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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