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 <-