From:

~~~ ~~~ Subject:

I have started to use "B" to indicate various aspects of the

conjugacy class generated by m'Xmc. The choice of B is sort

of an accident. I used "B" in the program fragment which

I posted to the list, and Dan Hoey analyzed the program fragment.

I have called it "B" in my mind ever since. However, I have used

B in several inconsistent ways. This is a proposal to

rectify that inconsistency.

Let X be any cube. Then the set of B-conjugacy classes of X is

the set of all m'Xmc for all m in M and all c in C. We denote

this set as BClass(X). B is the function B(X)=min(BClass(X)).

Note that we could have defined BClass(X) equivalently as the

set of all mXm'c, or as the set of all cm'Xm, or as the set of

all cmXm'. It is in general not the case that

m'Xmc = mXm'c = cm'Xm = cmXm' for any fixed value of m and c.

(Quite the contrary!). However, when we say "the set of all...",

the four ways of generating BClass(X) become equivalent. This is

the justification for the assertion in a previous note that

Gx\B = (Gx\M)\C = (Gx\C)\M.

Two cubes X and Y are B-equivalent if BClass(X) = BClass(Y).

Equivalently, two cubes X and Y are B-equivalent if B(X) = B(Y).

|X| is the length of X (the distance of X from Start). We have

|B(X)| = |X| for centerless cubes, but it is generally not the

case that |B(X)| = |X| for cubes with centers. In fact, let

X and Y be cubes with centers such that B(X)=B(Y). It is not

necessarily the case that |X| = |Y|. For example, consider

the set GC of cubes with corners with centers without edges.

We have B(RL')=B(I), but |RL'|=2 and |I|=0.

|BClass(X)| is the number of elements in BClass(X). If

|BClass(X)| = N, then X is said to have order-N symmetry. (I

sincerely regret ever using this terminology. As has been noted

on the list, it seems "backwards" somehow. But given that this

usage exists, the value 1152/N is generally more useful than the

value N.)

We note the following:

1. B(X) is a cube.2. BClass(X) is a set of cubes.

3. B(B(X)) = B(X) 4. BClass(B(X)) = BClass(X). 5. Both X and B(X) are in BClass(X).

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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?