Message #3651

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
Date: Wed, 15 Feb 2017 11:20:06 -0600

What an awesome puzzle and flurry of activity! This is really cool
Melinda, and there is so much to think about. I especially find the
limitations imposed by the magnets intriguing. I’ve been out of town and
missed much of the real-time fun, but read through the thread and watched
Mathologer’s video last night. I must confess I have not fully digested
all the great ideas flying around, but I had a few thoughts that still seem
useful to share.

Like Nan, my first instinct was to understand the lower dimensional case,
and I’m glad I tried. I started to think of a 4x2 block of squares, and it
took me a bit to realize that was making it more difficult to think about.
It may be obvious to folks, but my Aha moment was that each piece of
Melinda’s puzzle is really a face of the dual polytope, the 16-cell
<https://en.wikipedia.org/wiki/16-cell>. In the lower dimensional case,
each piece is a face of the octahedron. So it is natural to make the
2^3 analogue a set of triangles, a "net
<https://en.wikipedia.org/wiki/Octahedron#/media/File:Octahedron_flat.svg>"
of them in fact. This all made me realize Melinda’s 16 pieces are
representing tetrahedral faces, again a net
<https://en.wikipedia.org/wiki/16-cell#/media/File:16-cell_net.png> of the
16-cell.

I think the brilliance of Melinda’s design is that each tetrahedron is
represented by a cube, so the net becomes this nice 4x2x2 block. A less
elegant way to approach this puzzle would be to use the net of 16
tetrahedra directly, which should work even if more awkward. Using cubes
for tetrahedra is possible because the tetrahedral group is a subgroup of
the octahedral group. This doesn’t work in the lower dimensional case
because a triangle group is not a subgroup of a square group (look up cyclic
groups <https://en.wikipedia.org/wiki/Cyclic_group> if you want to
research). That is why it was unnatural to deal with squares for the 2^3
analogue.

So now I’m considering Melinda’s puzzle as a net of the 16-cell, which
makes it easier to think about what general rotations and twists are.

Given that, I thought I’d highlight a couple twists I don’t think I’ve seen
yet:

  1. A slight modification of Melinda’s reorienting move… Pull apart
    the two halves, but leave one fixed and rotate the second 90 degrees and
    reattach.
  2. If we were to allow interim jumbling, I think we can get 90-degree
    twists of the blue and orange faces. Instead of performing a
    180-degree rotation maneuver here, you would take the 4x2 block and
    translate it a step. The end result would be a 3x2x2 with two 2x1 blocks
    protruding off of it on opposite sides. But then you could just do a
    reorientation by rolling one of those protrusions around to meet the other
    and recover the original 4x2x2 block. Hope this is clear.

I want one Melinda! When are you going to set up a shop? :D

Cheers,
Roice


On Sun, Feb 12, 2017 at 9:35 PM, Melinda Green melinda@superliminal.com
[4D_Cubing] <4D_Cubing@yahoogroups.com> wrote:

>
>
> Yes, the reorientations were all 120 degree twists about the long
> diagonals through their black stickers. I knew the black stickers had to
> remain where they were because it was a twist of the black face.
>
> The magnets do not want to allow the twist or the piece reorientations,
> though the central 2x2x2 cube was happy once it was fully reorientated. The
> frames where you see my hand are the configurations that the magnets do not
> allow. Matt’s pattern should allow everything.
>
> -Melinda
>
> On 2/12/2017 7:23 PM, Christopher Locke project.eutopia@gmail.com
> [4D_Cubing] wrote:
>
> Melinda,
>
> Yes, that looks like a correct twist of the +y hyperface about the +z
> axis! If the colors are labelled: (-x brown, +x purple, -y gray, +y black,
> -z light blue, +z green, -w blue, +w red), then that 90 degree twist should
> move those middle 8 physical cubies around in a 90 degree twist just like
> you did, and the x/w stickers should move blue -> purple -> red -> brown.
> From step 3 to 4, I take it you did a 120 degree twist about a diagonal
> axis that goes through the center of each cubie and the total center of the
> physical puzzle (where black stickers are)?
>
> By the way, was the magnetic orientation okay after doing those twists?
> In the video move I pointed out (https://youtu.be/Asx653BGDWA?t=1410),
> you had problems doing some twists after the double inversion due to magnet
> positioning.
>
> Best regards,
> Chris
> On 2017年02月12日 17:55, Melinda Green melinda@superliminal.com [4D_Cubing]
> wrote:
>
>
>
> Matt,
>
> Christopher’s messages were a bit to opaque to me but I’m starting to get
> the gist. It’s good to know that this 90 degree twist is a valid move
> though it’s unfortunate that it’s far from pure. These almost look like
> gear-cube twists now. I even think I can guess how the orientations are
> supposed to end up after the appropriate reorientations of the black
> pieces. (Alternating CW and CCW twists of each piece about their black
> stickers.) I’ve attached a sequence of snaps showing the process. (Also
> here <http://superliminal.com/misc/twist90cp.jpg> in case the attachment
> doesn’t work.) The second snap shows the twist in progress. The third shows
> it completed, with me holding it in place against the magnets. You can see
> what I mean about the puzzle looking completely scrambled by this one
> twist. The fourth snap shows it with all 8 of the twisted pieces
> reoriented. The interesting thing is how it results in a much less
> scrambled looking puzzle.
>
> Christopher,
>
> I hope the photos helped. One interesting to note is that the end result
> of the sequence (plus a simple rotation) resembles the result of the double
> swap you highlighted in the video (https://youtu.be/Asx653BGDWA?t=1410)
> so maybe there’s hope for a more practical way to reach the full 2^4 state
> space.
>
> Thanks!
> -Melinda
>
> On 2/12/2017 3:46 PM, damienturtle@hotmail.co.uk [4D_Cubing] wrote:
>
>
>
>
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>
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>