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

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

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Section XIII - Quasi Hexa-Symmetry within the [ 24 Base Structural Patterns ]

Summary for the Section :

• In this Section , we shall take a look at [ Quasi Hexa-Symmetry ] found within the [ 24 Base Structural Patterns ] .

  And we shall attempt to split the [ 24 QSM-Patterns --- ALPHA Series ] into :

  • 4 x [ sets of 6-elements Quasi Hexa-Symmetric Patterns ] .

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Recalling the [ 24 Base Structural Patterns ] :

• Let us now bring-in again the [ 24 Base Structural Patterns ] and also the [ 24 QSM-Patterns --- ALPHA Series ] ,
  • for re-familiarization purposes :

  

• And we simply note here that :
  • [ Pattern A ] on the diagram of-the-right above is the same as the pattern [ Pattern AARON ] :

    

  And we shall be using [ Patter AARON ] and [ Pattern A ] interchangeably in the remainder of this Section .

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Setting up the [ 6 Master Structural Patterns ] for [ Quasi Hexa-Symmetry ] :

• Let us now specifically select the following 6 of the [ 24 Base Structural Patterns ] as our [ 6 Master Structural Patterns ] :
  • the Structural Pattern in { Row 1 , Column 1 } ,
  • the Structural Pattern in { Row 4 , Column 1 } ,
  • the Structural Pattern in { Row 4 , Column 4 } ,
  • the Structural Pattern in { Row 2 , Column 1 } ,
  • the Structural Pattern in { Row 2 , Column 4 } ,
  • the Structural Pattern in { Row 1 , Column 4 } .

  

  And the [ 6 Master Structural Patterns ] for our [ Quasi Hexa-Symmetry ] Expansion-Mapping procedure is then :

  

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Selecting the [ Anchor Pattern ] for our Expansion-Mapping Procedure :

• Let us now select one of the [ 6 Master Structural Patterns ] as the [ Anchor Pattern ] :
  • and we shall select the [ Pattern on-the-left ] here , at-this-first-instance .

  

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Expansion-Mapping using [ Pattern AARON ] as the [ Core Pattern ] :

• Let us now plug-in [ Pattern AARON ] / [ Pattern A ] as the [ Core Pattern ] for our Expansion-Mapping procedure :

  

  And the [ Colors-to-Numbers Assigement 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

• We then have this as a result :

  

  And the 6 patterns in the diagram on-the-right above are then :

  • from-left-to-right , [ Pattern A ] , [ Pattern U ] , [ Pattern X ] , [ Pattern G ] , [ Pattern K ]and [ Pattern D ] respectively .

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Confirming [ Quasi Hexa-Symmetry ] :

• Let us now double-check , to make sure that the [ 6 Master Structural Patterns ] do yield [ Quasi Hexa-Symmetric Patterns ] .

  First , let us plug-in [ Pattern A ] as the [ Core Pattern ] to arrive at this :

  

  Secondly , let us plug-in [ Pattern U ] as the [ Core Pattern ] to arrive at this :

  

  Thirdly , let us plug-in [ Pattern X ] as the [ Core Pattern ] to arrive at this :

  

  Fourthly , let us plug-in [ Pattern G ] as the [ Core Pattern ] to arrive at this :

  

  Fifthly , let us plug-in [ Pattern K ] as the [ Core Pattern ] to arrive at this :

  

  Sixth-and-finally , let us plug-in [ Pattern D ] as the [ Core Pattern ] to arrive at this :

  

• And from these results , we can now conclude that :
  • [ Pattern A ] , [ Pattern U ] , [ Pattern X ] , [ Pattern G ] , [ Pattern K ] and [ Pattern D ] do form a [ Quasi Hexa-Symmetric Set ] .

  • the [ 6 Master Structural Patterns ] we have above is indeed a [ set of quasi hexa-symmetric Structural Patterns ] .

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Breaking-down the [ 24 QSM-Patterns ] into 4 [ sets of 6-elements Quasi Hexa-Symmetric Patterns ] :

• Let us now plug-in [ Patterns A / Z / R / S ] as the [ Core Pattern ] ,
  • to see what we might come up with .

  Let us now plug-in [ Pattern A ] as the [ Core Pattern ] and do the Expansion-Mapping Procedure :

  

  Let us now plug-in [ Pattern Z ] as the [ Core Pattern ] and do the Expansion-Mapping Procedure :

  

  Let us now plug-in [ Pattern R ] as the [ Core Pattern ] and do the Expansion-Mapping Procedure :

  

  Let us now plug-in [ Pattern S ] as the [ Core Pattern ] and do the Expansion-Mapping Procedure :

  

• And we see here that :
  • we were able to separate the [ 24 QSM-Patterns ] into 4 sets of [ Quasi Hexa-Symmetric Sets ] , with 6 elements in each set .

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Locating the 4 sets within the [ 24 QSM-Patterns ] framework :

• Let us now take a quick look at the location of each [ set of 6 elements ] within the [ 24 QSM-Patterns ] framework :

  

  

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go to the next section : Section XIV - Quasi Octo-Symmetry within the [ 24 Base Structural Patterns ]

go to the last section : Section XII - Quasi Quad-Symmetry within the [ 24 Base Structural Patterns ]

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

Original dated 2010-3-06