From hoey@aic.nrl.navy.mil Fri Jan 21 18:32:30 1994
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Date: Fri, 21 Jan 94 18:32:15 EST
From: hoey@aic.nrl.navy.mil (Dan Hoey)
Message-Id: <9401212332.AA13137@Sun0.AIC.NRL.Navy.Mil>
To: Cube-Lovers@ai.mit.edu
Cc: Jerry Bryan
Subject: Re: Some Proposed Terminology
I welcome Jerry Bryan's efforts to
improve the terminology of the groups associated with Rubik's cube.
But there is some additional clarification I think is necessary.
> Let G be the standard cube group for the 3x3x3 cube....
> Let GC be the corners with centers without edges group....
> Let GE be the edges with centers without corners group....
That much will do, mod quibbles about what name is best.
> Let G\C be the corners with edges without centers group. I intend
> for the notation to indicate G reduced by C, where C is the rotation
> group for the cube....
> Let GC\C be the corners without edges without centers group....
> Let GE\C be the edges without corners without centers group....
First, these are not, strictly speaking, groups. Well, you can make
them groups, by defining what the group operation is. But I don't
know any way of doing that without losing the symmetrical nature of
the problem.
Second, I would suggest that G/C, GC/C, and GE/C are more standard
names for these objects. The elements are nominally 24-element sets,
each of which is an equivalence class when two positions are
considered equivalent when they differ by their position with respect
to the corners. The classes are called the cosets of C in G, GC, and
GE, respectively.
> Let G\M be the set of M-conjugate classes for G.....
> Let GC\M be the set of M-conjugate classes for GC....
> Let GE\M be the set of M-conjugate classes for GE....
The partition of a group into conjugacy classes is not at all the same
as the partition into cosets. So I would prefer to use different
symbology, like "\" for conjugacy and "/" for cosets, but....
> Recall that B is the function which calculates the canonical form
> for a cube under the composed operations of M-conjugation plus
> rotation. My programs calculate equivalence classes under B.
> Let G\B be the set of B-classes for G [ and likewise for GE, GC ].
Well, if you are using "\" for a generic partition into equivalence
classes, then we should really do something like G\Conj(M) for
partitions into conjugacy classes. At least then you can say
G/C=G\Cosets(C).
> Then, we have Gx\B=(Gx\C)\M=(Gx\M)\C. In English, we can decompose
> B into a multiplication by C and M (in either order).
No, that's _multiplication_ by C and _conjugation_ by M. A good
example of why it's important not to use confusing symbols. M and C
are not at all treated the same, except inasmuch as they are used to
induce partitions into equivalence classes. Say instead that
Gx\B = (Gx/C)\Conj(M) = (Gx\Conj(M))/C.
> If I wanted to model GC\C, I would have had to either model only
> seven of the cubies, or else modeled all eight but moved only seven
> of them. Since what I really wanted was (GC\C)\M, and since what I
> had was GC, I had to invent this funny B thing, where GC\B=(GC\C)\M.
> If I had been clever enough to model GC\C in the first place, I
> never would have had to invent B. Similar comments apply to my
> model for the edges.
Well, the part about moving only seven (corner) cubies is the approach
that's been taken before on this list to deal with cubes that don't
have face centers. It has the advantage that the object being treated
is a group. But the problem is that the group is no longer cubically
symmetrical (in some vague sense). This led me, at least, to lose
track of the structure that would allow analysis of M-conjugacy. So I
have to admire your tackling GE as a whole, instead of trying to stick
to GE/C. At first blush, it looks like GE/C is 24 times smaller. But
since GE/C\Conj(M) is almost 48 times smaller still, it's important to
work in GE at least enough to be able to use the conjugation. Which
is beside the point that I'm actually very interested in the structure
of Gx/Conj(M) itself. And that is what I was really getting at in
1984 when I asked about how many positions there really are.
Dan Hoey
Hoey@AIC.NRL.Navy.Mil