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

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

Top Of Page

Section XXVI - [ Quasi Hexa-Symmetric Sets ] within the [ 24 QSM-Configurations ]

Summary for the Section :

• In this Section , we shall be looking into the [ Quasi Hexa-Symmetric Sets ] found within the [ 24 QSM-Configurations ] .

go to Top Of Page

Recalling the [ Quasi Hexa-Symmetric Sets ] from Section XIII :

• Let us recall that we did identify , earlier in Section XIII of this paper ,
  • 4 sets of [ Quasi Hexa-Symmetric Patterns ] within the [ 24 Base Structural Patterns ] .

  

  

  And these 4 [ Quasi Hexa-Symmetric Sets ] with 6 elements each are :

  • [ A ] - [ U ] - [ X ] - [ G ] - [ K ] - [ D ] ,

  • [ Z ] - [ F ] - [ B ] - [ P ] - [ T ] - [ V ] ,

  • [ R ] - [ H ] - [ M ] - [ W ] - [ Y ] - [ N ] , and

  • [ L ] - [ Q ] - [ S ] - [ E ] - [ C ] - [ J ] .

• Let us now also try and see if we can identify [ Quasi Hexa-Symmetric Sets ] within the set of [ 24 QSM-Configurations ] .

go to Top Of Page

Recalling the 3 groups of 4 non-intersecting [ 6-elements loops ] :

• Let us recall that we have identified , in the previous section - Section XXV ,
  • 3 groups of 4 non-intersecting [ 6-elements loops ] .

  And these are :

  • The [ 1st Group ] :
    • [ 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 ] .

  • The [ 2nd 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 ] .

  • The [ 3rd 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 ] .

• And we see here immediately that the [ 2nd Group ] here
  • matches the above 4 [ Quasi Hexa-Symmetric Sets ] for the [ 24 Base Structural Patterns ] ,
    • on a [ set-for-set ] basis ;

    although the listing-order-for-the-6-elements within each pair of matching-sets are quite different .

    ( But the [ listing-order ] here is only a minor issue because :

    • the [ relative structural mapping relations ] between each pair of elements within the [ 6-elements set ] frameworks

      will remain in tact regardless of the listing-order-for-the-6-elements .)

  As such ,

  • [ 6-elements loops ] are now our prime candidates for
    • [ Quasi Hexa-Symmetric Sets ] within the set of [ 24 QSM-Configurations ] .

go to Top Of Page

Bringing-in the [ Linkage Diagrams ] :

• Let us now try to gain a better understanding of the [ 6-elements loops ] , so that :
  • we would be in a position to track the mechanisms that may give rise to the [ Quasi Hexa-Symmetric Sets ] .

  And for this purpose , let us now bring-in again this set of 3 Linkage Diagrams from the previous section , Section XXV :

  

  And we take special note here that :

  • The [ 1st Group ] of 4 non-intersecting [ 6-elements loops ] , 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 ] ;

      involves only [ 2-notch rotations ] in the [ -Y Axial-Direction ] and the [ -Z Axial-Direction ] ;

      • but not in the [ -X Axial-Direction ] .

  • The [ 2nd Group ] of 4 non-interescting [ 6-elements loops ] , i.e. :
    • [ 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 ] ;

      involves only [ 2-notch rotations ] in the [ -Z Axial-Direction ] and the [ -X Axial-Direction ] ;

      • but not in the [ -Y Axial-Direction ] .

  • The [ 3rd 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 ] ;

      involves only [ 2-notch rotations ] in the [ -X Axial-Direction ] and the [ -Y Axial-Direction ] ;

      • but not the [ -Z Axial-Direction ] .

• We can then bring-in this set of 3 diagrams for a [ structural view ] of above-said 3 different types of [ 6-elements loops ] :

  

  And we shall now make two (2) key observations here , for our usage to follow .

  • 1st Observation :

    For each of 3 types of the [ 6-elements loops ] above :

    • Only two (2) Axial-Directions are involved in each type of [ 6-elements loops ] ,
      • and the 2 said-[ Axial-Directions ] are always orthogonal / perpendicular to one-another .

  • 2nd Observation :

    For each of 3 types of the [ 6-elements loops ] above :

    • When we move around the [ 6-elements loop ] , from-position-to-position in the same onwards-direction :
      • the above-said 2 [ Axial-Directions ] alternates .

• And these 2 features just-afore-mentioned then provides for a very special property for these types of [ 6-elements loops ] ,
  • namely :
    • the [ Flipping-Over Symmetry ] property ,
      • which we shall explain in full details below .

go to Top Of Page

Structural Mapping relations within the [ 6-elements loops ] :

• Let us recall that in a [ Half-on-Half rotation ] of the [ 8-colors cube ] :
  • one-half of the [ 8-colors cube ] containing 4 [ colored-pieces ] remains stationary and does not move ;

  • and the other-half of the [ 8-colors cube ] containing the other 4 [ colored-pieces ] rotates ,
    • thereby causing an exchange-of-positions for these 4 [ colored-pieces ] ,
      • in a manner befitting the said [ Half-on-Half rotation ] .

  As such :

  • each-and-every [ Half-on-Half rotation ] of the [ 8-colors cube ] then depicts-and-defines a specific [ Structural Mapping relations ] :
    • between
      • the [ 8-colors-configuration after-the-execution-of-the-Half-On-Half-rotation ] ,

      and

      • the [ 8-colors-configuration before-the-execution-of-the-Half-On-Half-rotation ] .

