# Message #587

From: Jenelle Levenstein <jenelle.levenstein@gmail.com>

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

Date: Fri, 26 Sep 2008 10:43:39 -0500

If you implement moves as reflections then each move will turn some of the

pieces on the cube inside out. Do you think that if you implemented the 3^3

cube with your new definition of moves it would be easier or more difficult

to solve? Although it may be possible to do all the rotations on a 3D cube

using reflections it would take a long time for anyone to get used to and

the possibility of being able to put a piece in position inside out will

make solving more complicated. However it sound like a very interesting

puzzle.

On Fri, Sep 26, 2008 at 2:02 AM, Melinda Green <melinda@superliminal.com>wrote:

> This subject of dimensional analogy is very interesting to me. I think

> that the way I approach it is to ask a subtly different question than

> both of you guys do. You each seem to be asking what *is* the correct

> analogy in each dimension whereas I prefer to ask what *should* we

> choose the right analogy to be. What we’re looking for is the best

> definition of an N-dimensional set of puzzles, but "best" in this case

> is not the answer to a mathematical question, rather it *is* the

> question. "Best" is the question that gives the Rubik’s cube as the

> answer in 3D plus puzzles that please us the most in the other

> dimensions. From my perspective Lucas is making a suggestion which is

> entirely reasonable (I.E. not wrong) but which Roice does not find

> satisfying. It doesn’t work very well for me either but it is not wrong.

> Roice on the other hand is suggesting a definition based on

> rotations–one that I preferred too, at least maybe until now. My shift

> in thinking didn’t come from the realization that MC2D didn’t seem to

> fit perfectly into this definition. It would be nice if it did fit but I

> was perfectly happy for it to be an exception, mostly useful for

> illustrating state graph properties for these puzzles. By the way, I

> added a nice image of the MC2D state graph to the applet page along with

> some descriptive text. See http://superliminal.com/cube/mc2d.html.

>

> The thing that really struck me was Roice’s observation that rotations

> can always be described with pairs of reflections. This started me

> thinking that perhaps the "best" analogy might only involve reflection

> moves. Looked at this way, perhaps the original Rubik’s cube is the

> oddity which needed to use planar rotations to satisfy the practical

> demands of 3D objects in the physical world. It certainly makes for a

> fun and satisfying puzzle but perhaps we shouldn’t be more focused on

> the way that the plastic puzzle operates than the mathematical group

> that it operates upon.

>

> So now we have the basis for defining a new analogy that can reproduce

> our puzzles in N dimensions:

>

> "A valid move is any combination of reflections of a hyperface

> that leaves its orientation unchanged."

>

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

> and the Rubik’s cube is only physically implementable for even numbers.

> Looked at this way, perhaps the "best" version of MC3D would allow both

> odd and even numbers of reflections per move but with the option to

> restrict the available moves to even numbers of reflections in order to

> satisfy people with a nostalgia for physical reality. ;-) Looking at

> David’s implementation, I now see that this is exactly what he did

> although it is the default mode and that each reflecting click must be

> preceded by ctrl-q. David: I would love if you would add a new toggle so

> that plain clicks always perform reflection moves and ctrl-q clicks

> perform rotations.

>

> I purposely call all of these operations "moves" instead of twists

> because thinking about twisting drags in all the problems with rotations

> that we’ve been struggling with. I kept saying that it was better to

> think about planes of rotation rather than axes of rotation, but that

> seems unnatural for a lot of people. If we base the discussions on

> reflections, then this suddenly becomes quite natural.

>

> What do you people think? Is this a good basis for defining

> N-dimensional twisty puzzles? If so, the only things that remain to

> figure out are the best user interface for computer implementations

> based on this model, and how best to animate the moves, if at all. I’m

> not signing up to implement anything anytime soon but I do enjoy the

> thought exercise. I did not have any sense for what would make for a

> good user interaction model but I think that David may have pointed us

> in the right direction. I do have some ideas for animations that might

> work. Let’s start with a single reflection move. These can be

> reflections about a point, line, or any space of dimension lower than

> the puzzle itself. The simplest animation would seem to be a linear

> interpolation of the beginning and ending vertex positions. That would

> leave a moment of degeneracy in the middle when the part being moved

> gets flattened into that point or line, etc. but that’s fine. Imagine

> that happening in MC2D. A 3x1 slice would collapse into a 0x1 line at

> the midpoint of the motion. It is interesting to notice that that is

> exactly what you would see in the current projection if the motion was

> implemented as a 3D twist coming out of the plane and then back as some

> people have mentioned. Maybe an equivalent reflection move on MC3D would

> involve an affine 4D rotation in order to flip over a 2x2x1 slice,

> leaving it turned inside-out? It’s an interesting thought.

