   Date: Sat, 08 Jan 94 08:46:20 -0500 (EST)   From: Jerry Bryan <BRYAN%WVNVM.BITNET@mitvma.mit.edu >
~~~ ~~~ Subject: Some Terminology Concerning B

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     