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 Post subject: Expression of Rubik's Cube with Imaginary Pieces
PostPosted: Fri Jan 25, 2013 4:10 am 
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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).
0. Table of Contents
0. Table of Contents
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
0.png [ 1.84 KiB | Viewed 551 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
Situation.png [ 2.58 KiB | Viewed 551 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
Circular Operation.png [ 2.57 KiB | Viewed 551 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
Inversed Situation.png [ 3.76 KiB | Viewed 551 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
Multiply.png [ 33.9 KiB | Viewed 551 times ]

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ko 이 회원은 한국어가 모국어입니다.
en-2 This member is able to contribute with an intermediate level of English.


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 Pieces
PostPosted: Fri Jan 25, 2013 4:43 am 
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Joined: Mon Dec 19, 2011 2:07 pm
Location: Incheon, South Korea
But unfortunately the links are permitted the cafe's member. :D

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