• It then follows that :
  • each-and-every [ sequence of Half-on-Half rotations ] of the [ 8-colors cube ] also depicts-and-defines a specific [ Structural Mapping relation ] :
    • between
      • the [ 8-colors-configuration after-the-execution-of-the-Sequence-of-Half-On-Half-rotations ] ,

      and

      • the [ 8-colors-configuration before-the-execution-of-the-Sequence-of-Half-On-Half-rotations ] .

• Let us now bring-in again , from above , the [ structural view ] diagrams for the 3 types of [ 6-elements loops ] ;
  • and we mark-off , for our illustration here on the [ Structural Mapping relations ] ,
    • an [ Anchor Position ] , which may be any one of the 6 positions within the [ 6-elements loop ] .

  

  And we can see here then that :

  • each of the 6 [ positions ] witin the [ 6-elements loop ] then depicts-and-defines
    • a specific [ Structural Mapping relation ] :
      • between
        • the [ 8-colors configuration ] in that particular [ position ]

        and

        • the [ 8-colors configuration ] in the [ Anchor Position ] .

  ( The [ 6th Position ] above then defines a self-on-self [ Structural Mapping relation ] with respect to the [ Anchor Position ] . )

go to Top Of Page

Concentrating on the [ N-Y-H-R-W-M ] loop for the time-being :

• Let us focus our attention , for the time-being , on the [ N-Y-H-R-W-M ] loop ,
  • for our illustration purpose in this sub-section .

  And the applicable [ structural view ] diagrams here are then :

  • the 2 diagrams on-the-left and in-the-middle , below .

  

  Let us now plug-in [ N ] / [ Y ] / [ H ] / [ R ] / [ W ] / [ M ] into the [ Anchor Position ] ,

  • and do the expansion therefrom using the Linkange Diagram on-the-right , above ;

    to arrive at this set of 6 diagrams , below .

  

  

  And we can see here that :

  • the [ N-Y-H-R-W-M ] loop showed up in all 6 diagrams , but :
    • the direction-of-flow is counter-clockwise when [ N ] / [ H ] / [ W ] is in the [ Anchor Position ] ; and

    • the direction-of-flow is clockwise when [ Y ] / [ R ] / [ M ] is in the [ Anchor Position ] .

• Two (2) rather important observations here :
  • First-and-most-important-of-all :
    • The fact that :
      • [ N ] , [ Y ] , [ H ] , [ R ] , [ W ] , [ M ] did show up in all 6 diagrams , and

      • only these 6 [ 8-colors configurations ] showed up in the 6 diagrams ;

      is then an iron-clad confirmation that [ N ]-[ Y ]-[ H ]-[ R ]-[ W ]-[ M ] do form a [ Quasi Hexa-Symmetric Set ] .

    And we can now follow a similar procedure to identify the other [ Quasi Hexa-Symmetric Sets ] .

  • Secondly :
    • The fact that :
      • the direction-of-flow for the [ N-Y-H-R-W-M ] loop went counter-clockwise :
        • on [ N ] / [ H ] / [ W ] being in the [ Anchor Position ] ;

      and

      • the direction-of-flow for the [ N-Y-H-R-W-M ] loop went clockwise :
        • on [ Y ] / [ R ] / [ M ] being in the [ Anchor Position ] ;

      then raise an interesting question on the parity-issue for the [ 6-elements loop ] .

• Should we then treat the 3-elements [ N-H-W ] and the 3-elements [ Y-R-M ] as having opposite parity ?

  But this is an issue that can be readily counter-balanced / resolved by the fact that :

  • the [ 6-elements loop ] possess the [ Flip-Over Symmetry ] property :

  

  i.e. :

  • when we flip-over the [ 6-elements loop ] on any one of the above [ Flip-Over Axis ] ,

    the flipped-over [ 6-elements loop ] would look exactly-the-same as the [ 6-elements loop ] before the Flip-Over !

  A quick example here is in order :

  

  • When we flip-over the [ 6-elements loop ] in the diagram on-the-left , above ,
    • using the [ Flip-Over Axis ] as shown in the diagram in-the-middle , above ;

    we would then arrive at the [ 6-elements loop ] in the diagram on-the-right , above .

  And this then allowed us to switch the [ Configuration-in-the-Anchor-Position ] from [ N ] to [ Y ] in a jiffy .

  • Rather neat , I would say !

go to Top Of Page

Identifying the [ Quasi Hexa-Symmetric Sets ] :

• Let us now list-out the [ Quasi Hexa-Symmetric Sets ] found within the [ 24 QSM-Configurations ] :
  • For the [ 6-elements loops ] involving the [ -Y Axis ] and the [ -Z Axis ] , we have these 4 [ Quasi Hexa-Symmetric Sets ] :
    • [ 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 ] .

  • For the [ 6-elements loops ] involving the [ -Z Axis ] and the [ -X Axis ] , we have these 4 [ Quasi Hexa-Symmetric Sets ] :
    • [ 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 ] .

  • For the [ 6-elements loops ] involving the [ -X Axis ] and the [ -Y Axis ] , we have these 4 [ Quasi Hexa-Symmetric Sets ] :
    • [ 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 ] .

go to Top Of Page

What's Next :

• In the next Section , we shall be re-shaping each of the [ 6-elements loops ] here :
  • into 2 distinct-and-different tri-symmetric [ Triangles ] to come up with [ Quasi Tri-Symmetirc Sets ] .

go to Top Of Page

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

go to the last section : Section XXV - The Linkage Diagram for the [ 24 QSM-COnfigurations ]

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

Original dated 2011-3-06