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Subject: Re: Crazy Corner Pattern
From: mark.longridge@canrem.com (Mark Longridge)
Message-Id: <60.1167.5834.0C1E77E1@canrem.com>
Date: Mon, 19 Jun 1995 00:36:00 -0400
Organization: CRS Online  (Toronto, Ontario)

>>I'm trying to find the 1 pattern with M-class size of 4 of the
>>corners group.
>>The only pattern that I can find is 4 alternate corners twisted
>>clockwise which is in the twist orbit.

>>It does not seem to be any pattern with just corners twisted in
>>place.

>>Jerry, if you could *please* identify this pattern before I go
>>nuts...... !

>The position is called T-symmetric in Dan's taxonomy (actually,
>there are four T subgroups of M, T1 through T4).  The symmetry
>is related to opposite corners, e.g., the UFL-DRB axis,
>which is why there are four T subgroups.

>Also, the position is Q-transitive, so you can check it out
>in _Symmetry and Local Maxima_.  I quote ".... each edge on the
>girdle may be swapped with the diametrically opposite edge,
>provided that the corners on the girdle are swapped with
>their opposites as well."   Here, you would fix the edges and
>pay attention only to the swapping of the corners.

Hmmmmm, that's very interesting.

Below is my interpretation of "...corners on the girdle are swapped
with their opposites as well." Let's call it pattern X.

        D U D
        U U U
        D U D
F L B   R F L   B R F   L B R
L L L   F F F   R R R   B B B
F L B   R F L   B R F   L B R
        U D U
        D D D
        U D U

If I understand the terminology correctly, then for this pattern
X = |{m'Xm}|=3 and |Symm(X)|=16, same as the 4 spot.
Also X = |{c'Xc}|=3.

But perhaps this is not the same pattern....

Let's call this next cube arrangement pattern Y.

        F U U
        U U U
        D U L
R L B   R F B   D R R   B B D
L L L   F F F   R R R   B B B
U L L   F F U   L R F   L B F
        D D B
        D D D
        R D U

Y = |{m'Ym}|=4 and |Symm(Y)|=12, and Y=|{c'Yc}| = 4

This pattern was created by swapping 3 pairs of diametrically opposite
corners, which is in the *swap* orbit on a normal cube, but since we
are dealing with corners of Rubik's cube and ignoring edges we can
realize permutations with an odd number of pairs of corners swapped.

-> Mark <-