# Message #573

From: Roice Nelson <roice3@gmail.com>

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

Date: Wed, 17 Sep 2008 22:24:31 -0500

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

On 8/14/08, lucas_awad <lucasawad@gmail.com> wrote:

>

> So we are saying that MC2D does reflection moves and MC3D simple

> rotation.

>

> What I’m trying to see is if MC4D should have some different type of

> movement, not a reflection and not a simple rotation allowed. The same

> with MC5D, with moves that cannot be done in MC4D. Because I think

> that 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.

>

> This obviously would make MC4D and MC5D a lot more difficult, but it

> is ok if we compare the difficulties between MC2D and MC3D, where the

> first can be solved without trying to, I mean, with random moves.

>

> I know that a rubik’s cube was done to be a rotation puzzle, but if we

> go up or down in dimensions, I think that we shouldn’t allow (in

> higher) or we can’t (in lower) implement the simple rotation around an

> axe (one axe).

>

> Also, what I think is that we cannot know the exactly behavior that a

> higher dimensional puzzle would have, because we live only in a 3D

> space, and we cannot see anything higher than 3D, and cannot imagine a

> world in more than 3D, as I think that it cannot be imagined to live

> in a 2D world, as a 2D object would be invisible to our eyes. But, us

> I said, that is only what I think.

>

>

>