Date: Tue, 19 Dec 95 17:43:03 -0400
From: Jerry Bryan <jbryan@pstcc.cc.tn.us >
~~~ Subject: Physical Cubes and Models Thereof

The general subject of physical cubes and mathematical models thereof
has been discussed many times before, but I have never been totally
satisfied with all of the conclusions. I'm going to take one more
crack at it.

Let's start with the question of what constitutes a single move and
the argument between the quarter-turners and the half-turners. There
are good and valid arguments on both sides of the question, and there
is no one "right" answer. However, the strongest and most succinct
argument in favor of quarter turns is that they are conjugate. In
the case of the standard 3x3x3 cube, the set Q of twelve quarter
turns is M-conjugate, where M is the set of 48 rotations and
reflections of the cube.

A quarter-turner would normally generate G as G=<Q>. But given that
Q is M-conjugate, we could say equivalently that G=<{m'Xm | m in M}>
for any X in Q.

Question: for the 3x3x3, are there any elements X in G other
than those X in Q itself where we can generate G as G=<{m'Xm |
m in M}>? Remember that in most cases we would have 48
generators available.

Clearly, there are X in G such that <{m'Xm | m in M}> does not
generate G. For example, the M-conjugates of F2 do not
generate G. But I have a feeling that any group that is
generated by <{m'Xm | m in M}> is an "M-symmetric group" (using
the term "M-symmetric" very loosely and informally) and is
therefore a somewhat interesting group.

For the 4x4x4, I will use upper case letters for outer slab moves
(face moves) and lower case letters for inner slab moves. For
example, L'l'rR would rotate the entire cube away from you by 90
degrees, but the cube would otherwise look unchanged. If we denote
the set of outer slab moves as Q and the set of inner slap moves as
q, then we can generate a group as G4=<Q,q>. I am hesitant to say
that G4 is "the" cube group for the 4x4x4, because it is so hard to
agree on what "the" cube group is for higher order cubes. But in any
case, Q and q are not M-conjugate with each other. There is in fact
no way to have M-conjugate generators for the 4x4x4 and higher
physical cubes.

For a mathematical model, conjugacy can be repaired. For example,
there is an operation called Evisceration where inner slabs and
adjacent parallel outer slabs are exchanged. There is also an
operation called Inflection where inner slabs are exchanged with
their parallel inner slabs, and Exflection where outer slabs are
exchanged with their parallel outer slabs. We can use Rotations,
Reflections, Evisceration, Inflections, and Exflections to generate a
192 element symmetry group for the 4x4x4 called M4. We can then show
that Q and q are M4-conjugate, and conjugacy is repaired. That is,
we can generate G4 as G4=<{m'Xm | m in M4}> for any X in q or Q.
(See Dan Hoey's article "Eccentric Slabism, Qubic, and S&LM" dated
1 June 1983.)

In the previous paragraph, I used the term "symmetry group"
quite deliberately, although some of you may not agree with the
way I used it. I am still struggling to understand how
narrowly or loosely we should really construe the preservation
of a geometric property before we declare a permutation to be a
symmetry. In the case of M4 above, I think the designation of
"symmetry" is warranted, although it is a looser interpretation
than is typical.

But my purpose is to model physical cubes. Evisceration is not
possible on physical cubes. Conjugate quarter turn generators are
not possible for physical cubes larger than the 3x3x3 without
Evisceration (or its generalization to the NxNxN case). Therefore,
we abandon M-conjugation and its generalizations as a criterion for
modeling physical cubes.

Dan's Eccentric Slabism article talked about slab moves (a single
plane of cubies turning together) and cut moves (all the cubies on
each respective side of a plane cut of the cube turning together).
Evisceration convinced Dan to convert from a Cutist view of the cube
to a Slabist view of the cube. But Dan fully endorsed the Slabist
view only for even-sided cubes. His phrase "Eccentric Slabism" refers
to the fact that he still refused to make slab moves for the center
slabs of odd-sided cubes. The problem is that center slab moves
break M-conjugacy and its generalizations. But I've already given up
M-conjugacy and its generalizations. Given that, it seems unnatural
to leave out the center slab moves, so we leave them in.

We next confront the issue that physical cubes are rotated in space
with abandon. Different rotations of physical cubes are considered
to be equivalent, and/or rotations of physical cubes are considered
to be zero cost operations. But we desire a mathematical model of a
physical cube to be a group.

My preferred non-computing model of this situation is to treat the
various configurations of the cube as cosets of C, the set of 24
rotations of the cube. However, this model is awkward for computing.
For something like the 2x2x2, we more typically do something like
fixing a corner. We hereby adopt "fixing a corner" as the solution
for the general NxNxN case. See below for more details of how we
propose to do so in the general case.

