physical 2x2x2x2x2 or 2^5 design (warning, it isn't pretty)

[Grant S]

Hey!

Its another update from the discord server.
(Disclaimer, I'm going to try to credit people throughout describing the process / timeline of events, but in case I forget to mention someone or mess up some of the crediting, Akkei, Jedi, Hactar, and Luna have all been very critical to the process of having this happen).

(Disclaimer #2, I'm just in highschool, and do not know a lot of the correct vocabulary for all of this, I'll do my best to explain everything well, but it might not all be 100% correct)

In the past we had already considered the fact that something with icosahedral symmetry could be used to represent the possible orientations of a 5 dimensional 5C piece for a 2^5, however, we didn't have a nice way to color a piece, and icosahedrons don't stack well, so we never really thought too much about it.

We had also previously made renders of this design for a domino reduced 2^5 (one axis is restricted to 180 degree turns, so grey and black can't swap to any other axis)

Jedi / Anderson beta brought up this idea again a bit over a week ago, and akkei brought up the rhombic triacontahedron.

Which has sets of orthogonal faces, allowing them to stack on each other well.

Akkei took the design of the domino reduced pieces and kind of fit them into sets of 6 faces that are orthogonal

(grey here is still locked, and you can see the white, purple, green, and red still shaped like the normal 4C pieces from melinda's 2x2x2x2)
However, Akkei used the extra faces to make all of the combinations of these with different colors in the circles.

Although this elegantly builds off of our old design and is decently easy to read orientations from, I realize that the complicated coloring was technically redundant and that if we colored it correctly, we could still maintain the distinction between all 60 possible orientations a 5c piece can have.

Akkei and I both blindly put 32 of the pieces together to make renders of a possible 2^5, but we realized issues with them pretty quickly and moved on to later designs.
After many iterations and moving around how the pieces are oriented and looking at the symmetries this piece has and the symmetries of the 4Cs in the 2^4, we can to this design.

It is very messy, and very hard to tell what you are looking at at first, but I'll try to break it down as much as possible.
Pink and purple are on the "5 dimensional outside" of the pieces, analogous to the outside colors on the 2^4.
Each of pink and purple are a 2x2x2x2 hypercells, analogous to the 2 main 2x2x2 cells on the 2^4
Individual twists of the pink and purple hypercells can be done by doing any 2^4 rotation (simple rotations or gyros, and we stickered the pieces so that the gyros you know and love for the physical 2^4 will work for this!) This is much like how a valid move is any rotation of one of the 2 main 2x2x2 cells in the phys 2^4
Other twists of other hypercells can be done through some other things like the 180 twists and axial twists for the physical 2^4, but we haven't well established what would be allowed in terms of that.

In terms of reading the orientations of the pieces, it isn't very clear at first, but lets look at the U side of the grey half of the pink hypercell

Here white is on U, Blue is on F, red on I (the inside of the hypercell), orange on O (the outside of the hypercell), and green on B
Grey is on the L or left of the hypercell, (the other half of the pink hypercell has black here)

And pink is on the "5d outside"

If this 2x2x2 chunk of the cube we are looking down on were to have a Y rotation done on it (as part of a larger twist / rotation), you can see white, grey, and pink would stay as they are, green would go to where orange is, orange to blue, blue to red, and red to green, just like the side colors on half of a physical 2^4.
It is easiest to read the white / U color from this angle, meeting distinctly at the outer edges of the cell.

If you look at another angle from the same chunk of pieces, you can see blue is on F here.

(If you think you can move the stickers around to be in different places to maybe make it more readable, we tried, but you lose the symmetries that the 4d gyros we have used to reorient the pieces, so this stickering is forced, thanks to luna and hactar for helping to break down this process with me to find this stickering).

Its really hard to put all of how to read it into words, its just some intuition that you build up looking at these for a while.

A 5d gyro algorithm would have to be made to swap out the pink and purple with another axis, we don't have a sequence of moves or anything for this, jedi has talked about potentially some kind of off axis restacking, but nothing is figured out for sure yet.

I'm sure a lot of you will have questions, so I and the other people who were involved in making this happen will try to answer as well as we can. And I'm sure I or they will have some notes to clarify about the design.
And I know it is super confusing at first, but it is a working physical 5d puzzle!
And we thought it would be good to make all of you aware of this innovation that has been happening!

(Yes, I am likely going to end up building this at some point)

 - Grant

[Joel Karlsson]

Amazing! I've also noticed that dodecahedra or icosahedra could be used as pieces for a 2^5 before, although it's a big problem that they can't be stacked. Using rhombic triacontahedra is very clever and seems like it should work (haven't convinced myself that your suggestion works in detail yet).

Congratulations on this transdimensional achievement!

Best regards,

Joel

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[Roice]



I agree, this is incredible. 

It is especially cool to watch because it takes me back to 2006 when we were uncovering how to create MC5D virtually on the computer. That felt impossible at multiple points along the way, but different folks had ideas that unblocked being stuck and eventually things came together. In the end, it felt inevitable. It’s amazing to see another generation of folks taking things to new heights in a similar way. 

If you haven’t seen it, this makes me think (by analogy, not details) of Don Hatch’s 2-dimensional representation of arbitrary N-dimensional cubes. Y’all seem to be climbing a similar ladder with 3-dimensional representations! 

His code comments have ASCII pics worth studying and thinking about. 

http://www.plunk.org/~hatch/MagicCubeNdSolve/

I’m grateful to Grant and others for sharing the discord adventures here, since the bandwidth there quickly exceeds this old guy’s capacity :D (I would have been the oldest on the age survey if I had seen it in time 😆).

Also, have y’all thought about how the magnets are going to work? <Runs away!!!>

Cheers,

Roice

On Feb 20, 2023, at 2:41 AM, Joel Karlsson <joelkar...@gmail.com> wrote:



To view this discussion on the web visit https://groups.google.com/d/msgid/hypercubing/CAEohJcFJU_w1a2q-%2BPrXxzsOLmOnH_Tw%3DHY7zPFfX1PRfqtDfQ%40mail.gmail.com.

[Joel Karlsson]

I would also be interested in hearing about what magnet configurations you have thought of. Two parallel bar magnets to create a square of alternating N/S below each face would of course suffice but might not be the most efficient arrangement 😅

Thanks indeed for sharing these developments here.

Best,

Joel

[Grant S]

Our current thought is it would take 3,840 magnets. (4 per face x 30 faces x 32 cubes)

Turns out to be the same number of magnets as the physical 3^4. There is talk of some velcro stuff though, jedi, hyperespy, and I are all working on making our own physical 2^4s that use colored velcro instead of magnets!

[Joel Karlsson]

Okay, are you thinking of placing them with just one pole close to the surface, similar to Melinda's original design, then? Would that be easier to build than halving the number of magnets and placing them with both poles close to the surface (might require stronger magnets...)?

[Grant S]

Yes, it would be easier to build and also not require as strong of magnets. The plan would be a magnet layout with the polarities shown in the attached image.

[Joel Karlsson]

I see, thanks for the details!