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Date: Tue, 20 Jun 1995 13:48:51 -0400 (EDT)
From: "Jerry Bryan"
To: "Cube Lovers List"
Subject: Re: Re: A Third Way to Calculate the Real Size of Cube Space?
In-Reply-To: Message of 06/20/95 at 14:39:00 from ,
Martin.Schoenert@Math.RWTH-Aachen.DE
On 06/20/95 at 14:39:00 Martin Schoenert said:
>I can't figure out how the brute force computer search works.
>So I can't tell whether it is really different from the other methods
>(and if indeed it is a method to compute the real size of the cube ;-).
>Jerry, could you say a little bit more about this computation?
I look at corners and edges separately, and then combine the results.
I can't speak for how Dan did his corner search and his edge search,
but I can describe mine. Our results do match, which is always nice.
Dan did most of the figuring out of "combining the results" for corners
and edges.
Conceptually, I look at every single corner position X and calculate
Symm(X), and I look at every single edge position Y and calculate
Symm(Y). In practice, there are some important shortcuts.
I have a data base containing "every" corner position and a second
data base containing "every" edge position with "every" defined in
the following sense. The set of all positions is partitioned into
equivalence classes of the form {m'Xmc} for the corners and {m'Ymc}
for the edges, for m in M (48 rotations and reflections) and c in C
(24 rotations). The data bases contain a representative element
from each equivalence class.
For all the cases where |{m'Xmc}|=1152 and |{m'Ymc}|=1152, no searching
is required. For these cases, we know *a priori* that there are 24
M-conjugacy classes containing 48 elements each, and that for each
of the 1152 positions we have Symm(X)=I and Symm(Y)=I. Fortunately,
for the vast majority of the cases (over 97% for corners and well
over 99% for edges), the so-called B-class size is 1152.
Suppose the representative for {m'Xmc} is V and for {m'Ymc} is W.
For the remaining cases, the idea is to use Vc and Wc for
each fixed c in C as a base or representative element for an
M-conjugacy class of the form {m'(Vc)m} and {m'(Wc)m}. The tricky
part here is that while Vc and Vd are distinct when c and d in C
are not equal, nonetheless Vc and Vd may be M-conjugate. Hence,
we use each Vc and Wc which are distinct up to M-conjugacy as
a representative element for an M-conjugacy class.
Conceptually, we calculate Symm(T) for each T in {m'(Vc)m} for each
fixed c (and for Vc distinct up to M-conjugacy) and summarize the
results. In practice, it is sufficient to calculate Symm(Vc).
Here comes another tricky part (at least it was until I figured it out).
Initially, I assumed that Symm(T) was the same for all T in {m'(Vc)m}.
However, such is not the case. Rather, if you calculate each Symm(T), you
will discover that each one will be a group from a class of conjugate
groups, and that each group in the class of conjugate groups will appear
an equal number of times.
Given that you have picked an arbitrary representative from the M-conjugacy
class, you don't know which conjugate group you are going to get when
you calculate Symm(m'(Vc)m). I got around this issue originally by
calculating a representative group (the 98 subgroups of M then have
33 representative groups, one for each of the 33 symmetry classes).
In the work I am doing now, I map directly from each of the 98 subgroups
to their respective symmetry class.
The exhaustive search then consists of performing the above calculations
for each B-class of the form {m'Xmc} and {m'Ymc} and summarizing the
results (that is, counting all the M-conjugacy classes). In addition
to an overall total, you summarize by symmetry class, which gives you
the additional information I have talked about, over and above Dan's
Polya-Burnside results.
At this point, the summary results give you the real size of the corners
only problem and the real size of the edges only problem. While you are
at it, you have to count the M-conjugacy classes for even positions and odd
positions separately in order to put corners and edges together
properly.
Finally, you put the corners and edges together in all possible ways.
The putting together of the corners and edges is described in the
draft I have mentioned, so I will just wait until the draft is
ready before posting the rest.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
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