubies), perhaps
in the sense of Dan's CSymm function. For example, the corners
are properly positioned with respect to each other, even though
they are not properly positioned with respect to the fixed face
centers.

In the next few days, I intend to calculate Symm(X)
for the corners and edges separately for Mike's position, as
well as calculating CSymm(X) for the corners and edges
separately and combined. I think the results will be
enlightening.

``` = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)                        (304) 293-5192
Associate Director, WVNET                            (304) 293-5540 fax