# Message #596

From: David Smith <djs314djs314@yahoo.com>

Subject: Re: [MC4D] Something interesting and strange about permutations

Date: Sun, 28 Sep 2008 12:39:04 -0000

Hi David,

I would like to apologize for repetately implying that

you were incorrect about this topic in any way. By

doing so, I called into question your intelligence

and experience in these topics (which is clearly much

higher than my own), and demonstrated my own

lack of knowledge. I should have realized after you

corrected me that this confusion was based on a

misunderstanding, not that "I was correct and you

were not". You may think I am being too hard on

myself, but I must say what I feel.

— In 4D_Cubing@yahoogroups.com, David Vanderschel <DvdS@…> wrote:

>

> On Saturday, September 27, "David Smith" <djs314djs314@…> 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.

I was entirely wrong here! I completely forgot what the

basic notion of the group of a permutation puzzle was

(i.e. the set of sticker permutations).

> 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."

When I made this statement, I was basically saying

that you were incorrect, and I apologize for that.

> >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.

The basic source of our misunderstanding was what

Melinda meant when she said "reflection". I thought

it meant a single move, or a swap of two corners.

Thank you for reminding me about S4 and D4, and

how they relate to MC2D.

> >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.

Clearly what I said here was poorly phrased. It was

once again based on my assumption that you were

incorrect. I was thinking that you were confusing

the moves of MC2D for the moves of a face of MC3D,

which is of course absurd, as you would definitely

not make such an obviously wrong mistake!

> >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.

I was taking "move" to be a single face reflection,

and that when Melinda said an odd number of moves was

not interesting, that she was referring to a sequence

of moves.

> >> 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.

Yes, the Rubik’s Cube is a very simple and elegant

physical representation of a large group. Your

super-super-supercube concept was, as you explained,

just a further definition of how to rearrange

the cubies. Calculating the formulas for endless

redefinitions of the rules would be pointless! As

you said, we should concentrate on what we find to

be a worthwhile and satisfying concept.

> On Saturday, September 27, "Roice Nelson" <roice3@…> 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!

I too had never heard of such a reflection but it

does make sense, and is a completely analogous

way to define reflections about a space at least

2 dimensions lower than the object one is

considering.

Regarding my statements here, I think you may

feel that I should not be so critical of myself,

but I felt I must apologize for thinking and saying

you were incorrect (especially the second time),

as I should have realized that this was just

a misunderstanding since you explained yourself

so clearly in your last post.

All the Best,

David