# Message #3652

From: Melinda Green <melinda@superliminal.com>

Subject: Re: [MC4D] Re: Physical 4D puzzle achieved

Date: Wed, 15 Feb 2017 16:44:16 -0800

On 2/15/2017 9:20 AM, Roice Nelson roice3@gmail.com [4D_Cubing] wrote:

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> 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.

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> 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.

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> 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.

This is most clear when looking at Oskar’s lovely rendering <http://superliminal.com/cube/physical2x2x2x2oskar.jpg> of my previous concept. I was considering pieces as some sort of beads that would move along wire arcs, perhaps made out of some squishy material such as foam rubber. I don’t remember an "aha" moment but the current design probably resulted from staring at this picture and imagining various ways of squishing the parts while performing rotations.

> 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 what does the 2D analog look like using triangles?

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> 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.

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> * A rotation is any detachment of a connected subset of cells, and subsequent reattachment that preserves the structure of the net.

> * A twist is any planar cut of the net into two equal parts, followed by an arbitrary reorientation of one of the halves (and optionally one could also make net preserving changes to the half too), then a reattachment back along the planar cut.

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I like it!

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

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> 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.

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This is the same as a rotation followed by a simple 90 degree twist. The way I like to think about it is that you can pull the red-blue halves apart, reorient them, and connect them back together however you like. If the result can be reached by a sequence of simple rolling moves, then it’s a rotation, otherwise it’s a twist.

> 1. 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.

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I think the shearing step is clear enough though I’m not sure what orientation you intend the 2x1 block to end up in. Regardless, that would look like an even stranger result than anything I’ve seen so far because half of the pieces would have single stickers at the corners, and half would have 3-color junctions. I had been calling such oddly reoriented pieces "inverted", but is "jumbled" the more correct term? It would be nice if the terms jumbled and bandaged are the correct analogies.

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

Well the first question is which version do you want? The bandaged (current) version or Matt’s more general arrangement? I’m working with the dice guy right now to see if and how the 24 magnet pieces might be made and what they will cost. The magnets will likely need to be recessed so that it’s not too hard to turn. That means either an extra step to fill in the gaps, or just sticker over them. The nice thing is that none of the diagonal sticker cuts will cross any magnets.

I’m happy to produce these for group members at cost, at least until it becomes too much work. It took me nearly 4 hours to sticker this one (nearly 200 triangles!) but I’m sure that with practice, I can soon do it in under 2 hours.

-Melinda

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

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> 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.

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> 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.

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> -Melinda

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> On 2/12/2017 7:23 PM, Christopher Locke project.eutopia@gmail.com <mailto:project.eutopia@gmail.com> [4D_Cubing] wrote:

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>> Melinda,

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>> 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)?

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>> 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 <https://youtu.be/Asx653BGDWA?t=1410>), you had problems doing some twists after the double inversion due to magnet positioning.

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>> Best regards,

>> Chris

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>> On 2017年02月12日 17:55, Melinda Green melinda@superliminal.com <mailto:melinda@superliminal.com> [4D_Cubing] wrote:

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>>> Matt,

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>>> 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.

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>>> Christopher,

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>>> 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 <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

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>>> On 2/12/2017 3:46 PM, damienturtle@hotmail.co.uk <mailto:damienturtle@hotmail.co.uk> [4D_Cubing] wrote:

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