Date: Sat, 02 Apr 94 20:12:00 -0500
From: Mark Longridge <mark.longridge@canrem.com >
~~~ ~~~ Subject: Invariant Shifting

Something new to stop the drought of cube posts...

```Example of Invariant Shifting
-----------------------------
```

The resultant position generated by process p8 is invariant under
shifting, specifically 2 X on the Left and Right sides.

```P8   2 x ORDER 2:

shift
0                              D2 F2 T2 F2 B2 T2 F2 T2
1                           T2 D2 F2 T2 F2 B2 T2 F2
2                        F2 T2 D2 F2 T2 F2 B2 T2
3                     T2 F2 T2 D2 F2 T2 F2 B2
4                  B2 T2 F2 T2 D2 F2 T2 F2
5               F2 B2 T2 F2 T2 D2 F2 T2
6            T2 F2 B2 T2 F2 T2 D2 F2
7         F2 T2 F2 B2 T2 F2 T2 D2
```

This is the longest process I've found so far. Certainly this property
is not true of all squares group processes. I suspect there are no
processes in the full group with this property (of any significant
length). Perhaps the fact that the L and R faces never rotate will
give some clue on how to generate processes with this property.

Q: Is this the longest such process?

```Further Notes on Antipodes in the Square's Group
------------------------------------------------
```

I just realized some things about sq group antipodes which
I should have seen before...

The closest 2 antipodes can be is 2 square's group moves.
Take the position produced by p66:

```p66  Double 4 corner sw L2 B2 R2 F2 L2 F2 T2 R2 (T2 D2 F2 T2) F2 L2 D2
```

Any turn will reduce this to a position requiring 14 moves. Undoing
this move will regenerate the antipode. No single move can change
position p66 into another antipode, therefore the closest any 2
antipodes can be is 2 moves.

Futhermore any antipode can not be made into a local maximum
which is 14 moves deep with 1 half turn. I will conclude that
there are no local maxima in the square's group that
neighbour each other closer than 2 moves.

-> Mark <-
Email: mark.longridge@canrem.com