Hi
The way i prefer to explain (most common) commutators (for 3-cycles at least) is to show how it works on a "cutplane". The 2 different kinds of cutplanes available are outer and inner layers. Inner layers are of course not functioning for corner cycles. Cycling only 3 centers on a 3x3x3 cube is not possible.
The idea of almost all my commutators is to do the following:
1 - Replace a single cubie on a cutplane (A- commonly 3 or 4 turns)
2 - Rotate the cutplane (B - always a single turn!!)
3 - Return the cubie removed from the cutplane back into its new position on the cutplane (A' - inverse of A. To invert a sequence start from the back and move to the front. Invert every turn as you go along. The inverse of UR is NOT RU but R'U'.).
4 - Undo the cutplane rotation.
Easy example on corners (any size cube:
1 - R' D R (A-part. Replaces one corner on the cutplane)
2 - U (B-part, the cutplane rotation)
3 - R' D' R (inverse of A. Returns the corner removed from cutplane back into a new position on the cutplane)
4 - U' (inverse of B-part)
This becomes R' D R U R' D' R U' (indeed a 3-cycle)
(Easy example on edges:
1 - r' D r (A-part)
2 - U (B-part, the cutplane rotation)
3 - r' D' r (inverse of A)
4 - U' (inverse of B-part)
This becomes r' D r U r' D' r U'
Note the intended similarity between the 2 examples.)
This whole thing becomes ABA'B' where the total length becomes length of A times two plus 2 (since length of B is 1!!)
Now for bigger cubes edges belong to different ORBITALS.
On a 4x4x4 cube UFr, URb, UBl and ULf all belong to the same ORBITAL as those positions are only a single turn of the cutplane apart.
UFl, URf, UBr and ULb belong to the same (but different) orbital
What to do if you want to cycle 3 edges of different ORBITALS?
Simple answer: position the edges into same orbital first.
[UNFINISHED POST - lunch !!!!]
[edit - i will finish this post some other time, probably completely rewrite it by the look of it !!]
- Per
