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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.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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