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

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

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Section XXIV - Physical [ Tri-Symmetric Sets ] within the [ 24 QSM-Configurations ]

Summary for the Section :

• In this Section , we shall be looking into the 8 [ Tri-Symmetric Sets ] ,
  • found within the [ 24 QSM-Configurations --- Himalayan Series ] .

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Bringing-back the [ 24 QSM-Configurations --- Himalayan Series ] :

• Let us first bring-in again the [ 24 QSM-Configurations --- Himalayan Series ] :

  

  • for re-familiarization purposes .

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Identifying the Axis-of-Rotation for Tri-Symmetry :

• Let us now identify the axis-of-rotation that we shall be using for our rotational-symmetries below .

  And this axis-of-rotation is :

  • the axis running thru this particular pair of [ opposite corners ] of the [ cube ] , namely :
    • the [ 1, 1, 1 ] corner and the [ -1 , -1, -1 ] corner respectively ;

      as per the diagram on-the-left , below .

  

• And when we rotate the cube using the above-said axis as the axis-of-rotation ,
  • thru the angles of [ 120 degrees ] / [ 240 degrees ] / [ 360 degrees ] , as the case may be ;

  the following happens :

  • the [ ( 1, -1, 1 ) corner ] , the [ ( 1, 1, -1 ) corner ] and the [ ( -1, 1, 1 ) corner ]
    • exchange positions in a rotary fashion ;

  • the [ ( -1, 1, -1 ) corner ] , the [ ( -1, -1, 1 ) corner ] and the [ ( 1, -1, -1 ) corner ]
    • exchange positions in a rotary fashion ;

  • the [ ( 1, 1, 1 ) corner ] and the [ ( -1, -1, -1 ) corner ]
    • remain position-wise in their respective corners .

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Identifying the 8 [ Tri-Symmetric Sets ] :

• Let us now identify the 8 sets of [ Tri-Symmetric Configurations ] with 3 elements each .
  • For the 1st set , we have Configurations [ A ] , [ B ] and [ C ] :

    

    • And this set is indeed physically [ Tri-Symmetric ] .

  • For the 2nd set , we have Configurations [ D ] , [ E ] and [ F ] :

    

    • And this set is indeed physically [ Tri-Symmetric ] .

  • For the 3rd set , we have Configurations [ G ] , [ H ] and [ J ] :

    

    • And this set is indeed physically [ Tri-Symmetric ] .

  • For the 4th set , we have Configurations [ K ] , [ L ] and [ M ] :

    

    • And this set is indeed physically [ Tri-Symmetric ] .

  • For the 5th set , we have Configurations [ N ] , [ P ] and [ Q ] :

    

    • And this set is indeed physically [ Tri-Symmetric ] .

  • For the 6th set , we have Configurations [ R ] , [ S ] and [ T ] :

    

    • And this set is indeed physically [ Tri-Symmetric ] .

  • For the 7th set , we have Configurations [ U ] , [ V ] and [ W ] :

    

    • And this set is indeed physically [ Tri-Symmetric ] .

  • For the 8th-and-final set , we have Configurations [ X ] , [ Y ] and [ Z ] :

    

    • And this set is indeed physically [ Tri-Symmetric ] .

• Let us now summaries the above 8 physical [ Tri-Symmetric Sets ] of 3 elements each ,
  • via the diagram on-the-left below :

  

  • for the reader's perusal and further thoughts .

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A Quick Comparison here :

• Let us also bring-in , from Section XI , the diagram on-the-left , below :
  • arising from the 8 [ Quasi Tri-Symmetric Sets ] found within the [ 24 Base Stuctural Patterns ] .

  

  Yes , quite a bit of similarity here !

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go to the next section : Section XXV - The Linkage Diagram for the [ 24 QSM-Configurations ]

go to the last section : Section XXIII - Understanding Physical [ Quasi 24-Symmetry ] - Phase III

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

Original dated 2011-3-06