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 Post subject: Expression of Rubik's Cube with Imaginary PiecesPosted: Fri Jan 25, 2013 4:10 am

Joined: Sun Sep 09, 2012 4:00 am
Warning!-This is a paper but may contains 'bæd Ingɯliʃi'.
Because I'm 한국어 사용자(Korean user).
γ. Definitions
1. Orientation
2. Permutation
3. Methods to Express
4. Source
γ. Definitions
1. Situation means a set of all pieces' orientation and permutation.
2. Orientation of a piece means state of the piece. Here a state, it is a element of Symmetry Group of sphere. Especially, in Rubik's Cube, it is a element of O-Group. Orientation of a as a piece in situation S is O(S_a).
3. Permutation of a piece means a set of rotations that affects pieces. Permutation of a as a piece in situation S is P(S_a).
4. "A rotation affects a piece" after do rotations, the piece's orientation or permutation changes. Definite E(A_a) is true when rotation A affects piece a and else is false, E(A_a)↔(O(T_a)≠O(A_a))∨(O(T_a)≠O(A_a)).
5. Trivial Situation(or Solved Situation)(mark it T) is a "special" situation. (In general 333, because of center's orientation, there's many situations that is solved, but I'd definite only one is solved.)
6. A·B is a situation that is applied an formula if applies it to T, then became B to A.
7. Piece means permutation of T. Or P(T_a)=a where a is a piece.
8. Rotation means multiplying one of R, R' R2, ...B2 to a situation S.
9. Formula means permutations of rotation.(Algorithm has other meanings, so I substituted)
1. Orientation
Orientation of a piece is an element of O-group in Rubik's Cube, there's only needed names of elements or naming methods of it.
Here, O-group is a symmetric group of 3-hypercube, orientation is a quaternion.
It means, marking solved orientation ai+bj+ck(a, b, c∈R, a≠b≠c), other orientation can be marked like bi-aj+ck.
Or marking <a, b, c> is possible
Then changing orientation can be expressed like these.-(1)
Before applied orientation quadnion O, After one is (rotation symbol)o(O),
xo(O)=(-1+i)O(1/(-1+i))
x'o(O)=(1+i)O(1/(1+i))
x2o(O)=iO(1/i)
yo(O)=(-1+j)O(1/(-1+j))
y'o(O)=(1+j)O(1/(1+j))
y2o(O)=jO(1/j)
zo(O)=(-1+k)O(1/(-1+k))
z'o(O)=(1+k)O(1/(1+k))
z2o(O)=kO(1/k)
Rotation symbol of these is
Attachment:
File comment: Rotation Symbol

0.png [ 1.84 KiB | Viewed 384 times ]

and 2 using same in practice.-(1)
Here i, j, k axis is rotation center of z, x, y.
Then situation R can be written if R∈P(T_a), O(R_a)=zo(O(T_a))=(-1+k)O(T_a)(1/(-1+k)).
2. Permutation
Permutation of a piece means a set of rotations that affects pieces. And we can notice piece {U, R} is at {U, B} in situation R.
So, it needs to moving place at one rotation, it has to change like this.-(1)
F, B' --> (U)->(R), (R)->(D), (D)->(L), (L)->(U)
F', B --> (U)->(L), (L)->(D), (D)->(R), (R)->(U)
F2, B2 --> (U)->(D), (D)->(U), (R)->(L), (L)->(R)
R, L' --> (U)->(B), (B)->(D), (D)->(F), (F)->(U)
R', L --> (U)->(F), (F)->(D), (D)->(B), (B)->(U)
R2, L2 --> (U)->(D), (D)->(U), (F)->(B), (B)->(F)
U, D' --> (R)->(F), (F)->(L), (L)->(B), (B)->(R)
U', D --> (R)->(B), (B)->(L), (L)->(F), (F)->(R)
U2, D2 --> (R)->(L), (L)->(R), (F)->(B), (B)->(F)
Here left one is affecting rotation, right is if permutation of a piece contains like left element, substitute like right one.
3. Methods to Express
Situation S with imaginary piece can be express.
Attachment:
File comment: How to Express Situation

Situation.png [ 2.58 KiB | Viewed 384 times ]

Here, a ... b are pieces.
There's no matter in permutation of pieces.
Multiply of two situation can be define like this.
[img]Multiply.png[/img]
Here, circle is a operation with definition.
Attachment:
File comment: Definition of Circular Operation

Circular Operation.png [ 2.57 KiB | Viewed 384 times ]

Inversed situation of A is A^-1 and it's inverse element of A and it is same with
Attachment:
File comment: Expression of Inversed Situation

Inversed Situation.png [ 3.76 KiB | Viewed 384 times ]
.
4. Source
1. http://cafe.naver.com/cubemania/548906

ps. I do not have information of this paper
I need error list
ps2. This is not original version
Here is original-http://cafe.naver.com/cubemania/550793

 Attachments: File comment: How to Multiply 2 Situations Multiply.png [ 33.9 KiB | Viewed 384 times ]

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Last edited by FFFC on Fri Jan 25, 2013 4:39 am, edited 1 time in total.
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 Post subject: Re: Expression of Rubik's Cube with Imaginary PiecesPosted: Fri Jan 25, 2013 4:43 am

Joined: Mon Dec 19, 2011 2:07 pm
Location: Incheon, South Korea
But unfortunately the links are permitted the cafe's member.

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