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

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

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Section XV- Structural Considerations in Quasi Symmetry

Summary for the Section :

• In this Section , we shall take a deeper look at the [ Structural Mapping relation ] presented thus far .

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Setting up the set of [ 5,040 Base Structural Patterns ] :

• Let us bring-in again { Reduced Set of 5,040 [ 8-Colors Circular-Rings Patterns ] } and :
  • apply this [ Colors-to-Numbers Assignment Scheme ] below to all 5,040 members of the set ;

  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 a new set of 5,040 [ 8-numbers structual patterns ] .
  And the new 5,040 [ 8-Numbers Structural Patterns ] will be all distinct-and-different from one-another ,
  • because the original 5,040 members of the parent set are all distinct-and-different from one-another .

  Let us take a quick look now at a few examples of the conversion here , for the sake of clarity and precision :

  

  

• We can then regroup the entire set of the new 5,040 [ 8-Numbers Structural Patterns ] to form :
  • the set of [ 5,040 Base Strutural Patterns ]
    • for our use immediately below .

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Expansion-Mapping via the [ 5,040 Base Structural Patterns ] :

• Let us bring-in the [ 5,040 Base Structural Patterns ] and
  • select this [ Base Structural Pattern ] below as the [ Anchor Pattern ] for our Expansion Mapping Process , to follow .

    

• And we shall now select any one of 5,040 { [ 8-Colors Circular-Ring Patterns ] with the Red-Color in Fixed-Position [ FP-1 ] }
  • as the [ Core Pattern ] for our Expansion-Mapping Process .

• Let us now do the full Expansion-Mapping procedure as before ,
  • to arrive at a new set of 5,040 [ 8-colors circular-ring patterns ] :
    • all distinct-and-different from one-another ,
      • ( because the 5,040 [ Base Structural Patterns ] are distinct-and-different ) ; and

    • all will always have the [ red-color ] in Fixed-Position [ FP-1 ] .

  AND :

  • the new set of 5,040 members , must , necessarily by default , be the same as

    the { Reduced Set of 5,040 [ 8-Colors Circular-Ring Patterns ] } we originally started-out with .

    ( There simply is no other choice here ! ) .

• Thus , the { Reduced Set of 5,040 [ 8-Colors Circular-Ring Patterns ] } is indeed a [ Quasi 5,040-Symmetric Set ] .

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Recapping what we did in Part II / III / IV of this paper :

• Let us recap our course-of-actions in Part II / III / IV of this paper here :

  First :

  • We started-out with the { Reduced Set of 5,040 [ 8-Colors Circular-Ring Patterns ] } ,
    • and this is a [ Quasi 5,040-Symmetric Set ] .

  Second :

  • We split the above [ 5,040-members set ] into 210 x [ sets-of-24 ] ,
    • via the [ 24 Base Structural Patterns ] we have in Section III .

    And each of these [ sets-of-24 ] is a [ Quasi 24-Symmetric Set ] .

  Third :

  • We were able to identify the following within the above [ sets-of-24 ] :
    • [ Quasi Tri-Symmetric Sets ] ,
    • [ Quasi Quad-Symmetric Sets ] ,
    • [ Quasi Hexa-Symmetric Sets ] , and
    • [ Quasi Octo-Symmetric Sets ] .

• And what all this tell us is that :
  • There are specific [ Structural Mapping relations ] within the [ 5,040-members-set Framework ]
    • that will exhibit different types of [ quasi sub-symmetries ] .

  But , unfortunately :

  • our investigative work here in Part II / III / IV are not elaborative enough
    • for the fullest understanding of the said subject matter .

  And we must now bring-in the [ 8-Colors Cube ] , next , for our further studies .

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go to the next section : Section XVI - Setting up the [ 8-Colors Cube ]

go to the last section : Section XIV - Quasi Octo-Symmetry within the [ 24 QSM-Patterns ]

Original dated 2011-3-06