From @mail.uunet.ca:mark.longridge@canrem.com Sat Nov 5 23:49:54 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 AA02464; Sat, 5 Nov 94 23:49:54 EST Received: from portnoy.canrem.com ([198.133.42.251]) by mail.uunet.ca with SMTP id <91122-1>; Sat, 5 Nov 1994 23:50:20 -0500 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA17381; Sat, 5 Nov 94 23:47:01 EST Received: by canrem.com (PCB-UUCP 1.1f) id 1BB9FA; Sat, 5 Nov 94 23:24:55 -0400 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Shifty Invariance From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.846.5834.0C1BB9FA@canrem.com> Date: Sat, 5 Nov 1994 22:16:00 -0500 Organization: CRS Online (Toronto, Ontario) ---------------------------------------- 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) >> Martin asks: >Is UR11 the shortest process effecting the ``odd'' element in ? 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 , 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 rather than . -> Mark <- Email: mark.longridge@canrem.com