On Symmetry Through-put linking 3-4-5-6-7 Dimension's ,
Octo-Symmetry for the { 4-Dimension Quasi-Octahedron } ,
and 2 Identical { 8-D Structures } oriented Symmetrically

by Frank C. Fung ( 1st published in July, 2017. )

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Section XIV - The Tri-Symmetry Analysis for { Set D / E / F }

Summary for the Section :

• In this Section , we shall do the Tri-Symmetry Analysis for { Set D / E / F } .

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Recalling { Set D / E / F } :

• Let us first recall { Set D } , { Set E } and { Set F } :

  

  

  

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Using { Set P } as the Temporary { 4 Axial-Directions } :

• Let us now bring back this set of relations for { Set P } :

  

  Applying the relation on-the-right above to { Set D / E / F } , we have :

  

  

  

  yielding :

  

  

  

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Using { Set Q } as the Temporary { 4 Axial-Directions } :

• Let us now bring back this set of relations for { Set Q } :

  

  Applying the relation on-the-right above to { Set D / E / F } , we have :

  

  

  

  yielding :

  

  

  

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Using { Set R } as the Temporary { 4 Axial-Directions } :

• Let us now bring back this set of relations for { Set R } :

  

  Applying the relation on-the-right above to { Set D / E / F } , we have :

  

  

  

  yielding :

  

  

  

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Summarizing :

• Let us now summarize our findings here :
  • via this table below :

    Temporary 4 Axial-Directions Set used for evaluating the Relative Positions	* * *	Relative Position for			* * *
    		the original Fixed- Position Set D	the original Fixed- Position Set E	the original Fixed- Position Set F
    *****************************	*	******	******	******	*
    4 Axial-Directions of { Set P }	*	Type C matrix	Type A matrix	Type B matrix	*
    4 Axial-Directions of { Set Q }	*	Type A matrix	Type C matrix	Type B matrix	*
    4 Axial-Directions of { Set R }	*	Type A matrix	Type B matrix	Type C matrix	*

• Let us now make these statements :
  • Statement One :
    • When we switch from { Set Q } to { Set R } as the { Temporary 4 Axial-Directions Set } or vice-versa :
      • the { Type A matrix } does not move while the other two (2) matrices exchange positions .

  • Statement Two :
    • When we switch from { Set Q } to { Set P } as the { Temporary 4 Axial-Directions Set } or vice-versa :
      • the { Type B matrix } does not move while the other two (2) matrices exchange positions .

  • Statement Three :
    • When switch from { Set P } to { Set R } as the { Temporary 4 Axial-Directions Set } or vice-versa :
      • everybody moves in a { rotation-type order } .

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The Nagging Question here :

• There is a nagging question here , namely that :
  • Had we used a different order for the 4 lines in the { Temporary 4 Axial-Directions Set } ,
    • or for that matter ,
      • randomly used the { 2nd Position Vector } instead of the { 1st Position Vector } to create the 6 sets :
        • { Set A of 4 Vectors } , { Set B of 4 Vectors } , { Set C of 4 Vectors } ,

          and

        • { Set D of 4 Vectors } , { Set E of 4 Vectors } and { Set F of 4 Vectors } ;

    would our results here and in the last Section be different ?

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What's Next :

• It is necessary for us to look at this question closely , next .

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go to the next Section : Section XV - A Special Issue on Swapping { 4 Orthogonal-Lines Structures }

go to the last Section : Section XIII - The Tri-Symmetry Analysis for { Set A / B / C }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23