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On 13 Feb 1994, I proposed a model for centerless cubes which

I claimed met two criteria: 1) it was a group, and 2) it

maintained the symmetrical nature of the problem. On

23 May 1994, I retracted the claim that the proposed model

was a group.

I am now of the opinion that it is impossible to satisfy

both criteria simultaneously. I can make a very small modification

to the proposed model to make it a group, but the small

modification costs the model its cubic symmetry.

G is the full cube group, GC is the corners only cube group,

and GE is the edges only cube group. The proposed model

for centerless cubes consisted of partitioning any of G,

GC, or GE into sets of the form {Xc} for all c in C, where

C is the set of twenty-four rotations of the cube and X is

a cube. The sets are the elements of the proposed group.

The sets are called cosets and can also be denoted as

xC. The partitions are denoted as G/C, GC/C, and GE/C,

respectively.

Originally, the proposed group operator was {Xc} * {Yc} = {XYc}.

This operator fails to maintain closure, and hence fails to

define a group.

In order to illustrate the slight modification which will define

a group, we will start by restricting ourselves to GC. An

operator which works to define GC/C as a group is

{Xc} * {Yc} = {VWc}, where V is the unique element of {Xc} such

that the urf cubie is properly positioned in the urf cubicle,

and W is the unique element of {Yc} such that the urf cubie is

properly positioned in the urf cubicle.

Any other corner could have been used instead of urf, but once

you choose a corner the problem loses its symmetric nature.

Well, I guess it still has symmetry, but it is not the uniform

symmetry of the cube any more, because there is a preferred

orientation.

I have found only limited discussion in the archives, but

previous investigators have modeled a corners only, centerless

cube by leaving one corner fixed. Such a model is clearly

a group. For example, if we leave the urf corner fixed,

we can generate the group JC as JC=<L,L',D,D',B,B'>, where

we omit all twists of the U, R, and F faces from the set

of generators.

It is easy to find an isomorphism between GC/C and JC. I would

express it as something like {Xc} = {Wc} <--> W, where W is

defined as before. W is an element of JC, and as well is an

element of {Xc} = {Wc}. {Xc} = {Wc} is an element of GC/C.

But W is a particular element of {Xc} = {Wc}, whereas X is

an arbitrary element. Also, X is in GC, but X is not in JC

unless X = W. The mapping {Wc} <--> W is clearly one-to-one and

onto in both directions.

For the edges GE, we need to keep one edge cubie fixed, so the

centerless cube could be generated by something like

JE=<D,D',L,L',R,R',B,B'>, where we keep the uf cubie fixed by

omitting all twists of the U and F faces from the set of

generators. The isomorphism between GE/C and JE is expressed

as {Xc} = {Wc} <--> W, where X is an arbitrary element of

GE, and W is the unique element of {Xc} such that the uf

cube is properly placed in the uf cubicle. As before, any

edge cube would do as well, but once chosen, it is no longer

arbitrary.

For the whole cube G, at first blush it appears we could model

centerless cubes either by keeping a corner cubie fixed, or by

keeping an edge cubie fixed. But if we keep a corner cubie

fixed, the three immediately adjacent edge cubies are never moved

by any Q-turns. We could solve the difficulty by admitting

slice turns. But slice quarter-turns are odd on edges and

even on corners, so we have to restrict ourselves to slice

half-turns. I find this ugly, plus I would prefer to generate

G with Q-turns only. Hence, I would prefer to model a

centerless full cube as J=<D,D',L,L',R,R',B,B'>, where it

is an edge cubie which is held fixed rather than a corner cubie.

I said at the beginning that I thought it was impossible for a

model of centerless cubes both to be a group and also to

maintain cubic symmetry. The reason is as follows: it seems

to me that for any model which is a group, it should be

possible to find an isomorphism between the model and

J (or JC or JE, as appropriate). But J and JC and JE

do not have cubic symmetry because there is a preferred

orientation.

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