>

> And then what about those pairs of reflections that Roice says can

> produce rotations? How might we animate those? It seems like we would

> have the same two natural choices. We could perform a linear

> interpolation of the vertex positions, or maybe we could find pure

> rotation matrices that achieve the same results. Even if all rotations

> can be expressed as pairs of reflections, it might not follow that all

> pairs of reflections can be expressed as rotations, but if it is true

> then we will have found a way to redefine all of our puzzles, including

> the original Rubik’s cube. So now we have come full circle and it is

> time to ask what have we gained. First we might have gained a simpler

> way to way to define the puzzles we already know and with some new

> moves. Second, it might show us how to implement these puzzles in any

> number of dimensions. And finally, it might give us back all our

> familiar puzzles (Rubik’s cube, MC4D, Hyperminx, etc.) as special cases

> in which moves consist of pairs of reflections. Oh, and it gives us an

> MC2D that is *not* a special case! And I swear that was not my

> intention! :-)

>

> -melinda

>

>

> Roice Nelson wrote:

> > Hi Lucas,

> >

> > Sorry for the very long delay in responding to this. I didn’t want to

> > leave the possible issues you raised unresolved in the thread, but

> > hadn’t taken the time to write out a response until now. I believe we

> > can know the behavior of the higher dimensional puzzles exactly if we

> > are precise with our analogies. In a book I read recently,

> > Donal O’Shea wrote about mathematics "absolute precision buys the

> > freedom to dream meaningfully", and I agree!

> >

> > So anyway, I am afraid I have to dissent with the statement "if we go

> > up in dimensions we mustn’t be able to do the same kind of movements

> > that we do in a lower dimensional puzzle". It seems this is observing

> > a pattern that was the result of implementation choices that were made

> > rather than observing a trend through the sequence of dimensions while

> > explicitly controlling the analogies. To make MC2D interesting,

> > Melinda decided to allow reflection based twists, but there is nothing

> > fundamental about lower-d puzzles being able to do movements that the

> > higher-d puzzles can not. On the contrary, as one moves up the

> > dimension ladder, the capability for additional motions only

> > increases. There is no motion capable of being done in 2D but not 3D,

> > or in 3D but not 4D. The set of motions in higher dimensions is a

> > superset, containing all the lower-d motions plus more that are

> > available because of the extra space.

> >

> > I’d argue the reason for the higher difficulty of MC3D vs. MC2D has

> > much more to do with size of the state spaces of the two puzzles than

> > the motions allowed in these particular implementations.

> >

> > To figure out our options for making a twist, we can catalogue all the

> > possible "similarity" (or shape preserving) motions in any given

> > dimension of Euclidean space, and these are translation, scaling,

> > rotation, and reflection. There are no more I am aware of that show

> > up for higher dimensions, though rotations do get much more

> > interesting as we climb to higher spaces. Trying to use either

> > translation or scaling as a basis for twisting would only serve to put

> > the puzzle in quite a different, unusable form (imagine a 3D cube

> > "twisted" to have one face scaled to twice the size of all the

> > others). This leaves rotation and reflection as the only two motions

> > whereby the overall puzzle shape is the same before and after a

> > twist. One can’t physically reflect an object within a given

> > dimension without either (1) having short term access to a higher

> > dimension that the object could temporarily move through or (2) if the

> > space had a certain topology (e.g. a mobius strip or klein bottle),

> > moving the object through a path that flipped it (but a topology like

> > this of course has not been observed in our universe to date). Hence

> > the analogical argument for disallowing reflections on any of these

> > puzzles. But we can of course loosen the analogy and choose to

> > include them in software implementations if we want it as a unique

> > extension. And we can do this for puzzles of any dimension.

> >

> > Aside: If one chose to completely disallow rotations but allow a

> > minimum set of reflections for twisting, you could still get all the

> > possible permutations a puzzle would have with rotations alone (and

> > more actually). This is because of a property that previously came

> > up, that a rotation can equivalently be expressed as a set of 2

> > reflections. Writing this paragraph made me realize the 3D puzzle

> > reflection extension is more interesting than in the 2D case because

> > there are similarity reflections through diagonal axes of a face in

> > addition to coordinate aligned ones. I just checked David’s MC3D

> > implementation and saw that he handles this, distinguishing

> > reflections by whether an edge or corner is clicked. Nice! (maybe I

> > knew this in the past and my mind is just failing me)

> >

> > Well, I’ll stop prattling about this. I hope I wasn’t too

> > disagreeable on this topic and just as you said, this is only what I

> > think :) But I really do think MC4D has it right when comes to how

> > the twisting is performed.

> >

> > Take Care,

> > Roice

>

>

>