Date: Sat, 05 Nov 94 22:16:00 -0500
From: Mark Longridge <mark.longridge@canrem.com >
~~~ Subject: Shifty Invariance
```----------------------------------------
Even more thoughts on "Shift Invariance"
----------------------------------------

>>Mark continues
>>
>>    Equivalent to (U1 R1)^35= (R1 U1)^35 & Shift Invariant
>>    UR11 = U2 R1 U1 R1 U1 R3 U1 R3 U1 R3 U2 R1 U1 R1 U1 R3 U1 R3 U1 R3
>>           (22 q  or  20 h  moves)
>>
>Is UR11 the shortest process effecting the ``odd'' element in <U,R>?
```

After a bit of computer cubing I found:

```p183 6 Twist      R1 U3 R2 U3 R1 D3 U3 R1 U3 R3 D2 R3 U3 R1 D3 U3
(18 q  or  16 h  moves)
```

This requires using the larger group of <U1, R1, D1>, although I
expected a 16 turn process. Note the fact this larger group has face
index 3 (rather than 2). But now the process is NOT shift invariant
and we see the route itself can determine whether it will be
shift invariant!

I welcome any mathematical explanation!

With even more contemplation I noticed that the process for
the edge 3-cycle

```UR1 = U3 R1 U2 (R1 U1)^2 R2 U3 R3 U3 R2 U1     (16 q, 13 h)
```

...was reducible to

```UR1a= F1 U2 (F1 U1)^2 F2 U3 F3 U3 F2           (14 q, 11 h)]
```

Of course, now we are using <U1, F1> rather than <U1, R1>.

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