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

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

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Section XIX - Converting the [ 8-colors configurations ] into [ 8-numbers strings ]

Summary for the Section :

• In this Section , we shall list-out the procedure for :
  • first ,
    • converting the [ 8-colors configurations ] into [ 8-colors sequences ] ;

  • and then secondly ,
    • converting the [ 8-colors sequences ] into [ 8-numbers strings ] .

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Reasons for the deployment of [ 8-numbers strings ] :

• And the deployment of [ 8-numbers strings ] is indeed core-and-essential to our analysis for 2 reasons :
  • First Reason :
    • Given a fairly large amount of [ 8-colors configurations ] to work with ,
      • the use of computer-aided analysis is deemed both necessary and appropriate for our purposes here .

      And the current-state-of-affairs will only allow us to input [ numbers ] into the computer , and not [ colors ] .

    • It is therfore important that we understand fully the mechanism for converting the [ colors ] to the [ numbers ] .

  • Second Resaon :
    • In Part VII of this paper , we shall be looking into the subject of [ Physical Symmetry vs. Logical Symmetry ] .

      It is therefore essential that we are fully conversant with the 3 elements involved in the conversion process here , namely :

      • the [ Labelling Scheme for the 8 Fixed-Positions ] of the [ 8-Colors Cube ] ,

      • the [ Color Read-Off Order for the Fixed-Positions ] in the formation of the [ 8-colors sequences ] , and

      • the [ Colors-to-Numbers Conversion Scheme ] .

  And we shall be looking into these 3 items , next .

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Labelling the 8 Fixed-Positions :

• Let us recall that the [ 8-Colors Cube ] is always made-up-of 8 color-pieces ,
  • neatly arranged in 8 fixed-positions .

  

  Let us now label these 8 fixed-positions as :

  • the Fixed-Positions [ FP-1 ] , [ FP-2 ] , [ FP-3 ] , [ FP-4 ] , [ FP-5 ] , [ FP-6 ] , [ FP-7 ] and [ FP-8 ] ;
    • as per this set of 8 diagrams below :

  
  

• And we shall name this labelling scheme the [ Fu-Hei Scheme of Fixed-Positions ] :
  • 
    • for identification purposes .

• Three (3) quick comments here :

  Comment One :

  • We shall be using only one [ Labelling Scheme for the 8 Fixed-Positions ] throughout the entire paper ,
    • and this one-and-only [ labelling scheme ] is the above-said [ Fu-Hei Scheme of Fixed-Positions ] .

  Comment Two :

  • It is alright for us to stick with only one [ labelling scheme for the 8 Fixed-Positions ] .

    And this is because , in the analysis to follow :

    • we can always vary the [ Color Read-Off Order for the 8 Fixed-Positions ] , ( the next item on the agenda ) ,
      • to create the variations in the [ 8-colors sequences ] arising from any given [ 8-colors configuration ] ;

      for our further review and evaluation in our analysis process .

    It is therefore not necessary to birng-in another [ labelling scheme for the 8 Fixed-Positions ] .

  Comment Three :

  • There is a specific reason for using the [ Fu-Hei Scheme of Fixed-Positions ] here :
    • and this to maintain consistency with the author's other paper back in 2004 , namely :
      • An Approach to the I-CHING .

    And we bring-in this set of 3 diagrams for readers who are familiar with the above-said paper :

    • the diagram on-the-left is the [ Fu-Hei Scheme of the 8 Fixed-Positions ] from the I-CHING ,

    

    • the diagram in-the-middle is the author's interpretation of the [ Fu-Hei Scheme ] in 3-D , and

    • the diagram on-the-right is the current [ Fu-Hei Scheme of Fixed-Positions ] we are using here in this paper .

    And this is done :

    • just-in-case somebody find an interesting connection between our presentation here and the I-CHING !

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The [ Read-Off Order of the Fixed-Positions ] in the formulating the [ 8-colors sequences ] :

• In order to create and formulate a corresponding [ 8-colors sequence ] , for any given [ 8-colors configuration ] :
  • it is neccessary for us to read-off the [ color ] in each of the 8 Fixed-Positions in a [ specific order ] .

  And this [ specific order ] shall be known as the [ Color Read-Off Order for the 8 Fixed-Positions ] ,

  • or simply as the [ Read-Off Order ] for the 8 Fixed-Positions .

• Let us take a quick look at a few examples of the [ Read-Off Orders ] here :

  

  And a quick comment here :

  • Since we shall be working primarily with the [ Reduced Set of 5,040 configurations ] ,
    • where the [ Red-Color Piece ] is always in Fixed-Position [ FP-1 ] ;

    all our subsequent [ Read-Off Orders ] will also start with Fixed-Position [ FP-1 ] as the [ Lead-off Position ] ,

    • so that :
      • all the [ 8-colors sequences ] arising from the [ Reduced Set of 5,040 configurations ]
        • will always start-off with the [ red-color ] as the 1st-color of the [ 8-colors sequence ] .

    And this is done as a matter of convenience to facilitate our analysis and comparisons .

• Let us now do a quick example :
  • using the [ Fu-Hei Scheme of Fixed-Positions ] and the [ ASCENDING Read-Off Order ] .

  

  And the [ 8-colors sequence ] here is then :

  • [ Red-Cyan-Purple-Magenta-Green-Yellow-Orange-Blue ] .

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Numbering the Colors :

• In order to convert the [ 8-colors string ] into a corresponding [ 8-numbers string ] ,
  • it is necessary to set up a [ colors-to-numbers conversion scheme ] .

  And for the time-being , we shall use this scheme below as our primary [ Colors-to-Numbers Conversion Scheme ] :

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

  And we shall name this scheme here the [ FIRST-UP Colors-to-Numbers Conversion Scheme ] ,

  • for identification purposes .

• Let us also do a quick example here :
  • using the [ Fu-Hei Scheme of Fixed-Positions ] and the [ ASCENDING Read-Off Order ] , and

  • the [ FIRST-UP Colors-to-Numbers Conversion Scheme ] .

  

  And the [ 8-numbers string ] here is then :

  • [ 1-2-3-4-5-6-7-8 ] .

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go to the next section : Section XX - Quasi 24-Symmetry arising from the Half-On-Half Rotations

go to the last section : Section XVIII - The Reduced Set of 5,040 [ 8-colors configurations ]

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

Original dated 2011-3-06