Quaternion Cube

[@Mathologer]

Very cool puzzle :)

[@QuirkyCubes]

Thank you!

[@DadicekCz]

Visual group theory? Hell yeah

[@joespurway2678]

Linear 2^(1/2) regression 1.4142

[@ewthmatth]

(Rubik's) always has been 🧑‍🚀

[@LeoStaley]

Easier that using actual quaternions

[@QuirkyCubes]

lol

[@fotnite_]

Is it though? Maybe I'm just bad with shapes, but I find this thing more confusing.

[@typerys3805]

op is probably referring to the quaternion ring, which is a polynomial field over the quaternion group. as mentioned in the video, the quaternion group has only 24 elements with each product easily defined. ~~the field is more difficult to deciper without thinking about 4d rotations (mentioned in the video as special linear groups). (generally you can consider "3d rotation" subgroups generated by just 3 rotations)~~ also the cube in the video has each state representing an element in the group edits because im a fool and couldnt edit on mobile: for a good while ive assumed the unit quaternion field to represent rotations in 4D space in the way the unit complex numbers can represent orientations in 2D space. i dont really know how i came to this misconception (and looking back, its seeming like a silly misconception to fall for), but i am wrong. as the commenters below have shared, the quaternions are more cursed than i thought.

[@angeldude101]

Quaternions are mixtures (or weighted sums or linear combinations) of 4 basis 3D rotations. 3 of these rotations are 180° around 3 orthogonal axes (i, j, and k), and the remaining rotation is 0° around every axis and no axis (1). Mixing two quaternions (and renormalising) gives a rotation somewhere in between them, with their relative weights (or magnitudes or norms) determining how close the result is to one vs the other. Worth noting that quaternions are _rotations,_ but not _orientations._ Each quaternion can be thought of as either clockwise or counter-clockwise. The negative of a 90° clockwise rotation around a given axis and a 270° counter-clockwise rotation around the same axis. This is strongly related to the "double-cover" behavior of quaternions, since every orientation has two different representable rotations that end at it.

[@LeoStaley]

@@typerys3805 quoting Lord Kelvin, "Quaternions... though beautifully ingenious, have been an unmixed evil to those who have touched them in any way."

[@DeclanMBrennan]

Extremely cool. What a beautiful mechanism. I live near Dublin, Ireland. Every year in October, there is mathematical pilgrimage of sorts involving a walk along the Royal Canal from Dunsink Observatory, where Sir William Rowan Hamilton worked, to Broom Bridge, where he scrawled a graffiti of the quaternion algebra, when he invented it in 1843, while looking for a new approach to 3D mechanics. Among many other things, he also invented the Icosian game involving eponymous Hamilton cycles on a dodecahedron and icosian algebra to understand the symmetries of the same.

[@QuirkyCubes]

I see there's a plaque there now commemorating the grafitti - that's a great story, thanks for sharing.

[@sbares]

Not only does SL(2, 3) contain Q8 as a subgroup, it is actually isomorphic to the group of unit Hurwitz quaternions {±1, ±i, ±j, ±k, 1/2(±1±i±j±k)}

[@QuirkyCubes]

That's excellent, thanks for pointing it out. Going to look into this further and make a followup post of some sort.

[@tiddly5]

never seen theory expressed through twisty puzzles like this before, super cool

[@chsovi7164]

the main image on the wikipedia page for group theory is a rubiks cube :D

[@williejohnson5172]

Absolutely amazing. I know absolutely nothing about Q8 and quaternions groups. But I do know that any and every quaternion, unit or otherwise, must be expressed as some form of Euler's formula. Your cube gives me a practical way of seeing how they actually rotate (quaternions do not commute)

[@aroymart]

Your puzzles are so fascinating, thank you for sharing ! I love your explorations into visual group theory, and I think the lever mechanism is such a fascinating bit of engineering that twisty puzzles don't usually see.

[@alclelalclel]

i love it when puzzle makers are also group theorists!!

