QuirkyCubes

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@abhijeetghodgaonkar 15 сағат бұрын

Nice puzzle yp

@imbw267 2 күн бұрын

I like your funny words math man.

@lukostello 2 күн бұрын

I cant even concieve of this while I'm looking at it. Must be a very special kind of person to be able to imagine this thing into existence

@andreykant 3 күн бұрын

Very nice puzzle !!!

@asdfniofanuiafabuiohui3977 3 күн бұрын

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

@IdoN_Tlikethis 3 күн бұрын

damn, I understood none of that

@xyz.ijk. 3 күн бұрын

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

@alclelalclel 3 күн бұрын

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

@sidharthghoshal 4 күн бұрын

@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 2 күн бұрын

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 Күн бұрын

@@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.

@Kromiball 4 күн бұрын

It's a half gear redi cube!

@randommm-light 4 күн бұрын

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

@Rockyzach88 4 күн бұрын

Badass.

@AllenKnutson 4 күн бұрын

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 4 күн бұрын

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.

@_forg_ 4 күн бұрын

Why are you trembling lol

@QuirkyCubes 4 күн бұрын

the power of quaternions runs through my veins

@_forg_ 4 күн бұрын

@QuirkyCubes i see

@borb5353 4 күн бұрын

this is confusing me in a pleasant and mesmerizing way

@k4yd33yeah 4 күн бұрын

Bro really took "mathematical object" too seriously

@DiannaGold 4 күн бұрын

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

@infrabread 5 күн бұрын

Are you willing to sell this puzzle? I've never liked the tiny pieces on the mass produced MF8 puzzle.

@QuirkyCubes 4 күн бұрын

I have them up for sale on quirkycubes.com - it's a bit more expensive than I'd like but send me a message if you're interested.

@firefly618 5 күн бұрын

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

@sbares 5 күн бұрын

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 4 күн бұрын

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

@tobiasreckinger2212 5 күн бұрын

Really intuitive when you're used to quaternions

@gsestream 5 күн бұрын

stick to 3d, and keep it simple.

@aditya95sriram 5 күн бұрын

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

@dreamingvertebrate7162 5 күн бұрын

Fascinating! Bravo!

@isodoublet 5 күн бұрын

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

@crosswingrobots 6 күн бұрын

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

@QuirkyCubes 5 күн бұрын

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

@ArkanoidZero 6 күн бұрын

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

@multiarray2320 6 күн бұрын

thats not far away from a 3b1b video

@latefoolstalk676 6 күн бұрын

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

@abhijeetghodgaonkar 6 күн бұрын

Very cool puzzle yo!

@Randomstuff-m7p 6 күн бұрын

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

@QuirkyCubes 6 күн бұрын

@@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"

@r.pizzamonkey7379 6 күн бұрын

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 6 күн бұрын

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.

@NonTwinBrothers 6 күн бұрын

You have me intrigued

@MrRyanroberson1 6 күн бұрын

How much to buy one from you?

@QuirkyCubes 6 күн бұрын

I just put one up for sale @ quirkycubes.com They're a little more expensive than I'd like but this is a difficult one to build :/

@havenotchosenyet 6 күн бұрын

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

@wun_zee3599 6 күн бұрын

so it's a calculator

@Randomstuff-m7p 6 күн бұрын

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?

@Rhino-Flea 6 күн бұрын

cool stuff

@challox3840 7 күн бұрын

I wish i could give this to my math professors

@mananself 7 күн бұрын

Great job! I know the solving experience is quite unique. Can you comment on the difficulty? Are you able to solve this puzzle? Thanks!

@QuirkyCubes 7 күн бұрын

Sure. Yes, I've solved it twice using a spreadsheet to track side effects of the edge-preserving sequences, and I came up with a solution that's tough but gets the job done after a couple hours of fiddling. I think it's easier than biaxe-style puzzles, some of which express weird groups, since there are 4 options for turning axes instead of 2. I've also started working on an optimized solution using a computer and I think with better algorithms it will be easy to solve.

@hamish_todd 7 күн бұрын

As a computer animation engineer who has studied quaternions for a long time, this is incredible, a huge achievement. A little detail on the practical side for those interested. Forget about the cube he has made, for a moment. Quaternions are about rigid rotations (so, no weird gears - just rotations you might do with a spoon in your hand!) If you look at the quaternion diagram shown at eg 1:21 there are eight nodes. The way to think about these nodes is: 1. Top node: the "rotation by 0 degrees" / identity quaternion 2. Bottom node: the "rotation by 360 degrees" quaternion 3. i, j, k nodes: clockwise rotations by 180 degrees. If you like you could say i is around x axis, j around y axis, k around z axis 4. -i -j -k nodes: counter-clockwise rotations by 180 degrees Yes, it is weird that there is a "360 degree rotation" quaternion - you might have thought that would be the same as the identity; in fact the 1 and -1 quaternions, if you turn them into the matrix, will give you the same matrix (the identity matrix). But, this property matters hugely; this is why animation programming uses quaternions. The reason it matters is that it allows us to distinguish clockwise and counter. Animators need this. If they don't have it, their characters will bend their arms in a way that humans know is wrong. Back to the cube. Intuitively I find it very surprising that this thing represents the quaternions, but the picture at 2:48 proves it. I am not much of a cuber but what I find amazing is that it is "just" a half-cube. The quaternions (as you see above) are meant to in some sense "double" ordinary rotations. Another toy that represents them is Henry Segerman's octahedral maze, which covers a full sphere. The mapping from states to quaternions here is not straightforward, you have two moves. Still, it surprises me that this half-sphere gets a doubled-sphere somehow. I will need to spend time thinking about this. What a beautiful thing.

@jimmypatatoeyt4800 7 күн бұрын

wuh huh

@Dissimulate 7 күн бұрын

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.