Date: Wed, 8 Dec 93 16:28:29 EST
From: hoey@AIC.NRL.Navy.Mil (Dan Hoey)
To: firstname.lastname@example.org (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
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 <-