From @mail.uunet.ca:mark.longridge@canrem.com Sat Apr 2 21:21:55 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 AA12740; Sat, 2 Apr 94 21:21:55 EST Received: from portnoy.canrem.com ([198.133.42.251]) by mail.uunet.ca with SMTP id <88146(1)>; Sat, 2 Apr 1994 21:21:12 -0500 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA03000; Sat, 2 Apr 94 21:20:05 EST Received: by canrem.com (PCB-UUCP 1.1f) id 1993C2; Sat, 2 Apr 94 21:16:50 -0400 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Invariant Shifting From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.733.5834.0C1993C2@canrem.com> Date: Sat, 2 Apr 1994 20:12:00 -0500 Organization: CRS Online (Toronto, Ontario) 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