# Message #586

From: Melinda Green <melinda@superliminal.com>

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

Date: Fri, 26 Sep 2008 00:02:24 -0700

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 <br> 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