On 8-Colors Combinatorics and [ Quasi 24-Symmetry ]

by Frank C. Fung ( 1st published in March, 2011. )

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Section XXVII - [ Quasi Tri-Symmetric Sets ] within the [ 24 QSM-Configurations ]

Summary for the Section :

• In this Section , we shall identify the [ Quasi Tri-Symmetric Sets ] within the [ 24 QSM-Configurations ] .

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Spliiting the [ 6-elements loops ] :

• Let us recall that we did , in the last Section , ponder the possibility of :
  • splitting the [ 6-elements loop ] into 2 tri-symmetric [ 3-elements loops ] .

• And we shall do so now .

  For the [ 6-elements loops ] involving the [ -Y Axis ] and the [ -Z Axis ] , we have :

  

  • And the 2 diagrams in-the-middle and on-the-right , above , do exhibit [ Quasi Tri-Symmetry ] .

  For the [ 6-elements loops ] involving the [ -Z Axis ] and the [ -X Axis ] , we have :

  

  • And the 2 diagrams in-the-middle and on-the-right , above , do exhibit [ Quasi Tri-Symmetry ] .

  For the [ 6-elements loops ] involving the [ -X Axis ] and the [ -Y Axis ] , we have :

  

  • And the 2 diagrams in-the-middle and on-the-right , above , do exhibit [ Quasi Tri-Symmetry ] .

  We can now simply bring back the 12 [ 6-elements loops ] from the last Section , i.e. :

  • [ G-T-V-L-Q-X ] , [ A-N-Y-E-S-U ] , [ B-H-R-D-K-P ] , and [ C-W-M-F-Z-J ] in the 1st group ;

  • [ N-Y-H-R-W-M ] , [ A-U-K-D-X-G ] , [ B-P-Z-F-T-V ] , and [ C-J-S-E-L-Q ] in the 2nd group ;

  • [ U-K-P-Z-J-S ] , [ A-G-T-F-M-N ] , [ B-V-L-E-Y-N ] , and [ C-Q-X-D-R-W ] in the 3rd group ;

    and do the splitting as shown above to arrive at the [ Quasi Tri-Symmetric Sets ] .

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Listing of the [ Quasi Tri-Symmetric Sets ] :

• Let us now list-out the 24 [ Quasi Tri-Symmetric Sets ] for the record :
  • For the 1st group :
    • [ G-V-Q ] , [ L-T-X ] , [ A-Y-S ] , [ E-N-U ] , [ B-R-K ] , [ D-H-P ] , [ C-M-Z ] and [ F-W-J ] .

  • For the 2nd group :
    • [ N-H-W ] , [ R-Y-M ] , [ A-K-X ] , [ D-U-G ] , [ B-Z-T ] , [ F-P-V ] , [ C-S-L ] and [ E-J-Q ] .

  • For the 3rd group :
    • [ U-P-J ] , [ Z-K-S ] , [ A-T-M ] , [ F-G-N ] , [ B-L-Y ] , [ E-V-N ] , [ C-X-R ] and [ D-Q-W ] .

• And we bring back this set of 2 diagrams for the reader's easy reference and for checking purposes :

  

• Rather simple and straight-forward , eh !

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Listing-out the Structural Mapping relations :

• It is also appropriate to list-out the [ Structural Mapping relations ] for the above 3 groups of [ Quasi Tri-Symmetric Sets ] ,
  • and we shall use the [ Fu-Hei Scheme of Fixed-Positions ] as the referencing framework for the [ Fixed-Positions ] here :

    i.e. :

    • 

  And the [ Structural Mapping relations ] here are then :

  • For the 1st group :
    • The 2 colored-pieces in [ FP-1 ] and [ FP-5 ] do not move ,
    • The 3 colored-pieces in [ FP-4 ] , [ FP-7 ] , and [ FP-6 ] exchange positions in a rotary fashion ,
    • The 3 colored-pieces in [ FP-8 ] , [ FP-3 ] , and [ FP-2 ] exchange positions in a rotary fashion .

  • For the 2nd group :
    • The 2 colored-pieces in [ FP-1 ] and [ FP-3 ] do not move ,
    • The 3 colored-pieces in [ FP-6 ] , [ FP-4 ] , and [ FP-7 ] exchange positions in a rotary fashion ,
    • The 3 colored-pieces in [ FP-8 ] , [ FP-2] , and [ FP-5 ] exchange positions in a rotary fashion .

  • For the 3rd group :
    • The 2 colored-pieces in [ FP-1 ] and [ FP-2 ] do not move ,
    • The 3 colored-pieces in [ FP-7 ] , [ FP-6 ] , and [ FP-4 ] exchange positions in a rotary fashion ,
    • The 3 colored-pieces in [ FP-8 ] , [ FP-5 ] , and [ FP-3 ] exchange positions in a rotary fashion .

  And [ Structural Mapping ] consistency is always maintained throughout each of the three (3) groups .

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go to the next section : Section XXVIII - [ Quasi Quad-Symmetric Sets ] within the [ 24 QSM-Configurations ]

go to the last section : Section XXVI - [ Quasi Hexa-Symmetric Sets ] within the [ 24 QSM-Configurations ]

return to the 8-Colors Combinatorics / Quasi 24-Symmetry HomePage

Original dated 2011-3-06