What does the great Hamilton have to do with this?

[swipymam]

The Catch-24 encoding was based on the quaternion multiplication table:

where the units x(i), y(j), z(k) denote a 180° rotation along 3 axes, and the square root of the unit corresponds to a 90° rotation.
Multiplication of quaternions means concatenation of rotations.

We add "salts" (the inverse code of the original position) to the preliminary Catch-24 code using the "multiplication" table for quaternions, where the factors are 24 rotations of the entire cube, and get the final Catch-24 code:



The Catch-24 code allows the color code to be shortened from 54 to 18 characters. Another non-obvious property of the code is the ability to eliminate the human factor in automatic scrambling by a robot (in the absence of a solution in the form of scramble moves) at competitions.
20 randomly selected, positions (cube20.org):


Example of coding Catch-24 (position superflip-composed with four spot).

 

[eeeeeeeee]

what even is the great hamilton

 

[abunickabhi]

Can you explain the problem statement in simpler words?

 

[stoic]

what even is the great hamilton

I think it's this guy.

 

[eeeeeeeee]

Or maybe its the hamitolnian sequence

 

[Filipe Teixeira]

Ohhhhhh boyyy

Another complocated problamm

 

[swipymam]

Can you explain the problem statement in simpler words?

This is simply a look at position encoding from a different angle. You receive the position not as a sequence of rotations, and not even by the color of all 54 cube stickers, but as 18 characters of the English alphabet. In this encoding, there is no "solution" process inherent in standard scramble notation, only an exclusively static position. Based on such a representation, it is possible to solve the position using entirely new principles, employing, for example, operations on unit integer quaternions (which, like our Rubik’s cube, form a multiplicative group of 24 members), in contrast to existing approaches. Quaternion algebra is well-developed and, even after 182 years, it continues to evolve.

 

[Herbert Kociemba]

Another non-obvious property of the code is the ability to eliminate the human factor in automatic scrambling by a robot (in the absence of a solution in the form of scramble moves) at competitions.

I really don’t understand what you mean. Can you be a bit more specific or give some practical example?

 

[swipymam]

I really don’t understand what you mean. Can you be a bit more specific or give some practical example?

At competitions, you can transmit scramble positions using this 'open' code, without worrying that the sequence of turns (from traditional scramble notation) will be deduced. To put it simply, even competitors themselves can enter the code into a robotic scrambler. The rest is down to the equipment and procedure. Moreover, manual scrambling errors will be eliminated.
P.S. For existing robotic scramblers, all that's needed is to add a decoder to their software that converts the 'open code' into traditional face turn notation.

 

[pTr]

Damn this is a good idea.

 

[Herbert Kociemba]

Quaternions allow an elegant description of rotations in 3D concerning arbitrary rotation axes by arbitrary angles

but honestly I do not see the practical relevance for Rubik's Cube when describing cube states. There are other ways to describe a cube state in an even more compact manner and this quaternion stuff is just a bit of overkill.

 

[ProStar]

Damn this is a good idea.

No it isn’t it’s gibberish

 

[Herbert Kociemba]

it is possible to solve the position using entirely new principles, employing, for example, operations on unit integer quaternions (which, like our Rubik’s cube, form a multiplicative group of 24 members),

Please give some reference or explanation to your statements:
1. What are unit integer quaternions? I personally do not know this concept and I also did not find some definition elsewhere.
2. Why do these form a group of order 24?

 

[swipymam]

Please give some reference or explanation to your statements:
1. What are unit integer quaternions? I personally do not know this concept and I also did not find some definition elsewhere.
2. Why do these form a group of order 24?

A unit quaternion is a quaternion of norm one.

Do you read Russian? https://ru.wikipedia.org/wiki/Кватернион

 

[Herbert Kociemba]

Do you read Russian? https://ru.wikipedia.org/wiki/Кватернион

No, I do not read Russian but at least I now understand what you mean. There indeed is this subgroup with 24 elements I did not know before though it also contains half integers. It is quite interesting, here I found some more information
https://golem.ph.utexas.edu/category/2021/12/the_binary_octahedral_group.html
Still in my opinion this is only interesting for itself but quite irrelevant for Rubik's Cube.

 

[Herbert Kociemba]

@swipymam: And you should be aware that in the way quaternions are uses to describe rotations in 3D space a quaternion q and its negative -q describe the same rotation. So your 24 quaternions you refer to only describe 12 rotations of the cube and not 24 as it would have to be. This symmetry subgroup of the cube with 12 elements is the group with Schoenflies-Symbol T but you need the subgroup with 24 elements and Schoenflies symbol O (https://kociemba.org/math/symmetric.htm and https://kociemba.org/symmetric2.htm give more information for subgroups of cube symmetries).

 

[swipymam]

@swipymam: And you should be aware that in the way quaternions are uses to describe rotations in 3D space a quaternion q and its negative -q describe the same rotation. So your 24 quaternions you refer to only describe 12 rotations of the cube and not 24 as it would have to be. This symmetry subgroup of the cube with 12 elements is the group with Schoenflies-Symbol T but you need the subgroup with 24 elements and Schoenflies symbol O (https://kociemba.org/math/symmetric.htm and https://kociemba.org/symmetric2.htm give more information for subgroups of cube symmetries).

Both matrices and quaternions serve to represent elements of the rotation group SO(3) (the group of rotations in three-dimensional space). Quaternions (more precisely, the group of unit quaternions SU(2)) provide a "double cover" of SO(3) - here you are absolutely correct.

Matrices offer a direct way to represent the permutation of all Rubik's Cube stickers and can reduce the solving task to known linear algebra problems, such as matrix factorization.

Quaternions are particularly strong in representing rotations and orientations themselves, offering compactness and computational advantages for these tasks. They can be more intuitive for working with the composition of rotations and avoiding singularities.

Although there is not yet a unified solution methodology that combines quaternions and matrices as co-equal computational cores for the same aspect of solving a Rubik's Cube, they can be used within a single approach.

Both quaternion algebra and matrix calculations provide a powerful mathematical apparatus. Matrices are convenient for representing the complete state of the cube as a vector and moves as permutation matrices, reducing the solution to linear algebra problems. The algebra of quaternions possesses all the properties of a field, with the exception of the commutativity of multiplication. This makes quaternions a useful mathematical structure, especially in contexts where rotations in three-dimensional space are important.
Let's proceed gradually.

 

[Herbert Kociemba]

SU(2) (a group of special 2x2 matrices) is isomorphic to the group of quaternions but I do not see any need to introduce them here. Excuse me when I am wrong, but your answer seems at least to include some passages generated by ChatGPT or something similar. Btw. the additional 24 missing quaternions are 1/Sqrt(2)(±a ± b) where a and b are two different numbers from 1, i, j and k, for example 1/Sqrt(2)(±1 ± i). I am curious if you succeed to make something useful concerning Rubik's Cube with your quaternion approach.