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

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

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Section II - Basic Combinatorics for [ 8-Colors Circular-Rings ]

Summary for the Section :

• In this Section , we shall go thru the preliminaries :
  • for the particular type of [ 8-Colors Circular-Rings ] that we shall be using throughout this paper ,

    i.e. :

    • this type of [ 8-colors circular-rings ] with 8 positions .
      • 

  And we shall also state-out here at the out-set :

  • the General Rule governing the [ color-structure ] of the [ 8-Colors Circular-Rings ] ; and

  • the Convention that we shall be using for reading-off :
    • the [ 8-colors color-sequence ] permutations , and

    • the [ 8-colors circular-sequence ] combinations .

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Identifying the 8 Colors :

• Let us first list-out the 8 colors that we shall be using throughout this paper , and these are the colors :
  • [ Red ] , [ Cyan ] , [ Purple ] , [ Magenta ] , [ Green ] , [ Yellow ] , [ Orange ] , and [ Blue ] .

  ***********	***********	***********	***********	***********	***********	***********	***********
  RED	CYAN	PURPLE	MAGENTA	GREEN	YELLOW	ORANGE	BLUE
  ***********	***********	***********	***********	***********	***********	***********	***********

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The General Rule governing the [ Color-Structure ] of [ 8-Colors Circular-Rings ] :

• Let us now state the General Rule governing the [ color-structure ] of the [ 8-Colors Circular-Rings ] , as follows :
  • For any particular [ 8-Color Circular-Ring ] with 8 positions :
    • each color may appear once-and-only-once in the said Circular-Ring ;

    so that :

    • each-and-every [ 8-Colors Circular-Ring ] is always made-up-of exactly eight (8) colors .

  And this restriction here shall be applicable to all [ 8-Colors Circular-Rings ] throughout the entire paper .

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The Read-Off Convention for the [ 8-Colors Color-Sequences ] :

• Let us now take a quick look at a sample [ 8-Colors Circular-Ring ] ,
  • as per the diagram on-the-left below :

    

  We can then mark-off the 8 fixed-positions within the [ 8-Colors Circular-Ring ] as :

  • the Fixed-Positions [ FP-1 ] , [ FP-2 ] , [ FP-3 ] , [ FP-4 ] , [ FP-5 ] , [ FP-6 ] , [ FP-7 ] , and [ FP-8 ] respectively ;
    • as per the diagram on-the-right above .

• We then have this Standard Convention for reading-off the [ 8-colors color-sequence ] permutations :
  • First :
    • we shall always read-off the color-sequence on any [ 8-Colors Circular-Ring ] in a counter-clockwise fashion ;
      • as per the diagram on-the-right above .

  • Secondly :
    • we shall always start the read-off with the color in Fixed-Position [ FP-1 ] ;
      • and then move-on in a counter-clockwise fashion :
        • thru [ FP-2 ] / [ FP-3 ] / [ FP-4 ] / [ FP-5 ] / [ FP-6 ] / [ FP- 7 ] respectively ;

      and always end with the color in Fixed-Position [ FP-8 ] as the final-and-closing color of the sequence .

• Let us do a quick example and bring-back the [ 8-Colors Circular-Ring ] we have above , i.e. :
  • 

  And the [ 8-colors color-sequence ] permutation here , according the afore-said Standard Convention , is then :

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

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The 40,320 Permutations :

• Thus , the Total Number of Permutations for the [ 8-colors color-sequences ] is then [ 40,320 ] , i.e. :
  • [ 40,320 ] = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

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The 5,040 [ 8-Colors Circular-Sequences ] Combinations :

• Since we shall be working primarily with [ 8-Colors Circular-Sequences ] in this paper :
  • we take note , at this point , that the following 8 [ 8-Colors Circular-Rings ] :

    
    

    • being [ rotated images ] of one-another ;

    do indeed share the same [ 8-colors circular-sequence ] .

  And that-is-to-say that :

  • the 8 [ 8-colors color-sequences ] corresponding to the 8 [ 8-Colors Circular-Rings ] above ,
    • i.e. , the [ 8-colors color-sequences ] :
      • [ Red-Cyan-Purple-Magenta-Green-Yellow-Orange-Blue ] ,

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

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

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

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

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

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

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

    are always considered to be the same [ 8-colors circular-sequence ] .

• Thus , each-and-every [ 8-colors circular-sequence ] will take-into-account / take-command-of :
  • exactly 8 [ 8-colors color-sequence ] permutations ;
    • as we have just demonstrated above .

  And as such :

  • the Total Number of Combinations of distinct-and-distinguishable [ 8-colors circular-sequences ] is then [ 5,040 ] ;
    • i.e. :
      • [ 5,040 ] = [ 40,320 ] divided by [ 8 ] .

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The Read-Off Convention for the [ 8-Colors Circular-Sequences ] :

• And for the sake of consistency and clarity :
  • it is necessary for us to use the same Convention for reading-off the [ 8-colors circular-sequences ] , throughout the entire paper .

