From:

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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

of them.

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

I12-1.

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.