From:

~~~ Subject:

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 <-