[@aditya95sriram]

How am I discovering this channel just now! Really interesting stuff, thanks!

[@AsiccAP]

This video got recommended to me and I immediately subscribed! I would love to learn about group theory through Rubik's puzzles!

[@havenotchosenyet]

super interesting, hadn't thought of exploring such a large family of groups with Rubics cube like puzzles.. very clever

[@TranquilSeaOfMath]

Nice design. Smooth looking operation. Good theory discussion.

[@andrewsemenenko8826]

Wow. You really need to sell these :D Thank you for the STL file, I hope I can find a 3d printer somewhere close to me, this feels like it worth the effort ❤ The mere concept feels impressive to work with!

[@youtubehandlesux]

Remember to print with a resin printer, or a calibrated printer with a nozzle smaller than 0.3mm, or have a vapor smoothing chamber, or prepare to sand it manually for hours. Those things aren't the easiest to print.

[@QuirkyCubes]

I just listed them for sale @ quirkycubes.com . I do recommend getting a printer though, they're quite useful and you can find tons of puzzle designs to make:)

[@latefoolstalk676]

so great to combine beautiful group theory from algebra and visuals like the sube puzzle

[@MCSteve_]

I swear I've seen your videos many years ago, back when I was more into cubing Brilliant work Should totally explore more symmetries

[@firefly618]

This is way above my head (and I know how quaternions work) but very cool.

[@NonTwinBrothers]

You have me intrigued

[@dreamingvertebrate7162]

Fascinating! Bravo!

[@tobiasreckinger2212]

Really intuitive when you're used to quaternions

[@iwantedtohaveabigytnamepsi2007]

incredible design, great job mate!!

[@azaeldoesmusic]

The sound to makes when it turns is so nice 🤤

[@VagueHandWaving]

Amazing! Now I can model this in blender and have my computer software calculate vector rotations by turning the gears of the cube!

[@andreykant]

Very nice puzzle !!!

[@ArkanoidZero]

Have you considered adding a gear shift mechanism to allow for a 6 axis/8 corner version of this? It seems like it would be a very interesting puzzle

[@TheBabelCorner]

Amazing idea

[@abhijeetghodgaonkar]

Very cool puzzle yo!

[@DiannaGold]

something about this makes me want to scream in pain that I didn't know I had.

[@randommm-light]

so cool! pls make one to model quantum electrodynamic particles.. look up Cohl Furey and e8

[@hrishikeshaggrawal]

Incredible!

[@sampyuays]

good demonstration, also, which printer do you use?

[@QuirkyCubes]

Thanks, Bambu P1S

[@YT-gv3cz]

​@@QuirkyCubes Wow, looks like it turns pretty well. I just got a P1S and want to print some fun mechanical puzzles. May I ask what filament and setting do you use for these twisty puzzles? Thanks!

[@YT-gv3cz]

​@@QuirkyCubes Oh nevermind just saw the detailed print file in description : )

[@QuirkyCubes]

@@YT-gv3cz good question, I'll add some more info to the post but I use Polymaker ABS (bambu ABS does not vapor-smooth properly), print everything at half speed, rafted on Bambu support for ABS

[@pranavsetpal]

Yoo this is hella cool

[@AllenKnutson]

I'm pretty curious now if you can get PSL(2,7) = PSL(3,2), of order 168, as the group of some cubey puzzle.

[@QuirkyCubes]

Yes, in fact it's found in the same geometry as this puzzle, with the gearing reversed. So one corner turns the diagonal corner in the opposite direction. I'm hoping to find a nice way to construct it and go into deeper linear algebra theory in another video.

[@isodoublet]

My days of not understanding group theory are certainly coming to a middle

[@jamiepayton1574]

Wow. Cool stuff. Sadly even I as a massive nerd don’t have many friends who both know group theory and care about twisty puzzles. Therefore I can’t share it with anyone. Awesome stuff tho

[@YellowBunny]

Nice!

