Date: Sat, 17 Jun 95 17:18:00 -0400
From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~ Subject: A Third Way to Calculate the Real Size of Cube Space?

We define the real size of cube space to be the number of M-conjugate
classes {m'Ym} for m in M, set of 48 rotations and reflections
of the cube, and for Y in G.

Dan Hoey has calculated the real size of cube space using the
Polya-Burnside theorem.

Dan and I (mostly Dan) have also calculated the same result using
exhaustive computer search. The computer search is much less
elegant than the Polya-Burnside results, but the search does
provide additional information, such as the number of positions
associated with each symmetry group. The results from the
computer search have not yet been posted to the list, but a draft
paper is in progress.

In the meantime, it occurs to me that perhaps -- but only perhaps --
there is a third way to calculate the real size of cube space.
The third way would not require (much) computer searching, but would
provide the same level of detail about number of positions per
symmetry group as does the full blown search.

The idea is based on a posting from Mike Reid. Mike calculated the
number of positions in G whose symmetry preserves the U-D axis.
Such positions have a symmetry group which is called X1
in Dan's taxonomy. For these positions, we say Symm(Y)=X1,
where in general for Y in G we have Symm(Y) is the set
(and group) of all m in M such that Y=m'Ym.

X1 contains sixteen elements (eight rotations and eight reflections),
and preserves the U-D axis. X2 and X3 are conjugate subgroups of X1
and similarly preserve the F-B and the R-L axes, respectively. If Y is
X1-symmetric, then we have {m'Ym}={Y1,Y2,Y3}. One of the Yi is Y
and is X1-symmetric, one of the Yi is X2-symmetric, and the other
Yi is X3-symmetric.

Mike determined (without computer search) that there are 128
X1-symmetric positions. Since four of the positions are also
M-symmetric, we have 124 positions Y for which Symm(Y)=X1.
Similar results hold for X2 and X3. Hence, there are 124
M-conjugacy classes containing cubes for which Symm(Y)=Xi, or
perhaps we might say for which SymmClass(Y)=X. The important
fact here is that we have determined that there are
124 M-conjugacy classes for symmetry class X without having
to do a computer search.

If we could similarly determine the number of K-symmetric positions
for each of the 98 subgroups K of M without computer search, then
we could calculate the real size of cube space. You really only
have to determine the size of 33 subgroups. Just as the solution
for X1 also gave us the solution for X2 and X3, similarly the
solution for any subgroup provides the solution for all conjugate
subgroups, and there are 33 classes of conjugate subgroups.

I usually get myself in trouble when I delve too much into things
I don't understand, but let's try a few examples. The subgroup
HV={i,v} is easy to understand, where v is the central inversion.
For the edges, the number of HV-symmetric positions should be
24*20*16*12*8*4. That is, put the first cubie anywhere (24
possibilities) which dictates the location of the respective
"opposite" cubie. There are then 20 possibilities for the
location of the third cubie which again dictates the position
of the respective "opposite" cubie, and so forth. In the same
manner, the number of HV-symmetric corner positions is
24*18*12*6. The number of HV-symmetric positions is then
(24*20*16*12*8*4)*(24*18*12*6)/2 to take parity into account.

Now we have the rub. In order to calculate the positions for
which Symm(Y)=HV, we must subtract out the HV-symmetric
positions which have stronger symmetry than HV, just as we
subtracted out the M-symmetric positions in Mike's X1 case.
But to do so, we cannot take the subgroups of M in isolation.
We have to do them all, starting with M and working our way
down. (And HV is pretty far down the food chain.)

Some of the subgroups I can do pretty easily, and for others I have
not a clue. Recall that A is the subgroup of M consisting of the
24 even rotations and reflections and that C is the subgroup of
M consisting of the 24 rotations (12 even and 12 odd). As long
ago as _Symmetry and Local Maxima_, Dan Hoey and Jim Saxe
determined that there are only four A-symmetric positions and
only four C-symmetric positions, namely the four that are also
M-symmetric. Hence, there are no positions for which Symm(Y)=A
nor for which Symm(Y)=C. But I haven't a clue how they knew,
nor how to go about constructing an A-symmetric or a C-symmetric
position from scratch. You can't get very far with my proposal
unless you can figure out how to construct K-symmetric positions
for any K.

For one more example, consider H, the set of 12 even rotations and
12 odd reflections. I know from computer search and also from
_Symmetry and Local Maxima_ that there are 24 H-symmetric positions,
of which 4 are M-symmetric and 20 are H-symmetric without also being
M-symmetric. The 20 H-symmetric but not M-symmetric positions form
10 M-conjugacy classes for which we would say SymmClass(Y)=H. It
ought to be easy to derive this result without a computer search, but
again I confess I haven't a clue as to go about constructing the 24
H-symmetric positions from scratch.

Well, I could cheat and look up the Class H positions
in _Symmetry and Local Maxima_, but what about the classes
that haven't been figured out yet? Also, I could cheat and
use the results from computer search, but that's hardly the
point.

One final point: just as Mike's 128 X1-symmetric positions formed
a group, similarly the set of K-symmetric positions form a group
for all 98 possible values of K. We have to be a little careful
with our terminology. The X1-symmetric positions form a group,
as do the X2-symmetric and the X3-symmetric positions. But if
we want to talk about the X-symmetric positions, we no longer
have a group. For example, we do not in general have closure
when forming the composition of X1-symmetric positions with
X2-symmetric positions.

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Robert G. Bryan (Jerry Bryan)                        (304) 293-5192
Associate Director, WVNET                            (304) 293-5540 fax