From ccw@eql12.caltech.edu  Tue Dec  7 08:25:59 1993
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From: ccw@eql12.caltech.edu (Chris Worrell)
Message-Id: <931206185340.20400b26@EQL12.Caltech.Edu>
Subject:  Re: Unique antipode of edges only
In-Reply-To: Your message <9312070232.AA10000@life.ai.mit.edu> dated  6-Dec-1993
To: BRYAN%WVNVM.WVNET.EDU%WVNVM.WVNET.EDU@mitvma.mit.edu,
        cube-lovers@ai.mit.edu

On 12/06/93 at 10:45:00 mark.longridge@canrem.com said:
>-> I was somewhat startled to see the unique antipode of the 3x3x3 edges
>-> in the quarter-turn metric.  Do you know what pattern that is?
>->
>-> Dan

>It's got to be all edges flipped in place.


Unfortunately, this is wishfull thinking.
This antipode is 15 qtw from Home, an odd distance.
All edges flipped is an even distance from Home in the qtw metric.

Looking at Jerry Bryan's pictures, I see 5 two edge swaps.

>
>          *6*              *6*
>          6*6              3*4
>          *6*              *1*
>          *2*              *5*
>          2*2              3*4
>          *2*              *2*
>       *3**1**4*        *1**1**1*
>       3*31*14*4        5*23*42*5
>       *3**1**4*        *6**6**6*
>          *5*              *2*
>          5*5              3*4
>          *5*              *5*
>
>         Start          Antipodal
>

If we assume face 1 is F, I get
          (FU) (BD) (FD,BU) (FL,LU) (FR, RU) (LD,BL) (RD,BR)


Is the 1152 number the result of factoring out the 24 spatial rotations
and 2 reflections of the centers?

Are there any estimates of how many distinct sequences actually generate
this Antipodal Class?
Ideally, it would be interesting to have a total list of these sequences.