From @mail.uunet.ca:mark.longridge@canrem.com Fri Oct 28 11:50:15 1994 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA29859; Fri, 28 Oct 94 11:50:15 EDT Received: from portnoy.canrem.com ([198.133.42.251]) by mail.uunet.ca with SMTP id <86790-2>; Fri, 28 Oct 1994 11:48:59 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA01388; Fri, 28 Oct 94 11:44:45 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1B76FC; Fri, 28 Oct 94 10:52:00 -0400 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Shift Invariant Part 2 From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.825.5834.0C1B76FC@canrem.com> Date: Thu, 27 Oct 1994 21:56:00 -0400 Organization: CRS Online (Toronto, Ontario) 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 . > 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