This message may be overdo, but I figuered it ought to be
sent now since ALAN has done some work on this subject and may be able
to use my results.
Recently I have found many Identity transformations and this
message is basically a catalog of them.
I12-1 FR'F'RUF'U'FRU'R'U I12-2 L'D2F'D'FLD2BDB' I12-3 FR'F'RUF'UL'U'LFU' I14-1 D'L'DRD'LR'DF'D'RDR'F I14-2 D'L'D'F'DFLDF'R'D'RDF I14-3 LD2RL'F2L'F2LR'D2 I14-4 F'R'D'RDFUF'D'R'DRFU' I14-5 (LRD2L'R'D')^2 NOT GOOD IN SUPERGROUP DOES <D2> I14-6 LR'FRL'U'DFB'R'F'BUD' " " <FR'> I16-1 (LRD2L'R'D2)^2 I16-2 F'R'D'RDFU2F'D'R'DRFU2 I16-3 F'R'D'RDFD2B'D'L'DLBD2 I16-4 F'R'D'RDFUDR'D'B'DBRU'D' I16-5 F'R'D'RDFU'DR'D'B'DBRUD' I16-6 UF2U'DR2D'U'R2UD'F2D I16-7 LR'D2RL'F2RL'F2LR'D2 I16-8 FDLD'F'LDL'F'L'D'LFD'L'D I16-9 LD'L'D'F'DFUF'D'FDLDL'U' I16-10 FL'F'LF'D'FUF'DFL'FLF'U' I16-11 F'D2R'D'RD'FLD2BDB'DL' I16-12 (F2B2R2L2)^2 I16-13 LR'FRL'U2D2L'RB'LR'U2D2 NOT IN SUPERGROUP <FB'> I16-14 LR'F2RL'U'DFB'R2F'BUD' " " <F2R2> I18-1 F'BD2F'D'FD2B'LBDB'L2FL I18-2 LRD'L'R'D'LRDL'R'DLRDL'R'D' I18-3 D'RD'R'DBDB'DBD2B'D'RD2R' I18-4 LR'F2RL'U2D2L'RB2LR'U2D2 I18-5 LDR'L'D'LRDL'R'D'RLDR'L'D'R I18-6 (RBL'R'B'L)^3 I18-7 LDR'L'D'LRDL'R'DRLD'R'L'DR NOT IN SUPERGROUP <D2> I18-8 (F'D'FD'RD2R'D)^2 " " <D2>
These are not all of the identities that I have found but are generators
How a generator generates other identities: 1. Inversion (2) 2. Rotation (24) 3. Reflection (2) 4. Shifting (N) where N is the length of the identity
The numbers in () are the number of different ways that can be gotten
for each of these methods. Combining gives 96N.
For example an identity of length 12 generates a possible 96*12=1152.
However this number is usually not reached becauseof inherent symmetry.
If you take the inverse of I12-1 --> U'RUR'F'UFU'R'FRF'
Then its reflection -->UL'U'LFU'F'ULF'L'F
Then a rotation U->F,L->R,F->U -->FR'F'RUF'U'FRU'FR'U
You get back what you started with.
When shifting is included in this process there are a total of 6 different
ways this can be done giving 1152/6=192 different identities generated by
Shifting: Basically you chop the transform in its interior and append the first part to the second part. For instance. I12-1 FR'F'R / UF'U'FRU'R'U Becomes UF'U'FRU'R'U FR'F'R Note that this is just a rotation away from the origional. >From these some equivilences may be deduced: DL'F'D2R'D'R = L'D2F' = BD'B'D2L'F'D DF'R'DRD'FD' = FL'F'L =D'LD'BDB'L'D FR'F'RUF' = U'RUR'F'U = UF'L'ULU'
Unfortunatly these equivilences only generate the 3 identities of
length 12 , using the idea that midpoint of an identity must be the
unique maximum along the path of the transform.