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 XII - Setting up for an Analysis on Tri-Symmetry

Summary for the Section :

• In this Section , we shall do the preparation work :
  • for our analysis on Tri-Symmetry in the next 2 Sections .

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Switching to Vectors :

• Our { Analysis on Tri-Symmetry } will be based on { Dot Products of Vectors } , and as such :
  • it is necessary for us to switch to vectors .

  Let us now set up the six (6) sets of vectors :

  • { Set A of 4 Vectors } , { Set B of 4 Vectors } and { Set C of 4 Vectors } ,

    and

  • { Set D of 4 Vectors } , { Set E of 4 Vectors } and { Set F of 4 Vectors } ;
    • corresponding to the { 1st Position Vectors } for each of the 6 sets of { 4 lines mutually orthogonal to one-another }
      • from the last Section .

  
  
  

  Special Note :

  • { Vector UV-1 } , { Vector UV-2 } , { Vector UV-3 } and { Vector UV-4 } are
    • the 4 unit-vectors in the 4 Axial-Directions defining the { 4-Dimension Vector Space } ,
      • as previously set up in Section VIII .

  And we shall refer to this { orthogonal unit-vector basis } set as { Set UV } in our discussions to follow .

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The Approach for our Analysis on Tri-Symmetry to follow :

• What we intend to do for { Set A / B / C } in the next Section is that :
  • First ,
    • we shall take the { 4 orthogonal Line-Directions for Set A } as our { 4 temporary Axial-Directions } for the time-being :
      • to see how { Set B } and { Set C } will be oriented with respect to this { 4 temporary Axial-Directions } set .

  • Secondly ,
    • we shall take the { 4 orthogonal Line-Directions for Set B } as our { 4 temporary Axial-Directions } for the time-being :
      • to see how { Set C } and { Set A } will be oriented with respect to this { 4 temporary Axial-Directions } set .

  • Thirdly ,
    • we shall take the { 4 orthogonal Line-Directions for Set C } as our { 4 temporary Axial-Directions } for the time-being :
      • to see how { Set A } and { Set B } will be oriented with respect to this { 4 temporary Axial-Directions } set .

• And we shall do the same for { Set D / E / F } in the Section to follow thereafter .

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Setting up the Temporary Orthogonal Basis :

• We then set up , for { Set A / B / C } :
  • the Orthogonal Unit-Vector Basis { Set X } with the mapping matrix on { Set UV } being in replica of { Set A } ,
  • the Orthogonal Unit-Vector Basis { Set Y } with the mapping matrix on { Set UV } being in replica of { Set B } ,
  • the Orthogonal Unit-Vector Basis { Set Z } with the mapping matrix on { Set UV } being in replica of { Set C } ;

    as shown on-the-left , below :

  
  
  

  • And the reverse mapping relations are then as shown on-the-right , above .

  We take note here that :

  • for a { 4 x 4 matrix } mapping one set of { orthogonal unit-vector basis } onto another set of { orthogonal unit-vector basis } ,
    • the [ inverse ] of the said { 4 x 4 matrix } is its [ transpose ] ;

    i.e.:

    • for { Set X } vs. { Set UV } , we have :

      

    • for { Set Y } vs. { Set UV } , we have :

      

    • for { Set Z } vs. { Set UV } , we have :

      

• We then set up , for { Set D / E / F } :
  • the Orthogonal Unit-Vector Basis { Set P } with the mapping matrix on { Set UV } being in replica of { Set D } ,
  • the Orthogonal Unit-Vector Basis { Set Q } with the mapping matrix on { Set UV } being in replica of { Set E } ,
  • the Orthogonal Unit-Vector Basis { Set R } with the mapping matrix on { Set UV } being in replica of { Set F } ;

    as shown on-the-left , below :

  
  
  

  • And the reverse mapping relations are then as shown on-the-right , above .

  We take note here again that :

  • for a { 4 x 4 matrix } mapping one set of { orthogonal unit-vector basis } onto another set of { orthogonal unit-vector basis } ,
    • the [ inverse ] of the said { 4 x 4 matrix } is its [ transpose ] ;

    i.e.:

    • for { Set P } vs. { Set UV } , we have :

      

    • for { Set Q } vs. { Set UV } , we have :

      

    • for { Set R } vs. { Set UV } , we have :

      

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What's next :

• We shall now do the Tri-Symmetry Analysis for { Set A / B / C } , next .

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

go to the last Section : Section XI - Reconciling the { 12 Lines vs. 8 Lines } Difference

return to the Symmetry Through-put HomePage

Original dated 2017-7-23