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