Message #595

From: David Vanderschel <DvdS@Austin.RR.com>
Subject: Re: [MC4D] Something interesting and strange about permutations
Date: Sun, 28 Sep 2008 05:15:49 -0500

On Saturday, September 27, "David Smith" <djs314djs314@yahoo.com> wrote:
>I like to think of the super-supercube as a mathematical
>entity, rather than a physical reality. I don’t really
>think of the hypercubies as having stickers at all. They
>are just uniquely identifiable in any position or
>orientation.

Then what is the group? For me, the associated group
is the resulting set of sticker permutations. If you
just think about permuting the cubies, then it is
difficult to take into account orientation.

Melinda:
>> >> >MC2D moves only do interesting things with odd
>> >> >numbers of reflections,

Me:
>> >> ? I do not understand the above statement. Why is a
>> >> 4-cycle (Rotate corner positions.) not interesting?
>> >> [(1,2)(3,4)] * [(2,4)] = [(4,3,2,1)]

Smith:
>> >You are definitely correct here David, but your example
>> >is unfortunately not (it actually contains 3
>> >reflections).

>I am quite familiar with cycle notation! I believe
>you are making an error as to what a move is in MC2D.

Quite likely. Remember, this started with my "I do
not understand the above statement."

>You seem to be talking about reflecting the entire
>square!

No. I am talking about permutations and symmetries of
the permutation patterns themselves - i.e., about the
group. MC2D can be thought of as implementing S4 (all
permutations of 4 things). The objects being permuted
are just the 4 corner cubies. The rest of the puzzle
does not really matter except to the extent that it
constrains what constitutes a twist. (Actually, any 3
of the puzzle’s 4 adjacent-corner flips will generate
the entire group.) D4, corresponding to rigid motions
of a square, is a subgroup of S4. D4 has permutations
considered to be reflections. Those are what I am
talking about.

So when I say that [(1,2)(3,4)] is a reflection in the
y-axis, I am talking about its effect on the positions
of the corner cubies, independent of the fact that the
edge cubies do not move. (As far as the group is
concerned, the edge cubies do not even exist.)

In the statement that confuses me, I took the
reference to "reflections" as referring to certain of
the group elements. I now realize that she is
probably using the word in the very low level sense
for which I have been saying "twist" or "flip",
because that basic move is a reflection of a single
slice.

When I was still thinking that "reflection" referred
to a group element as opposed to an action on a slice,
I inferred that she must be talking about permutations
that result from more than one flip. So I provided an
example with an even number of group elements (2)
which are reflections and which, on composition,
produce what I consider to be an "interesting"
product. Well, it is interesting, but the exercise
does miss Melinda’s point.

>As in MC3D, a move is any rotation or reflection of a
>face that preserves the general shape (i.e. takes
>positions of cubies to positions of previous cubies).
>I believe you are thinking about the entire puzzle as
>a face of MC3D, because the moves you are describing
>are reflections of a face in that puzzle.

I’ve lost you here. There are a lot more than 4
things being permuted in MC3D. I have the feeling
that you may be thinking more about a transformation
of n-space, where I am talking about permutations on 4
things.

>If you launch MC2D, it will be immediately clear that
>there is only one move, and that is to swap two
>adjacent corners. Thus, (1,2)(3,4) is actually two
>moves, one reflection of the North face and one of
>the South face.

OK. I see another source of my confusion. I had
taken "move" to refer to the result of any sequence of
flips, because it was clear to me that repeating the
same flip merely cancels the effect of the previous
occurrence of it. But I think I misunderstood and
that she intended "move" to refer to what you could do
with a single slice by composing multiple reflections
with respect to it. This is meaningful in higher
dimensions; but, in MC2D, it is trivial, as there is
only one "reflection" of a slice - the basic flip. So
you cannot combine _different_ reflections when
reorienting a given slice. Now I think I understand
that Melinda was just pointing out the degeneracy of
MC2D in the context of composing multiple reflections
on a single slice. The remark may well have been
tongue-in-cheek, since the only reasonable odd number
in this context is 1. The point is so simple that I
was looking for something more profound for it to
mean.

>> After I posted my previous message it occurred to me
>> that I missed yet another possibility and an even
>> larger group. We might call it a SUPER–super-
>> supercube. …

>A very interesting idea, and congrats for thinking
>of it! Who knows, I may end up doing my permutation
>formulas for these puzzles as well! :)

When I wrote,
>> Anyone for slice-swapping? ;-)
I was sort of poking fun at myself and anyone who
would take these variations too seriously. I was
pointing out that one could continue to invent new
rules for rearranging the cubie pile ad infinitum. At
some point one must draw the line and concentrate on
the variations that appear to be more pleasingly
elegant.

Carrying it to the ultimate extreme (for the order-3
3-puzzle): Suppose we had 27 cubical blocks with
unique identifiers on all their faces. Take a
particular 3x3x3 pile as being the initial state.
Then any other way of piling them determines a
permutation of the identifiers relative to the
original pile. Now suppose we start making random
stacks (busy monkeys implementing the most liberal
rules for rearranging the stack) and noting the
resulting permutations. On average, how many times
must we scramble it before the permutations we have
collected will generate the entire group?

Regarding elegance: I think the big success of the
original Rubik’s Cube arose from the fact that it is
an elegant object. Even folks who have no hope of
solving it can marvel at the cleverness of the
mechanism.

On Saturday, September 27, "Roice Nelson" <roice3@gmail.com> wrote:
>Melinda’s thought of allowing reflections across any subspace of
>intermediate dimension is interesting and worthy of further study! I had
>only previously considered reflections across hyperplanes of dimension n-1
>only as you describe, but the behavior of the other cases also appears to be
>uniquely determined. Consider point
>reflections<http://en.wikipedia.org/wiki/Point_reflection>,
>which are easy to think about in any dimension (and only
>orientation-reversing in odd dimensions).

I must confess that I had never contemplated these
other kinds of "reflection" at all. Reflection in an
(n-1)-dimensional hyperplane is the only kind of
reflection that I ever thought about when hearing the
word "reflection". Thus I misinterpreted Melinda’s
reference and assumed that reflecting about a given
hyperplane of dimension less than n-1 amounted to
picking a hyperplane of dimension n-1 which contained
the given one. I had never even heard of "point
reflection". But I do see that meaningful
transformations of n-space can be specified this way.
New insight!

Note that a point reflection in 2-space is not a
reflection at all in the sense in which we have been
using "reflection" here; so, in our mirroring context,
you would not really expect it to come up. This is
probably true for other reflection-variation/dimension
combinations as well.

>However, I do have the feeling (no proof
>unfortunately) that the n-1 hyperplane reflections
>are the most fundamental operations. …

>Anyway, my intuition says if we limit ourselves to
>reflections across hyperplanes as you’ve described,
>we can reach all mirror puzzle permutations, …

My own intuition is similar. I think I might even be
able to formalize it; but I am not sure it would worth
it, as it seems clear enough. There are only so many
ways to reorient an n-cube (or an n-slice), mirrored
or not. If you already have all the non-reflecting
reorientations, then you only need one of the
reflections to generate all the rest. (You can get
that by an enumeration and uniqueness argument.)

>Btw, one suggestion for naming the puzzle types we’ve
>laid out would follow the language of topology:
>orientable or nonorientable, the latter being the
>case where an odd number of reflection moves was
>allowed.

I may not understand what you are driving at here.
If the issue is "reflection allowed or not" why would
"nonorientable" be preferable to "reflection
allowed". At present I don’t get a suitable semantic
reaction from "nonorientable".

Regards,
David V.