Message #546

From: lucas_awad <lucasawad@gmail.com>
Subject: Re: Something interesting and strange about permutations
Date: Sat, 09 Aug 2008 18:35:25 -0000

Yes, that was what I wanted to say.

About the faces not affected (not changing stickers with others) I
will compare MC3D (with the called "unreal movements") with MC4D.

If we have an MC3D with that kind of moves, we will be doing moves in
two ways because we have 4 2-color cubies per face, as we do in 3 ways
in MC4D, with 6 2-color cubies per face.

So moving in two different axes is:

1 2 3 —> 3 2 1
4 5 6 —> 6 5 4 (from 4th and 6th cubies)
7 8 9 —> 9 8 7

1 2 3 —> 7 8 9
4 5 6 —> 4 5 6 (from 2nd and 8th cubies)
7 8 9 —> 1 2 3

1 2 3 —> 3 2 1 —> 9 8 7
4 5 6 —> 6 5 4 —> 6 5 4 (we can also get a U2 move)
7 8 9 —> 9 8 7 —> 3 2 1

If we pick the movement from 3-color pieces, we can do also U and U’
moves, like a move mirroring from both 1st and 2nd cubies (but this is
going out the 2-color pieces possible moves, so I don’t see it really
possible, but is what we do in a rubik’s cube, moving like that).

Extending it to MC4D and doing always three movements from 4-color,
3-color and then 2-color cubies, we always get a move that is like a
4-color move or a U2 (a combination of two 4-color moves).

So what I think, perhaps I am wrong, is that those limited moves are
the really possible, but as I said, perhaps I am wrong.

— In 4D_Cubing@yahoogroups.com, "Roice Nelson" <roice3@…> wrote:
>
> Hi Lucas,
>
> It took me a bit, but I think I’m now mostly following what you are
saying
> here. If I am interpreting correctly, I think what you have effectively
> discovered is how the MC2D rotations are not analogous to any of the
other
> puzzles (except David Vanderschel’s extended MC3D functionality)
because the
> MC2D face rotations allow mirroring. The MC4D and MC5D puzzles
don’t permit
> mirroring twists, which is in more strict analogy with the physical 3D
> Rubik’s cube. There is flexibility in how exactly we want to carry
over the
> analogy of twists, but I like the MC4D
> FAQ<http://www.superliminal.com/cube/FAQ.txt>description of what it
> means to make a twist, which says "Take the face you
> want to twist and remove it from the larger object. Turn it around
any way
> you like without flipping it over, and then put it back so that it fits
> exactly like it did before.". If we were to adhere to this in MC2D, no
> scrambling twists would be possible, and hence it would be
degenerately easy
> to solve
<http://www.gravitation3d.com/magiccube5d/2d_rubik’s_cube.jpg> :)
>
> I found your observation about MC4D twists "only affecting 4 faces"
> intriguing! All the MC4D twists (except the identity) do in fact
affect all
> 6 adjacent faces, but it sounds like you are making a distinction
with the
> 3D case where there is no possibility to make a twist and have all
stickers
> on an adjacent face remain the same color. In MC4D, all the adjacent *
> cubies* are getting shuffled around, but some twists (not all!)
allow the
> sticker colors to remain the same on 2 of the 6 adjacent faces.
This was a
> cool point for you to make, as I have never explicitly focused on that
> contrast with the 3D puzzle before. Likewise in MC5D, the cubies on 8
> adjacent faces are always affected with every twist, but some twists
allow
> stickers on up to 4 of the 8 adjacent faces to not change color (in
our MC5D
> implementation, this is actually the only possibility since the
twists are
> not fully worked out). I don’t think that we are understanding the
higher
> dimensional puzzles wrongly, but that this different behavior arises
due to
> the extra space in the higher dimensions.
>
> Also, it is possible on the 3D, 4D, and 5D cubes to build a 3-color
series
> based on two 2-color series, although the 2-color series require 4 moves
> instead of 2. An example in the 3D case (with the 2-color series in
> parenthesis):
>
> (R’FRF’) B’ (FR’F’R) B
>
> Anyway, I hope I was on the right track and that these ramblings are
> usefully related to your thoughts…
>
> Roice
>
> P.S. As a short aside, it is an interesting fact that the motion of any
> rotation can equivalently be described as a set of two reflections,
which is
> why your U2 example is achievable as two of the "unreal movements".
Visual
> Complex Analysis <http://www.usfca.edu/vca/> is a fantastic source
to learn
> much more about this.
>
>
> On 8/6/08, lucas_awad <lucasawad@…> wrote:
> >
> > After solving the MC5D, I have discovered something a bit strange
> > about permutations.
> >
> > As everyone who read the solution for MC4D know, we can permutate the
> > 4-color hypercubies by doing the 3-color series two times (one of them
> > the reverse).
> >
> > But, why we cannot permutate the 3-color pieces with doing two times a
> > 2-color permutation with 2 moves on MC2D?
> >
> > Because the face rotation is different.
> >
> > When rotating a "face" in MC2D, the move is like this:
> >
> > 1 2 3 –> 3 2 1
> >
> > In 3D, the same movement should be:
> >
> > 1 2 3 –> 3 2 1
> > 4 5 6 –> 6 5 4
> > 7 8 9 –> 9 8 7
> >
> > But that’s not what we really do with a rubik’s cube, it is this (it
> > would we for example U2, if it is "U" face):
> >
> > 1 2 3 –> 9 8 7
> > 4 5 6 –> 6 5 4
> > 7 8 9 –> 3 2 1
> >
> > If you see, this algorythm (2-color permutation in MC2D) doesn’t only
> > do 4-6 permutation, also 2-8, which don’t happen in MC4D with 3-color
> > series.
> >
> > By doing the previous movement (the unreal one) we only affect two
> > faces which change their stickers (the same as MC2D), but with a
> > rubik’s cube (and also MC4D and 5D) we are affecting 4 adjacent faces
> > (the other keep still the same stickers). So with the unreal movement
> > we would be able 3-color pieces by doing the sequence: ( F - R ) U ( R
> > - F ) U
> >
> > However, in MC4D we do movements that only affect 4 faces, and that
> > allows us to easily permutate the 4-color hypercubies by doing the
> > 3-color series algorythm. The fact I’m thinking now is if in MC4D and
> > MC5D all adjacent faces should be affected to make the rotation real,
> > and we are understanding higher dimensional puzzles wrongly.
> >
> > I hope that you understand what I have said.
> >
> > Greetings
> > Lucas
> >
> >
> >
>