We can note several things about the "fixing the corner" model:

1. It breaks M-conjugation. But we gave up M-conjugation
anyway. Consider the 2x2x2 as a good example. If we
insist on treating different rotations as equivalent,
then the 2x2x2 really isn't M-conjugate. I am simply
suggesting that the NxNxN physical cube really isn't
M-conjugate, no matter the value of N, if we treat
different rotations as equivalent.

2. With the "cosets of C" model, we can make the cosets into
a group by taking as a representative for each coset the
unique element which fixes the same corner. There is
then an easy isomorphism between the "cosets of C" model
and the "fixed corner model". My only trouble with the
"cosets of C" model is that I keep wanting to call it G/C,
and you can't call it that. C is not a normal subgroup
of G, and we cannot speak of G/C as a factor group of G.

3. We can have conjugation and we can have symmetry with a
"fixed corner" model. It is just not M-conjugation.
Rather, it is the symmetry that preserves the fixed
corner, and conjugation within that symmetry group.

The 4x4x4 is a good example of how we propose to "fix the corner" for
the general NxNxN case. Consider our status after L'l'r. A physical
cubist would say that you were only one move from Start, and would
"solve" the cube simply with R. But R would yield L'l'rR, which would
leave the cube rotated. This is fine for our physical cube, but not
so fine for a mathematical model of a physical cube which seeks to
fix a corner. Hence, we define R as R=Llr', and similarly for the
other slab moves which would otherwise move the fixed corner. The
generalization to cubes higher than 4x4x4 is obvious.

Actually, I would prefer a slightly different but equivalent
definition for those slab moves which fix a corner. Frey and
Singmaster use script letters for whole cube moves (those moves in
C). I would implement R as follows: perform R in the normal sense of
the operation composed with Script-R' (and similarly for other slab
moves that would move the otherwise fixed corner). So for the 4x4x4,
let's suppose we fixed the TRF corner. Our generators would be,

```L',l',r,(R)(Script-R'),
B',b',f,(F)(Script-F'),
D',d',t,(T)(Script-T')
```

and their inverses.

Clearly, the same technique works not only for the 4x4x4 and above,
but also for the 2x2x2 and for the 3x3x3.

I am thinking of this in a Slabist interpretation. However, a case
could be made that the (R)(Script-R') type of moves are really Cutist
moves.

I think all the other problems associated with a mathematical model
of a physical cube can be unified under the heading "Invisible Moves
of Facelets". The most obvious example is that the Supergroup is
invisible on the 3x3x3 unless the orientations of the face centers
are marked somehow or other. But with larger cubes (e.g., the face
centers of the 4x4x4), it is not just changes in orientation that are
invisible; there are also invisible changes in location.

In all cases, I would propose initially modeling the "larger group"
(call it L), where invisible changes in location and orientation are
visible. Number all 16 facelets of each face on the 4x4x4, for
example. You do have to decide how "large" you wish your larger group
L to be. For example, to make invisible orientation changes visible,
you have to give a facelet four numbers rather than just one. The
set of all positions that are equivalent when the "invisible" changes
are ignored is a subgroup K. Your final model is then the cosets of
K in L.

The "cosets of K in L" model will always work, but it may be difficult
to deal with computationally. Ideally, you would be able to find a
subgroup G of L for which you could find an easy isomorphism with the
cosets of K.

As an example, consider the Supergroup of the 3x3x3 and call it L.
Within L, there is a subgroup K which fixes the corners and edges. K
is just all the legal face center reorientations. Therefore, if we
wish to ignore face center orientations our model can be the cosets
of K in L. There is an easy isomorphism between the cosets of K in
L, and our standard model for the 3x3x3 which we call G.

In truth, we would never model the 3x3x3 in such a convoluted
fashion. We would just use G and be done with it. But for the 4x4x4
and larger cubes, I am not sure there is any choice.

As the cubes get larger, you would generally find that there was
a nested sequence of subgroups -- K_0, K_1, etc. -- for which the
cosets of K_n in the larger group L would produce a useful model.
For example, on the 4x4x4 one of your K's might be the group of
permutations that fixed everything but the positions of the center
facelets within a face (keeping them the proper color, of course).
But a more stringent K might be the group of permutations that fixed
everything but the orientations of the center facelets within a face.

I will end by pointing out that Goldilocks would really like the
3x3x3. Papa Bear's 4x4x4 is too large and Baby Bear's 2x2x2 is too
small. But Mama Bear's 3x3x3 is just right. The physical 3x3x3 is
the only physical NxNxN which can be modeled with M-conjugate
generators (assuming we fix the face centers). And the 3x3x3 is the
NxNxN (physical or mathematical) with the nicest isomorphism between
the cosets of K in L and some reasonable group G.

``` = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)                jbryan@pstcc.cc.tn.us
```