[@sidharthghoshal]

@QuirkyCubes Can every finite group/monoid be realized as some combination puzzle in our 3 dimensional world? I would love to have a model of many of the known finite groups and monoids

[@QuirkyCubes]

I don't know the answer to that. I have been working on a catalog of groups that can be realized as points on 3D geometries, which will be a browsable online interface with 3D renders. I will include this with an upcoming Mathieu M11 puzzle video. I can say that some groups have been elusive - for instance it is much easier to find the SL(2,3) supergroup than the pure Q8 group. M10 has yet to appear in any of my searches but I see the related A6 and A6.2^2 frequently. Same with PSL(2,8) vs. Ree(3). I think there's probably a very clear explanation for this, but I don't know what it is yet. I've spent a lot of time looking for the sporadic group Janko J2 in 3D, and I'm tempted to say that these 2nd generation sporadic groups don't have any mechanically feasible 3D representations. Hopefully I'm wrong and in time they'll be found.

[@sidharthghoshal]

@@QuirkyCubes unclear if this is trustworthy but GPT-o1 seems to think that every finite group can be realized (but for any particular group the engineering might be ridiculously hard). The set up for my question was to take a sphere, consider any ORIENTED circle C on the sphere, consider an arbitrary angle E, and then rotate the hemisphere to the LEFT of that circle C by the angle E. Using this you get a continuous version of all rubiks cube like puzzles. After describing this group to it O1 claims this is group is essentially equivalent to the set of ALL homeomorphisms of the sphere. Then O1 claims that ANY finite permutation group can be embedded, by selection M congruent regions of the sphere and finding a homeomorphism that interchanges them.

[@tanvach]

This is so cool

[@asdfniofanuiafabuiohui3977]

Quaternions are actually not the fundemental object that they are. In fact, QTNS are a kind of object called a "bivector" (an oriented area, made from the wedge product of two vectors). People have confused them for being a special kind of vector or algebra as an extension of complex numbers, due to the fact that people have misinterpreted complex numbers as being a vector because it has 2 components, when in fact complex numbers are a scalar + a bivector. Then, the QTNS are the natural extension of this because the two axies of 2D goes to 3 in 3D, making 3 combinations of axies into 3 bivectors. Because people are not used to bivectors, they assume there must be an in between with 3 components, however due to them being bivectors, its obvious (1 scalar + 3 bivector parts). Its also why they work for rotations (bivectors being oriented areas naturally relate to rotations); people assume rotations happen around an axis, but they occur in a plane. Because of the 4 components of QTNS, people assume they're 4D when they're in fact 3D, and use them thinking that its rotated around a 4D axis, when its rotated around a 3D plane (bivector). The formula for a circle has 3 components (ax^2 + by^2 - r^2 = 0); but is clearly 2D. It also explains why the 3 "vector" parts behave differently but identically to one another vs the scalar part, because the whole number is scalar + 3 bivector parts. It also explains the strange multiplication; which comes from the fact that when you square axies together, they square to 1, but square bivectors they square to -1. (swapping the components inverts the number: xy = -yx) x * x = 1, y * y = 1, z * z =1; xy * xy = -yx * xy = - y * y = -1, yz * yz = -zy * yz = - z * z = -1; zx * zx = -xz * zx = - x * x = -1; xy * yz = xyyz = xz, yz * xy = yzxy = -zyxy = zyyx = zx = -xz

[@borb5353]

this is confusing me in a pleasant and mesmerizing way

[@aloading444]

cool cube! would a version of this with all eight corners able to turn be possible to make? if so, would it be trivialized or would it be harder than this? this cube also reminds me of a 2x2 gear shift, the way the corners interact with each other.

[@QuirkyCubes]

The 8 corner version is perhaps more trivial - the edges have 12 possible states, but the corners don't solve themselves and need to be 'un-twisted' so there are 36 states.

[@QuirkyCubes]

Someone actually just made this with internal gears - search Quartet Redi Cube

[@aloading444]

@ wow that’s cool, thanks for letting me know about it!

