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Date: Sat, 18 Dec 1993 14:02:01 EST
From: "Jerry Bryan"
To: "Cube Lovers List"
Subject: Second Addendum - Size of Cube Group under M
I feel like I am pestering the list to death with corrections.
I still believe that the figure that I proposed for the size of
the cube group under M is correct. The first post included
a "correct" but I think unsatisfactory explanation. The second post
improved upon one point that was unsatisfactory in the first post.
Now, let's see if I can get it completely correct.
The size of the corner group under (my version of) M is known.
The size of the edge group under (my version of) M is known as
well. Let C be the size of the corner group, and E be the size
of the edge group. Remember, the elements of the groups are
equivalence classes induced by (my version of) M. Here is an incorrect
formula for G, the size of the entire cube group under (my
version of) M.
G = (C*24) * (E*24) / 2
The division by 2 is introduced to account for parity between the
corner group and the edge group. But the value for G produced by
this formula is only half as big as it should be. The problem is
that M induces equivalence classes based on both rotations and
reflections, not just base on rotations. Hence, we are led to the
following (still incorrect) formula:
G = (C*24*2) * (E*24*2) / 2
As before, the division by 2 takes care of parity between the corner
group and the edge group. In addition, the multiplication by 2 takes
care of reflecting each group. But the value for G produced by this
formula is twice as big as it should be. The problem is that while
any corner rotation can occur with any edge rotation (subject to
parity), you must either reflect both groups, or else reflect neither
group. Thus, we have the following (correct) formula:
G = ((C*24) * (E*24) / 2) * 2
The division by 2 takes care of parity between the groups, and the
multiplication by 2 takes care of reflection of the two groups
as a unit. If we wish, we can cancel the multiplication and the
division to yield
G = (C*24) * (E*24)
This is the same formula I originally posted, and I did say in the
original post that the division by 2 cancelled out. However, I
think that this post provides a better explanation of the
cancellation than did the original post.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
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?