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

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

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Section XX - [ Quasi 24-Symmetry ] arising from the Half-On-Half Rotations

Summary for the Section :

• In this Section , we shall attempt to establish :
  • the [ quasi 24-Symmetric ] nature of the [ 24 QSM-Configurations --- Himalaya Series ] .

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Recalling the [ 24 QSM-Configurations --- Himalaya Series ] :

• Let us bring-in again the [ 24 QSM-Configurations --- Himalaya Series ] , from Section XVIII :
  • for re-familiarization purposes .

  

  And we take special note here that :

  • the Linkage Diagram on-the-right above is made-up-of 36 links and 24 positions .

Coverting the 24 [ QSM-Configurations ] to [ 8-Colors Sequences ] :

• Let us now quickly do the conversion to the [ 8-colors sequences ] ,
  • using the [ Fu-Hei Scheme of Fixed-Positions ] and the [ ASENDING Read-Off Order ] ;

    i.e. :

    • 

  to arrive at this set of 24 [ 8-colors sequences ] , as per the diagram on-the-right below .

  

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Converting to the [ 8-numbers strings ] :

• Let us now convert the above 24 [ 8-colors sequences ] into the [ 8-numbers strings ] ,
  • using the [ FIRST-UP Colors-to-Numbers Coversion Scheme ] ; i.e. :

  Color #1	Color #2	Color #3	Color #4	Color #5	Color #6	Color #7	Color #8
  ***********	***********	***********	***********	***********	***********	***********	***********
  RED	CYAN	PURPLE	MAGENTA	GREEN	YELLOW	ORANGE	BLUE

  • to arrive at the diagram on-the-right below .

  

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A Quick Comparison :

• It is appropriate to bring-back , at this point , the [ 24 Base Structural Patterns ] from Section III for a quick comparison :

  

  And we can see here that :

  • the 24 [ 8-elements Circular Sequences ] on the left-hand-side diagram and

    the 24 [ 8-Numbers Strings ] on the right-hand-side diagram

    • do match exactly , on a grid-for-grid / grid-on-gird basis .

• Since we have already demonstrated in Part II of this paper that :
  • the Left-Hand-Side Diagram does exhibit [ Quasi 24-Symmetric ] properties ;

  we can now safely assume that :

  • the Right-Hand-Side Diagram will also exhibit [ Quasi 24-Symmetric ] properties .

• But the real question here is this :
  • Is the set of [ 24 QSM-Configurations --- Himalaya Series ] also [ Quasi 24 Symmetric ] in nature ?

  And the answer here is YES !

  • as explained below .

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[ Quasi 24-Symmetry ] via Physical Relations :

• Let us bring-in again this set of 2 diagrams from above ,
  • and concentrate , for the moment , on the right-hand-side diagram .

  

• Let us recall that :
  • we originally started-out with [ Configuration Annapurna ] as [ Configuration A ] ,

  • and expanded via [ 2-notch rotations in the -X / -Y / -Z Axial-Directions ] ;
    • to arrive at the set of [ 24 QSM-Configurations --- Himalaya Series ] .

  And the diagram on-the-right above is then the Linkage Diagram for the above-said expansion process .

• And we take note here that , in this Linkage Diagram ,
  • each of the 24 positions has exactly three (3) links coming-out-of-it .

  And as such ,

  • the [ 2-notch rotations in the -X / -Y / -Z Axial-Directions ] ( totaling 3 rotations )

    for each-and-every one of 24 positions on the said Linkage Diagram ,

    • are fully accounted for .

  And what this means is that :

  • No matter how we might want to move-around within this Linkage Diagram :
    • we can only reach the 24 positions in the diagram and no more .

  Thus , the [ 24 QSM-Configurations --- Himalaya Seires ] may now be considered :

  • a self-contained set of 24 [ 8-colors configurations ] .

• Let us suppose , for the moment , that :
  • we plugged-in any of the 24 above-said [ QSM-Configurations ] into [ Position A ] in the Linkage Diagram ,

    and do the full expansion via the 36 links as shown in the Diagram .

  Would we then get the exact-same set of [ 24 QSM-Configurations ] , each and every time , regardless ?

  • Clearly we would .

  And as such , each of the [ 24 QSM-Configurations --- Himalaya Series ] is ranked on par with one-another ;

  • and therefore the notion of [ Quasi 24-Symmetry ] .

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What is next :

• And we shall look at , next , another set of [ 24 QSM-Configurations ] ,
  • arising from a new-and-different [ 8-colors configuration ] --- [ Configuration Mount Robson ] .

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go to the next section : Section XXI - Understanding [ Quasi 24-Symmetry ] - Phase I

go to the last section : Section XIX - Converting the [ 8-colors configurations ] into [ 8-numbers strings ]

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

Original dated 2011-3-06