[@G.Aaron.Fisher]

Do you know if the standard 3x3 has a subgroup isomorphic to the quaternions? I worked on this question for a bit, and was stumped. I was hoping there'd be an easy proof if there were no subgroups of order 8, but there are plenty. One example being the group generated by RR and L (obviously not the quaternions though). I also tried to find a set of generating moves for a quaternion subgroup and came up short, I think largely due to my lack of cubing knowledge. I think the most useful avenue of attack would be considering possible candidates for -1. It needs to be its own inverse, so something like the checkerboard pattern would work. But it also needs to be reachable by a repeated set of moves in three different ways.

[@QuirkyCubes]

That's a really cool question and I thought about the 3x3 geometry a bit when I was trying to come up with this puzzle. I think the answer is definitely yes - since you can take the edges of this puzzle and make identical sequences for i,j,k on the Rubik's Cube edges, that's one way to do it. A more elegant way is to do it over the 8 corners. It definitely works considering solely permutation, I don't think it works considering orientation. The -1 state is just every corner swapped, then i, j, and k are formed by drawing an x on each opposing pair of faces and doing a 4 cycle along one diagonal / and a 4 cycle on the other \. You get 3 different pairs of 4 cycles and they generate Q. Maybe this is a subgroup of, say the Trajber's Octahedron.

[@Randomstuff-m7p]

What is the full group for the 3x3. Is it O(h) or O(3)? The full octahedral group. Or the Hyperoctahedral group of dimension 3? And where can I learn to use what ever group that answer is?

[@QuirkyCubes]

@Randomstuff-m7p The Rubik's cube group is a composition of various symmetric/alternating groups and cyclic groups. Considering solely the edge pieces, the group description is C2^11:Symm12. The symmetric group 12 tells you you can permute the edges around any point on the cube in any combination, and the C2s represent an edge being flipped in either direction. There are only 11 C2s because the orientation of edge 12 is determined by the 'parity' of the others. Likewise, when you introduce other pieces back into the cube, the edge permutation no longer expresses Symm12 but Alt12, with only half of the possible positions. I'm not sure that the Rubik's cube group is especially useful outside of analyzing the Rubik's cube, but understanding how it's a composition of other groups can be useful. The Rubik's cube group almost contains the octahedral group as a subgroup - it's expressed as the class of patterns with one dot in the middle of each face (or 4 faces). You can only produce 12 of the symmetries though due to parity rules. There are surely ways to express the full group on a subset of pieces.

[@crosswingrobots]

Can't you get the same thing with just a solid cube that you rotate on your desk?

[@QuirkyCubes]

Good question - the rotations of the faces or corners of the cube, say the numbers on a cubic die or a D8 die, also form a group with 24 states. This is a different group called Symm4 or Oh, with different algebra rules, so you won't find any way to compose a common 'inverted state' there. Here is its cycle graph: commons.wikimedia.org/wiki/File:Subgroup_of_Oh;_S4_blue_red;_cycle_graph.svg

[@challox3840]

I wish i could give this to my math professors

[@Rockyzach88]

Badass.

[@Rhino-Flea]

cool stuff

[@Gergenhimer]

as someone who loves group theory yet understands none of it, this is awesome

[@Dissimulate]

I wonder whether the octonion group can be made into a puzzle. The graph of transformations fits in 2d, but it might not work with the same kind of gear configuration.

[@Kromiball]

It's a half gear redi cube!

[@Randomstuff-m7p]

Now do the monster group... What would that even look like?

[@QuirkyCubes]

@@Randomstuff-m7p Sadly we'll probably never know - noone has even come up with a permutation representation of the monster group. It's almost certain there's not a clean representation of it in 3D space. As John Horton Conway said, "[These things] look nice... So do these things in higher dimensional space except that I haven't gotten 196,833-dimensional eyes, so I'll never see them"

[@terdragontra8900]

How did you check that its group is SL(2, 3)? It’s hard to think about this without one in my hand. “L” is an element of order 3, so apparently it has a “square root” (an sequence of order 6, that gives you L if performed twice), what sequence is that?