  And the standard [ Read-Off Convention ] here for reading-off the [ 8-colors circular-sequences ] is as follows :

  • First ,
    • we shall always start the read-off with the [ Red-Color ] ;

  • Secondly ,
    • once we have identified the position of the [ Red-Color ] within the [ 8-Colors Circular-Ring ] :
      • whether it be the fixed-position [ FP-1 ] , [ FP-2 ] , [ FP-3 ] , [ FP-4 ] , [ FP-5 ] , [ FP-6 ] , [ FP-7 ] or [ FP-8 ] ;

      we shall then move-on in a counter-clockwise fashion to read-off remaining colors in a sequential manner ,

      • until the full count of 8 colors has been accounted for .

• And that-is-to-say that :
  • Under this Convention ,
    • each-and-every one of the 5,040 [ 8-colors circular-sequences ] shall always start with the [ Red-Color ]
      • as the first-color of the sequence .

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Setting up the Reduced Set of 5,040 [ 8-Colors Circular-Ring Patterns ] :

• Let us now bring-in the full global set of 40,320 [ 8-colors circular-ring patterns ] and select therefrom :
  • a sub-set of [ 8-colors circular-ring patterns ] satisfying this particular [ selection criterion ] ,

    namely that :

    • each-and-every one of the selected [ 8-Colors Circular-Ring Patterns ]

      must carry the [ red-color ] in Fixed-Position [ FP-1 ] .

  There are therefore exactly 5,040 distinct-and-distinguishable [ 8-Colors Circular-Ring Patterns ] in this sub-set ,

  • i.e : [ 5,040 ] = 1 x 7 x 6 x 5 x 4 x 3 x 2 x 1

    ( Fixed-Position [ FP-1 ] will always spot the [ red-color ] but the colors in the remaining 7 Fixed-Positions may vary . )

  And we shall name this sub-set :

  • the { Reduced Set of 5,040 [ 8-Colors Circular-Ring Patterns ] }
    • for identification and for our easy-referrals in later Sections .

  And a few quick examples here :

  

• And we take special note , at this point , that :
  • This { Reduced Set of 5,040 [ 8-Colors Circular-Ring Patterns ] } here

    and

    the set of 5,040 [ 8-colors circular-sequences ] we have above

    • do exhibit a [ one-to-one correspondence ] relation with one-another .

  And the explanation for this is as follows :

  • First :
    • For each of the { Reduced Set of 5,040 [ 8-colors Circular-Ring Patterns } ,
      • we can always derive an exact [ 8-colors color-sequence ] associated with it ,
        • via using the standard [ Read-Off Convention ] we have above for reading-off the [ 8-colors color-sequences ] .

      And each of these 5,040 derived [ 8-colors color-sequences ] is unique-and-distinct ,

      • because the parent 5,040 [ 8-colors Circular-Ring Patterns ] are distinct-and-distinguishable .

      Also , each of these 5,040 derived [ 8-colors color-sequences ]

      • will always spot the [ red-color ] as the 1st-color of the sequence .

  • Secondly :
    • For each of the { Reduced Set of 5,040 [ 8-colors Circular-Ring Patterns } ,
      • we can also derive an exact [ 8-colors circular-sequence ] associated with it ,
        • via using the standard [ Read-Off Convention ] we have above for reading-off the [ 8-colors circular-sequences ] .

      And each of these 5,040 derived [ 8-colors circular-sequences ] :

      • will always spot the [ red-color ] as the 1st-color of the sequence .

  • As such :
    • the [ 8-colors circular-sequence ] and the [ 8-colors color-sequence ]
      • derived from the same { 8-Colors Circular-Ring Pattern with the [ red-color ] in Fixed-Position [ FP-1 ] }

      will always match exactly ,

      • since both read-off's will start-off at the same color-spot on the Circular-Ring ,
        • i.e. : at Fixed-Position [ FP-1 ] spotting the [ red-color ] .

    And :

    • Since we have already established above that :
      • the derived 5,040 [ 8-colors color-sequences ] are unique-and-distinct ;

      it then follows that :

      • the matching 5,040 [ 8-colors circular-sequences ] are also unique-and-distinct .

  • Let us recall , at this point , that :
    • the original set of 5,040 [ 8-colors circular-sequences ] is in-itself-and-by-itself an exhaustive set ,
      • covering all [ 8-colors circular-sequences ] arising from the global set of 40,320 permutations .

    It then follows that :

    • the 5,040 unique-and-distinct [ 8-colors circular-sequences ] arising from
      • the { Reduced Set of 5,040 [ 8-Colors Circular-Ring Patterns ] }

      must match the origial set of 5,040 [ 8-colors circular-sequences ] on a [ one-to-one ] basis .

      ( There is simply no other choice here . )

  • This then establishes the [ one-to-one correspondence ] we needed
    • for :
      • the { Reduced Set of 5,040 [ 8-Colors Circular-Rings Patterns ] }

      and

      • the original set of 5,040 [ 8-colors circular-sequences ] .

• And we shall be using primarily this { Reduced Set of 5,040 [ 8-Colors Circular-Ring Patterns ] }
  • as the core-and-basic set for our discusssions and analysis in Part II and Part III of this paper .

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go to the next section : Section III - Setting up the [ 24 Base Structural Patterns ]

go to the last section : Section I - Introduction

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

original dated 2011-3-06