User:Dave3457/sandbox/Conjugates Sandbox
A conjugate is a series of 3 sequences the middle of which, if performed alone, would manipulate a group of pieces at a specific location on the cube. The first sequence however maneuvers an otherwise unaffected group of pieces into that specific location. After the substituted group of pieces are manipulated by the middle sequence, the third sequence, which is the inverse of the first, returns the altered substituted group, back to its original location. Because the third sequence is the exact inverse of the first, the entire cube, with the exception of the substituted group of pieces, is completely restored. The middle sequence of a conjugate is most often a commutator. The short form for the conjugate A B A' is [A: B]
which has it origins in group theory
xxxxxxxxxxxxxxxxxxx A conjugate is a sequence of moves that consists in doing a sequence A, then a sequence B, and finally the inverse of the sequence A. In other words, a conjugate is an algorithm like A B A', where A and B can be any sequence of moves. In intuitive terms, it means: do something to set up another task which does something useful, and undo the setup.
Contents
Mathematical definition
Given a group, a conjugate is an element of the form ghg', where g and h are elements of the group with inverses g' and h'.
Cube notation is very close : A B A'
It is sometimes written as [A: B] = A B A'
Effect
In A B A', the sequence A is the setup move or sequence and A' puts the cube in its original state. In between, there is an embedded sequence B. Alone, this sequence would affect some set of locations K on the cube. Because of the conjugation, the affected pieces will be KA', which means that the locations will be transformed by the inverse of A.
Indeed the sequence B will be applied on a cube which space has been modified by A. The sequence B is supposed to change some pieces in the locations K and A consists in bringing the piece you want to change to the location K. Once you've done the sequence B, you bring those pieces back with A'.
Trivial case
If the set of locations K is not modified by A, then the conjugation does not change anything, and A B A' = B.
General case
Pieces can be affected by the following subsequence of ABA':
- by B only: this part is outside of the conjugation
- by AB: this is the first semi-conjugated part
- by BA': this is the second semi-conjugated part
- by ABA': this is the conjugated part
Let's call J the set of pieces modified by A, and N the intersection of J and K.
The initial location of the outside part is: O = ((K \ N)B \ N)B'
The initial location of the first semi-conjugated part is: H1 = (NB \ N)B'A'
The initial location of the second semi-conjugated part is: H2 = NB' \ N
The initial location of the conjugated part is: C = (NB inter N)B'A'
The slot location is: NA'
Usage
The conjugation is often used to provide a setup for some transformation. The conjugation ABA' is very similar to B, except that B is applied on other pieces as it would be without conjugation. For example, if a sequence exchange some pieces, or rotate some pieces, you can choose to exchange other pieces or rotate other pieces :
[F': [M2D2M2, U]]
where [M2D2M2, U] exchange the four top edges, so applying F' exchange the front-right edge instead of the front-top edge.
The intersection is N = front-top edge
The slot is NA' = NF = front-right edge
The conjugated part is:
C = (NB inter N)B'A' = empty
Surprisingly, in this example, the conjugated part is empty. It means that no individual piece is actually following the whole conjugation. F' and [M2D2M2, U] are only connected so that some pieces are transfered through the intersection N. The transfer occur between both semi-conjugated parts.
In this simple case, we do not really distinguish the semi-conjugated parts. But it is possible to take this difference into account, and also to use a conjugated part.
Semi-conjugated parts
Semi-conjugated parts can be used to move pieces inside or outside a slot. This is widely used to insert or extract a pair of pieces in F2L or to insert or extract corners or edges individually.
H1 is extracted from the slot and H2 is inserted into the slot.
Example: RU'R'
N = top-right edge, top-right-back and top-right-front corners
Slot location : NA' = NR' = top-right-front corner, front-right edge and front-right-bottom corner
H1 = (NU' \ N)UR' = top-right-front corner and front-right edge
These pieces are extracted from the slot and replace the top-back edge and top-back-left corner.
H2 = NU \ N = top-front-right corner and top-front edge
These pieces are inserted into the slot, replacing front-right edge, and front-right-bottom corner.
Example: RUR'
N = top-right edge, top-right-back and top-right-front corners
Slot location : NA' = NR' = top-right-front corner, front-right edge and front-right-bottom corner
H1 = (NU \ N)U'R' = front-right edge and front-right-bottom corner
These pieces are extracted from the slot and replace the top-front edge and top-front-left corner.
H2 = NU' \ N = top-back-left corner and top-back edge
These pieces are inserted into the slot, replacing front-right edge, and front-right-top corner.
Conjugated part
The conjugated part is used to move pieces inside the slot (the pieces do not enter nor leave the slot):
RUR'
where the intersection is N = top-right edge, top-right-front corner and top-right-back corner
and the slot is NA' = NR' = front-right edge, front-right-top corner and front-right-bottom corner
The conjugated part is :
C = (NB inter N)B'A' = top-right-front corner
This piece is moved to the front-right-bottom corner and rotated so that the right facelet is moved to the bottom facelet.
Link with commutators
The outside part O is only affected by B', so it can be put back into place by adding B' to the sequence, giving ABA'B', which is the commutator of A and B.
The first semi-conjugated part H1, which was affected by AB, is then only affected by A, becoming the container of pieces PA.
The second semi-conjugated part H2 is split into two parts:
- one part which is affected by B', so affected by BA'B', and is then a conjugated part of the commutator
- one part which is not affected by B', so affected only by BA', becoming the container of pieces PB.
The conjugated part is split into two parts:
- one part which is affected by B', becoming the quirk part
- one part which is not, becoming a conjugated part of the commutator
Pieces that were not affected by the conjugate, but that are affected by the last B' move, are the PN pieces of the corresponding commutator.
