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Date: Mon, 17 Jan 1994 09:06:59 EST
From: "Jerry Bryan"
To: "Cube Lovers List"
Subject: Number of M-Conjugate Classes for GC\M
On 4 December 1993, I posted results from a breadth-first exhaustive
search of GC\M, the corners of the 3x3x3, reduced by M-conjugation.
The posting included a summary of how many conjugate classes there
were at each level of the search tree (i.e., distance from Start).
It occurred to me that I had not also posted a summary of M-conjugates
for the corners of the 3x3x3 by the size of the conjugate classes. I
searched my records, only to discover that I had never calculated
such sizes. If I had, I probably would have been forced to analyze
properly the distinction between M-conjugation and B-conjugation,
because B-conjugation makes no sense for the corners of the 3x3x3.
B-conjugation *can* be performed for the corners of the 3x3x3, but you
end up with the 2x2x2 instead because B-conjugation effectively
removes the centers.
Anyway, I have now calculated M-conjugate class sizes for GC\M via
computer search, and here are the results.
M-Class Number
Size of
Classes
1 1
2 1
3 3
4 1
6 34
8 33
12 301
16 104
24 9064
48 1832428
Total 1841970
Notice that with M-conjugation, the maximum class is size is 48,
rather than 1152 as it is with B-conjugation. Hence, my posting
of 4 December 1993 incorrectly identified the results as being
for "1152 fold symmetry". The results are correct, but they
should be labeled as being for "48 fold symmetry", i.e., for
M-conjugation rather than for B-conjugation.
In calculating M-conjugate class sizes for GC\M, I did not "start
from scratch". Rather, I used the existing results for B-conjugate
classes as a base. In the case of B-conjugate classes of order
1152, no calculations are required. Each such B-class can simply
be partitioned into 24 M-classes of order 48. Hence, I had to
perform calculations for less than 4% of the B-classes. Here is
a summary matrix, showing for each B-class size the number of
each M-class size which are derived.
M-Class Size
1 2 3 4 6 8 12 16 24 48 Total
24 1 0 1 0 2 1 0 0 0 0 5
B-Class 48 0 1 0 0 1 2 2 0 0 0 6
Size 72 0 0 2 0 11 0 2 0 5 0 20
96 0 0 0 1 0 1 3 0 2 0 7
144 0 0 0 0 20 0 42 0 30 14 106
192 0 0 0 0 0 29 0 8 73 16 126
288 0 0 0 0 0 0 252 0 682 406 1340
384 0 0 0 0 0 0 0 96 0 224 320
576 0 0 0 0 0 0 0 0 8272 22360 30632
1152 0 0 0 0 0 0 0 0 0 1809408 1809408
Total 1 1 3 1 34 33 301 104 9064 1832428 1841970
The first row of the matrix exemplifies the process of calculating
M-Class sizes from B-Class sizes. In the case of corners, there
is only one B-class of order 24, namely Start. The 24 elements of
the B-class are the 24 elements of the form Ic, where c is in C,
the 24 rotations of the cube. Under B-conjugation, these 24
elements are equivalent (i.e., in a centerless cube such as the
2x2x2, the 24 rotations of I are indistinguishable). But in a cube
with centers, such as the corners of the 3x3x3, the 24 elements
are not equivalent.
For example, the M-class of order 1 is {I}. One of the M-classes
of order 6 is {FB', UD', RL', LR', BF', DU'}. The M-class of order 3
is {FFB'B', RRL'L', UUD'D'}. That's as many as I can do in my head,
but I think the pattern is clear.
M-classes are a partition of the B-classes.
In the case of B-classes of order 1152, the partition is regular --
i.e., you get exactly 24 M-classes of order 48. However, all
partitions are not regular. In the partition of the B-class of I
which we just discussed, there is 1 M-class of order 1, 1 M-class
of order 3, 2 M-classes of order 6, and 1 M-class of order 8, for
a total of 24 M-classes. Many other partitions are not regular,
as well.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) (304) 293-5192
Associate Director, WVNET (304) 293-5540 fax
837 Chestnut Ridge Road BRYAN@WVNVM
Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU
If you don't have time to do it right today, what makes you think you are
going to have time to do it over again tomorrow?