From @mail.uunet.ca:mark.longridge@canrem.com Mon Apr 24 04:12:38 1995 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 AA28938; Mon, 24 Apr 95 04:12:38 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <212875-8>; Mon, 24 Apr 1995 04:13:36 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA11604; Sun, 23 Apr 95 22:40:35 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1DD97C; Sun, 23 Apr 95 22:33:19 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Orders of Symmetry From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1104.5834.0C1DD97C@canrem.com> Date: Sun, 23 Apr 1995 23:29:00 -0400 Organization: CRS Online (Toronto, Ontario) >Date: Wed, 8 Dec 93 16:28:29 EST >From: hoey@AIC.NRL.Navy.Mil (Dan Hoey) >Message-Id: <9312082128.AA23718@Sun0.AIC.NRL.Navy.Mil> >To: mark.longridge@canrem.com (Mark Longridge), CRSO.Cube@canrem.com >Subject: Re: More corrections > >> * Hmmm, what are all the possible orders of symmetry? * > > M has subgroups of order 48, 24, 16, 12, 8, 6, 4, 3, 2, 1. Some of > these subgroups (e.g., A, C) are not symmetry groups of any position, > so I can't be sure there are positions of all these symmetry orders. Quite a while ago I asked Dan the question above, and I've thought a lot about the answer. So I decided to look at certain cube positions and I wrote a module to perform C and C + Sm where C = 24 rotations of the cube Sm = Central Reflection on any pattern I had in my database, and count how many different patterns resulted from the 48 operations. The following are some patterns which I found: Number of different Pattern patterns ------- --------- 48 R1 U1 24 L2 U2 16 Mark's Pattern 1 (18 q+h, 22 q) R2 U3 R1 D1 F1 B1 R3 L3 U1 D1 F3 U1 F3 U2 D3 B2 R2 U1 (Also 7 clockwise + 1 anticlockwise corner twist) 12 2 dot, 2 T, 2 ARM (sq group antipode, see p108) 8 6 Dot (a slice pattern) 6 2 DOT, 4 ARM (sq group antipode, see p99) 4 ???? 3 4 Dot pattern (slice pattern) 2 6 H pattern type 2, T2 B2 L2 T2 D2 L2 F2 T2 1 Pons Asinorum (6 X order 2) or all edges flipped It took a while to find a pattern which could be transformed 16 different ways. Still trying to find a pattern which will result in 4 distinct ways, but I am not optimistic. A random walk through the cube resulted in a pattern which would transform 48 ways in every case I tried. >> A) What is the next most commutative element (the pancentre?) >> after the 12-flip? > > (presumably, start excluded as well) > > i'll guess that these four conjugacy classes are tied for next. > > corner cycle structure: (1+)(1+)(1+)(1+)(1+)(1+)(1+)(1-) > edge cycle structure: (1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1) Here's a small followup to the pancentre question. The reason why the 7 clockwise + 1 anticlockwise corner twist is the next most commutative element after the 12-flip & start is because it has the most number of cube elements (in this case corners) the same as possible without all the elements being the same, as with the 12-flip. It must be 7 clockwise + 1 anticlockwise corner twist because the next most commutative element effecting edges only would be the 10-flip and that would have 2 elements not the same as the rest instead of just 1. -> Mark <-