Melinda's 2x2x2x2 - Canonical Moves
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- เผยแพร่เมื่อ 28 มิ.ย. 2024
- The canonical moves:
0:35 1) Simple rotation
0:39 2) 90 degree twist of outer face
0:47 3) 180 degree twist of side face
0:54 4) 90 degree axial twist of central face
1:26 5) Outer face twists
1:44 6) Gyro rotation
Useful non-canonical moves:
3:10 Clamshell move
3:24 Restacking moves for scrambling
5:05 Flipping a single piece
6:01 Reassembling the whole puzzle
This is a follow-up to • Melinda's 2x2x2x2 - In... - เกม
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I absolutely love this! A long while ago I saw the Mathologer video on your software. It's genuiely amazing!
Thank you so much! He now has a couple of my new puzzles and I'm hoping he'll create a video about it too. He's a pretty amazing guy.
I think the restacking moves should be legal in solves - while they might not correspond to a "real turn" in 2x2x2x2 space, they are used in a few of the moves you show in the video. For example, the central face turn is just a short end restack followed by a outer face twist followed by a short end restack back. Similarly, the gyro rotation is really just a short end restack followed by a long end restack followed by two clamshell moves.
I would think of them as just rotating the cube around its 4th axis.
The gyro move is only a 4D rotation "in total". So long as you achieve that result, it literally doesn't matter how you accomplish it.
Regarding the axial twist, you can get the same effect with just the simpler canonical moves, but it's also perfectly legal in its own right. You can see this from the fact that you can undo it using a single 90 degree twist after making the appropriate gyro move. Shortcuts such as the clamshell move can be considered acceptable or not, similarly to the way people argue whether a center slice move on a 3x3 should be considered 1 move or 2. It's a matter of taste, but clearly if you take this too far it doesn't feel right, and that's how I feel about slice moves in isolation. Fine for scrambling, but I personally wouldn't allow them in a competition.
So, which moves or sequences of moves in the puzzle are in direct one-to-one correspondence to a single move of the true 4D equivalent? What reasons are there that such shouldn't be the sole determiner of which moves/sequences are canonical?
Is there a better more technical/mathy video talking about the 4D-ness of this puzzle and directly comparing it to other virtual renderings of the 2x2x2x2?
I'd be interested in this as well.
Direct 1-1 matches of moves between virtual and physical puzzle versions include all 6 canonical moves although #1 and #6 are pure rotations and don't count towards move totals. They're equivalent to turning a regular Rubik's cube around in your hands.
I'm curious. Canonical move 2 states that you can perform 180° twists of either half of the puzzle, however canonical move 5 states you can separate the halves independently and recombine them in any orientation you wish (from my understanding). Would it not stand to reason that one could just separate the halves, rotate one side 90° in the same way move 2 rotates 180°, then put them back together?
Yes indeed. There is some redundancy in the move set. That is an artifact of the way the canonical move set was produced. In short, it was created by a consensus of the people on the mailing list who had opinions. We had a healthy discussion, and perhaps nobody got exactly the final set exactly the way they wanted, but this was the set that nobody objected to.
Part of the idea here is that simple rotations or otherwise homeomorphic transformations in 4d permute the faces of a tiled object in ways that cannot be replicated by such kinds of transformations in 3d; if we're creative, however, we can dream up suitably strange 3d moves that would yield similar permutations. Here's my question: it seems like, as we increase n, we get more and more of these simple transformations... so I wonder if there is a topology whose simple twists and rotations will eventually exhaust *all* permutations, losing the Rubik's group structure. In other words: as n in R^n approaches infinity, does the Rubik's group of an n-cube just approach S_k (where S is the symmetric group over the k numbered faces or hyperfaces of our cube)? Or perhaps approach S_(k-x), i.e. the Rubik's group as dimensional n tends to infinity (asymptotically) would just be a full group of permutations over the faces of a hypercube in some lower dimension?
I don't fully understand. Are you asking whether this approach applies to still higher dimensions? If so, I don't know, but I doubt it.
