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

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

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Section III - Setting up the [ 24 Base Structural Patterns ]

Summary for the Section :

• In this Section , we shall bring-in the set of [ 24 Base Structural Patterns ] ,
  • which will serve as the back-bone for our Expansion-Mapping Process here and beyond .

  And it is noted here , at the very beginning of Part II , that :

  • it is thru the [ Expansion-Mapping Process ] making use of the [ 24 Base Structural Patterns ]
    • that we were able to identify the [ quasi 24-symmetric sets ] found within

      the { Reduced Set of 5,040 [ 8-Colors Circular-Ring Patterns ] with the Red-Color in Fixed-Position [ FP-1 ] } .

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Bringing-in the [ 24 Base Structural Patterns ] :

• Let us now bring-in the set of [ 24 Base Structural Patterns ] , as follows :
  • in this { 4 rows x 6 columns } arrangement .

  

• Several Key Features to take note of here :
  • Each of the 24 [ Patterns ] is made-up-of a [ Circular-Ring ] with 8 positions ;

  • Each of 8 positions within the [ Pattern ] carries a [ number ] , from [ 1 ] to [ 8 ] :
    • with each of the 8 positions carrying a different-and-distinct [ number ] ;

  • The position at-the-top of each [ Circular-Ring ] always carries the number [ 1 ] .

  ( Issues related to the design of the [ 24 Base Structural Patterns ] will be dealt with in Part V of this paper ) .

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The Expansion-Mapping Process explained :

• In our Expansion-Mapping Process , we simply carry-out this 4-steps procedure below :
  • Step 1 :
    • We select any one of the [ 24 Base Structural Patterns ] as the [ Anchor Pattern ] for our Expansion-Mapping process .

  • Step 2 :
    • We bring-in one of the 5,040 { 8-colors circular-ring-patterns with the Red-Color in Fixed-Position [ FP-1 ] } ,
      • and use this as the [ Core Pattern ] for our Expansion-Mapping Process .

  • Step 3 :
    • We create a [ Colors-to-Numbers Assignment Scheme ] , via :
      • matching the 8-colors of the [ Core Pattern ] against the 8-numbers of the [ Anchor Pattern ] ,
        • on a position-for-position basis .

  • Step 4 :
    • We do the [ expansion-mapping process ] on the [ 24 Base Structural Patterns ] ,
      • making use of the [ Colors-to-Numbers Assignment Scheme ] ;

      to arrive at the resultant set of 24 [ 8-colors circular-ring-patterns ] .

• Let us now do a first-run of the Expansion-Mapping Process , next .

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Selecting an [ Anchor Pattern ] for our Expansion-Mapping Process :

• Let us now select one of the [ 24 Base Structural Patterns ] as the [ Anchor Pattern ] .

  And we shall select the [ Base Pattern ] in { Row 1 , Column 1 } as our [ Anchor Pattern ] , at this first-instance ,

  • i.e. :

  

  • the [ Base Structural Pattern ] as marked-off in [ dark-grey color ] in the diagram above .

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Bringing-in the pre-selected [ 8-colors circular-ring pattern ] :

• Let us now bring-in the pattern [ Pattern AARON ] :

  

  • and use this as the [ Core Pattern ] of our Expansion-Mapping Process .

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The Colors-to-Numbers Matching Process :

• Let us now do the [ Colors-to-Numbers Matching Process ] :

  

  • matching the [ Anchor Pattern ] on-the-left against the [ Core Pattern ] on-the-right .

  And the [ Colors-to-Numbers Assignment Scheme ] here , based on the above , is then :

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

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The Expansion-Mapping Process :

• Let us now do the Expansion-Mapping process , using the [ Colors-to-Numbers Assignment Scheme ] from above ,

  

  • to arrive at the resultant set of 24 patterns , as per the diagram on-the-right above .

  And we shall name this particular [ set of 24 patterns ] as the [ 24 QSM-Patterns --- ALPHA Series ] ,

  • for comparisons with other [ sets of 24 patterns ] in later Sections to come .

    ( QSM then stands for Quasi SymMetric . )

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Identifying the pattern [ Pattern KENT ] for our use :

• Let us now also identify the [ 8-colors circular-ring-pattern ] in { Row 2 , Column 4 } in the diagram on-the-right above ,
  • i.e. , this pattern :

    

    as the pattern [ Pattern KENT ] , for our use in the next Section .

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go to the next section : Section IV - Demonstrating [ Quasi 24-Symmetry ] - Phase I

go to the last section : Section II - Basic Combinatorics for [ 8-Colors Circular-Rings ]

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

Original dated 2011-3-06