[@QuirkyCubes]

Well... with a computer, kind of in reverse. You can assign a number to each edge and check it with GAP, which I did exhaustively over icosahedral vertices to come up with the various gearings that I'm working on. I hadn't made the square root connection before, that's really interesting. It seems R'LR' is the root of L, and LR is the root of RLR'. Those forms cover all 16 states. I suppose that makes sense because the order 6 states are just (order 3)' × quaternion inverse, so L' × LR'LR' (-1) is R'LR'

[@terdragontra8900]

@@QuirkyCubesAh I see, very cool

[@r.pizzamonkey7379]

I don't really understand. Could you do a more thorough explanation of how the state relates to quaternion states? Is each side the product of 4 quaternions? Do the patterns of the edges represent quaternion states? I feel like you skipped over the most interesting part.

[@QuirkyCubes]

On a permutation group the states are described as cycles over points, which on the puzzle are the patterns / positions of the edges. The puzzle's quaternion states are formed by compositions of SL(2,3) states. So the move L on the puzzle produces two 3-cycles over the edges, and the move R̅ produces a different dual 3-cycle. When composed as L R̅ these form a quaternion state that appears as a dual 4-cycle. So a state of the edges and the sequence used to get there are one and the same - and some of these states/sequences can be assigned to i,j,k such that the quaternion algebra works. The point permutations are a different way to represent the quaternion group then the usual matrices, but they have exactly the same relationships, so they're called isomorphic.

[@andriypredmyrskyy7791]

Would actually buy just so I can understand quaternions more tangibly

[@QuirkyCubes]

@@andriypredmyrskyy7791 it's up on quirkycubes.com!

[@xyz.ijk.]

I must have one of those. (Read in a diabolical monster voice.)

[@dranorter]

Holding out for SL(2,7)

[@QuirkyCubes]

I made another comment earlier but I got my groups mixed up. PSL(2,7) has been found but not sure about this one. Is there something especially interesting about it?

[@dranorter]

@QuirkyCubes I think I meant PSL(2,7) anyway, I misremembered and thought they were the same! The one of order 168. I have a puzzle axis design for it myself but I wouldn't know where to start with making the gears. PSL(2,7) is cool because it's the Fano Plane symmetries. It also has some fun numerical coincidences with Tarot cards. I think one challenge with these small groups is finding representations that are good puzzles.

[@QuirkyCubes]

@@dranorter Oh, cool! Send me a message on discord or twistypuzzles and maybe I can suggest something with the gears. Interestingly this cube with the gearing reversed (so that opposing axes rotate in opposite directions) is PSL(2,7). Jabberwock's Photonic Crystal also expresses it on the small pentagons. I'll research Fano planes and the tarot connections, it would be interesting to explore more concrete analogies in future puzzles.

[@Placeholder-sq9gs]

I have no idea what any of this meant but nice cube

[@imbw267]

I like your funny words math man.

[@k4yd33yeah]

Bro really took "mathematical object" too seriously

[@multiarray2320]

thats not far away from a 3b1b video

[@XVYQ_EY]

That's just gear skewb without 1 layer.

[@GigaminxSolver33017]

im the 2000th like of this video

[@pncka]

big words scary

[@Yoctogon]

cube

[@halfasleeptypist]

TheGrayCuber is definitely gonna love this ^^

[@wun_zee3599]

so it's a calculator

[@Randomstuff-m7p]

They are all calculators. Valid Math mirrors the material universe. If you cant make a calculator like this out of the math then how can you say your math represents anything?

[@IdoN_Tlikethis]

damn, I understood none of that

[@gsestream]

stick to 3d, and keep it simple.

[@_forg_]

Why are you trembling lol

[@QuirkyCubes]

the power of quaternions runs through my veins

[@_forg_]

@QuirkyCubes i see

[@jimmypatatoeyt4800]

wuh huh