Message #545

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Something interesting and strange about permutations
Date: Fri, 08 Aug 2008 19:30:19 -0500

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@gmail.com> 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
>
>
>