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