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.
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.
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.
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.
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!
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.
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?
@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.
@@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.
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.
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.
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
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.
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.
@@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 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
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.
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?
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'
Very cool puzzle :)
Thank you!
never seen theory expressed through twisty puzzles like this before, super cool
Visual group theory? Hell yeah
Linear 2^(1/2) regression 1.4142
Easier that using actual quaternions
lol
Is it though? Maybe I'm just bad with shapes, but I find this thing more confusing.
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.
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.
@@typerys3805 quoting Lord Kelvin, "Quaternions... though beautifully ingenious, have been an unmixed evil to those who have touched them in any way."
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.
I see there's a plaque there now commemorating the grafitti - that's a great story, thanks for sharing.
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.
This video got recommended to me and I immediately subscribed! I would love to learn about group theory through Rubik's puzzles!
Nice design. Smooth looking operation. Good theory discussion.
The sound to makes when it turns is so nice 🤤
incredible design, great job mate!!
Amazing idea
I swear I've seen your videos many years ago, back when I was more into cubing
Brilliant work
Should totally explore more symmetries
This is so cool
Incredible!
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!
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.
Nice!
Amazing! Now I can model this in blender and have my computer software calculate vector rotations by turning the gears of the cube!
Holding out for SL(2,7)
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?
@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.
@@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.
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.
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.
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
TheGrayCuber is definitely gonna love this ^^
as someone who loves group theory yet understands none of it, this is awesome
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.
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.
good demonstration, also, which printer do you use?
Thanks, Bambu P1S
@@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!
@@QuirkyCubes Oh nevermind just saw the detailed print file in description : )
@@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
That's just gear skewb without 1 layer.
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.
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?
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'
@@QuirkyCubesAh I see, very cool
cube
wuh huh