Interesting question! I've been thinking about it but curious if you could clarify something. You mentioned k being the number of faces, but I'm wondering if the question should be with k the number of stickers? Since Rubik's cubes permute stickers around, The Rubik's group R is a subgroup of the symmetric group over the numbered stickers, S_s. Are you wondering if the order of the two groups S_s and R converge as the dimension increases? Or are you intentionally wanting to compare the Rubik's group (on stickers) to the symmetric group on cube faces?
@@RoiceNelson Thanks for the correction! Yes, I was thinking outward-facing faces of the "tiles" or "cubies" as opposed to the faces of the cube itself. Stickers is a much better way of putting it. Melinda, does that make it any clearer to you?
One upshot of this question in the world of creating and manipulating physical puzzles is that as you ascend dim(n), you'd be able to extract and reorient a sticker arbitrarily without making the puzzle unsolvable (as can happen with the 3d case; here, I believe, the magnets prevent such a reconfiguration but still need to restrict the possibilities thereof to do so.)
@@otherwords1375 I have a little better idea but definitely can't say that I fully understand. When you say "reorient a sticker", I think you mean a cubie, sort of like how a cubie can be flipped 180 degrees in isolation in a 4D puzzle but nothing like that in 3D. I think you area asking about such extra freedoms in higher dimensions, and if so, then yes, I believe that's the case. The way I like to express that is that as you go to higher and higher dimensions, you simply have a lot more "space" to deal with, somewhat analogous to what happens when you go from a Rubik's cube to a Megaminx. They are extremely similar puzzles, but the Megaminx is easier in some ways because there are more places where you can stash stuff you don't want to mess up. If that's not what you had in mind, then please try to clarify.
@@MelindaGreen Yes, that's what I meant by "reorient." The intuition about similar added freedoms in lower dimensions is helpful.
I'm realizing that my question is somewhat removed from cubing itself. What I'm ultimately wondering is whether or not you could prove out the eventual convergence of higher-dimensional Rubik's groups over k cubies to a symmetric group over k cubies. If indeed true, this result seems like it could provide interesting tools for moving between discrete and continuous mathematics in the context of other difficult problems (while the Rubik's transformations are indeed discrete in the sense that they act over "cubies," they must be continuous operations over an n-dimensional face to permute "cubies" legally.) This line of thinking strikes me as motivated, given the creative proofs that a look to continuous mathematics has often supplied to otherwise intractable combinatorial theorems.
In any event, the design you've developed here is truly ingenious, and I'm sure it'll take me a good while to appreciate it fully. Thanks for the reply.
I don't understand how rotating a single cublet could be even technically legal
It's should not be used in a solution. I'm just pointing out (perhaps badly) that the resulting position will still be solvable, unlike with the 3D version.
I don't understand the move either. Could you explain where the move originates from / how you found out it is legal? In the video you just say "so it turns out that this is legal" which is really unsatisfying for such a weird move. Wouldn't it be possible to use computer simulations to find a legal move sequence that flips just one corner like that? (I mean, if not, the move should not be legal)
It's been known for a long time that you can get a single 4D piece flipped like this. There are MagicCube4D macros that will do this, so in the virtual puzzle you could use this in a valid solution at the cost of a large number of simple twists. You can find such a macro in the Mathologer's solution here: th-cam.com/video/Ph6P1Ixfqzk/w-d-xo.html and linked in its description. I shouldn't have called it "legal" in the physical puzzle, but I did say it would be outrageous to do it directly like that. My point was only that the resulting position is still solvable, unlike what happens in the 3x3.
Canonical move #5 is topologically the same as scrambling an 02 cube puzzle.
I'm not sure how you figure that, but my feeling is that when reorienting a face in place, (rather than also moving it elsewhere), it's the most legitimate move. That's because I think the best way to extrapolate the puzzle to N dimensions is to 1) pick a cell to twist, 2) pull that cell off, 3) reorient it into any orientation, and 4) reattach it to where it began. In the 3D cube that equates to twisting about a major axis. You can't flip the face over because the inside isn't part of the puzzle. For the 4D puzzle that means you only need this move and a gyro.
