From:

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The net is so wonderful about answering questions, here are a few

more:

1. Take a standard 3x3x3 Rubik's cube, and remove the corner and

center labels to make an Edges Cube. (I am assuming that the

underlying plastic is black. If the underlying plastic is

white and one of the colors on the labels is also white, the

Edges Cube is not so pretty). Scramble the cube. Give it to

a cubemeister to solve. How will the cubemeister know if the

cube is solved? In other words, how will the cubemeister

distinguish Start from Pons Asinorum?

One answer is that the cubemeister cannot. Unless the cubemeister

saw the cube before it was scrambled, or unless the cubemeister

was told which reflection of the colors was Start, there would

be no way to tell. Another answer is that either one is Start --

that there are two Starts. However, if you like this answer,

and if you identify the identity with Start, you are in the

disquieting situation of having a group with two distinct

identities (grin!).It is obvious that this problem does not arise if the labels are

left on the centers. Almost as obvious is the fact that the

problem does not arise if the labels are left on the corners, even

if the labels are removed from the centers. The corner group

cannot be turned inside out by a reflection as can be the edge

group.

2. As silly as my second answer is, it leads to a second question.

Just what is the 2x2x2 cube? Or more correctly, how do you

know when it is solved? With any size of cube, if you restrict

yourself to quarter-turns, by definition you cannot rotate the cube

in space as a single operation. Yet, a simple quarter-turn sequence

such as RL' does rotate the 2x2x2 cube because it is faceless. Is

Start of the 2x2x2 operated on by RL' solved? If so, you can argue

that the 2x2x2 has 24 Starts. Most people would not. They would

argue that there is only one Start, and that 2x2x2 cubes that differed

only by a rotation are equivalent.

3. Combining #1 and #2, I *think* that most people would argue that

Start and Pons Asinorum on the Edge Cube are not equivalent,

but that simple rotations of the 2x2x2 are equivalent. If I am

correct about "most people", why? Is a rotation symmetry

intrinsically a stronger or weaker symmetry than a reflection

symmetry?

4. When I was first posting my results about the Edge Group, and

particularly when it first began to sink in what the four equivalence

classes with only 24 elements really were, I had a moment of

panic. Since Start and Pons Asinorum differ only by a simple

reflection, why had not my version of M-conjugation declared them

to be equivalent? (I speak of "my version of M-conjugation", but

the question is no different if you look at Dan Hoey's original

M-conjugation). I think I know the answer, but I will leave the

problem as an exercise for the student. Furthermore, I think the

answer to #4 is really the same as the answer to #3.

5. What is a reflection, really? Here is an exercise to illustrate

the question. Take two identically colored and oriented 3x3x3

cubes. On one, perform F and on the other perform F'. Examine

the two cubes, plus their images in a mirror. Why are there

four distinct cubes rather than only two? At one level of

abstraction, the answer is simple. Of the four, one is not

reflected, one is pre-reflected, one is post-reflected, and one

is both pre- and post-reflected. Is this a sufficient answer,

or is there something deeper?

At this point, I can't help but note Martin Gardner's famous

mirror question in Scientific American many years ago: why

does a mirror reverse left and right but not up and down?

6. I found Dan Hoey's postings about the four special states of the

Edge Group to be delightful. Some of the results were based on

a computer search of the group, for example the fact that

f(I)=(0,9,12,15) could only reasonably be determined from a

computer search. However, the thought occurred to me that most

of Dan's results were independent of the computer search, and

I was curious precisely which results would stand without the

search? For example, if we identified the group as being

rectangular, would we be led to saying which of the four special

states were diagonally opposed without the computer search?

Without the search, I might be tempted to say that Start and

Pons Asinorum were diagonally opposed.

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