Date: Thu, 27 Oct 94 21:56:00 -0400
From: Mark Longridge <mark.longridge@canrem.com >
~~~ Subject: Shift Invariant Part 2

Continuing the previous discussion on shift invariance...

Mark writes:
>> This is the longest process I've found so far.

Martin writes:
>How about (UR)^140 or (UR)^1400? As mentioned above, you can make the
>processes as long as you wish.

```...or (U1 R1)^35 ? And indeed, (U1 R1)^(35 * 40) is shift invariant.
I meant to say (and should have said):
```

"This is the longest optimal process I've found so far."

Although I was inspecting (U1 R1)^N patterns in the quest for shift
invariance, (U1 R1)^35 = (R1 U1)^35 escaped me. In fact it was my
mistaken belief that the < U , R > group had no shift invariant
processes.

I did not realize the connection between the centre of a group and
shift invariance until Martin's message of Mon Oct 24 17:10:27 1994.
I actually did use GAP on the < U, R > group but I couldn't resolve
the resulting position (can GAP use letters? I should have used
letters).

The missing insight was realizing that, although the full group had
a unique centre, other subgroups have different centres.

So without further adieu:

```6 Counterclockwise Twist,
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)

(U3 R3)^35 would execute a 6 clockwise twist.
```

Martin writes:
> 4) The ``odd'' element.
> The element (UR)^140 lies in the centre of the subgroup <U,R>.
> It is the only shift invariant element of odd order (hence the name).
> Thus this process and its inverse are shift invariant.
> There are 24 such elements in the entire group (two for each edge).

Is this odd due to ( U1 R1 )^35?
Actually everything about the above description appears even.
It is an even number of quarter turns...

Martin writes:
> For me the most amazing elements were the ``odd'' element and the
> ``diagonal square'' element.

I concur completely, although the all-commuting 12-flip is definitely
interesting too. I was surprised to see the process was shift invariant.

Martin writes:
> Thus at this time all non-trivial such elements had been found, except
> for the ``odd'' element.

For which I refer to the process UR11, 22 q turns.

Martin, you will be pleased to hear that I like GAP, but I need a
bigger hard drive for that beast!

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