@@MelindaGreen I don't know if topologically the same was the correct term for me to use, but it can share the same state space as the 02 puzzle as long as you only use canonical move #5 and treat it like a 1x1x1x2. Rowan Fortier talks about it a little bit in his unboxing video here: th-cam.com/video/Q90jYLQsHcg/w-d-xo.html
@@SilverWingedOne Rowan showed a 1x1x2x2. "Same state space" means "Same puzzle". AKA isomorphic. Helpful, but I don't get the extra stickers needed. The outer face twists of my puzzle are the only ones that allow all 24 possible twists. That's the same number as the cube symmetry group. It's interesting to know that the symmetry group for the 4D cube is 192. Exactly half the number of magnets.
Woah, so convoluted geometry.
It's a minor miracle that it's possible at all.
3D mechanical version of 3^4 cube. Can it be mechanically constructed?
@@abhirishi6200 I don't know.
I want one now but I probably can't afford it :(
I am getting a 2x2x2x2 in the mail today!
Before I use it, I have just one question.
Abiut the axial center face twist done at 0:57, is it possible to reorient the orange face on any axis, like the x, y, and z rotation axes, instead of just one?
Thanks!
Sorry for the late reply. You probably already found your answer, but here you go: You can't orient the central face any which way. Aside from the 90 degree rotations shown, you can also flip it 180 degrees such that it ends up with the same vertical axis as it started with. Anything else and the cubies would be in the right positions but each of them would be twisted around their corners. So that vertical flip is not useful because you can get the same effect from one simple rotation and a 180 degree twist of one of the side faces.
@@MelindaGreen apparently i didnt know this but it isnt possible to buy the whole puzzle at once, so it turned out that i now only have 10 white pieces. Guess i didnt read the website well!
Now I have decided I will just save up for the other colors and the magnets and buy them when I habe enough
@@okboing I'm sorry you only got the white pieces. Shapeways doesn't allow me to sell a bundle of multiple parts. Note that it's cheaper to buy a fully built puzzle from me than to build it yourself because I buy parts and magnets in bulk. I also hope to have an even better and cheaper mass-produced version soon(ish). So just use the quote request form on the main project page and I'll make sure you are set up as you prefer.
@@MelindaGreen can you provide the link so I can buy it all in one from you?
Thanks!
@@okboing Sure, just go here, read the instructions, and click the email link, tell me who you are, and hit "Send" - superliminal.com/cube/2x2x2x2/#how
Can a 3x3x3x3 be made from magnets and dice?
What about 5 dimensions?
There may be clever ways to do these things, but I'm pretty sure this design can't be naturally extended to the 3^4 or higher dimensions. I feel like this design was incredibly lucky.
Is there a plan to make it mass produced?
There is not a plan, though it definitely is my hope.
Why do #1 and #2 need to be listed separately from #5? Is there something I'm missing about these?
They don't need to be there for practical reasons. #1 & 2 seem fundamental by comparison with 3D counterparts, whereas #5 seems fundamental by the math, and you can use either one to derive the other. In the end, rule set choice is somewhat arbitrary, and this set gives a nod to both approaches, even though it seems a little redundant. Great question though!
@@MelindaGreen Thanks for the clarification! I finally ordered one of these last month and it is now assembled! Holding it in my hands and turning it myself I'm really beginning to appreciate this puzzle on a new level beyond just watching the videos, somehow your design makes 4-dimensional rotations intuitive and approachable. Truly innovative! I love it.
Is there a way to turn the top or inside face?
The axial twist turns the inner face--which is the same as twisting the top/bottom face since that's the other half of the puzzle.
@@MelindaGreen I think get it now. Can the inside face be turned in any other way, such as 90 degrees in the Z or X axis?
How much money would buying one of these be? (Aproximately)
wondering the same
As cheap as I can make them. Currently $124. Most of that is the 3D printing cost.
Melinda Green ok thanks
I've been able to lower the cost a bit and improve the magnet quality, but it's still over $100.
I want a physical 3x3x3x3 so bad.
This is beautiful. 🤍
There is one 3x3x3x3 that exists, but the maker hasn't even solved it because it's just not practical. See it here: th-cam.com/video/QTc-rG-nunA/w-d-xo.html