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 XXII - Orientation Symmetry for { Structure A } vs. { Structure B }

Summary for the Section :

• In this Section , we shall demonstrate :
  • the nature of the orientation symmetry for { Structure A } vs. { Structure B } .

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Re-Creating the { Structure Fixed-In-Space } :

• Let us bring-in again the 8 unit-vectors defining the { 8-Dimension Vector Space } from Section XIX ,

  i.e. , the 8 [ physical unit-vectors ] :

  • { Vector PUV-1 } , { Vector PUV-2 } , { Vector PUV-3 } , { Vector PUV-4 } ,
    { Vector PUV-5 } , { Vector PUV-6 } , { Vector PUV-7 } , and { Vector PUV-8 } .

• Let us now again construct { Structure A } using :
  • the { 8-D Orthogonal Unit-Vector Basis SET ZERO }

    and

  • the co-ordinates of the 128 Position Vectors for the 64 Lines in :
    • { Class 8-0 / 0-8 } , { Class 6-2 / 2-6 } and { Class 4-4 / 4-4 } .

• And we shall now again treat this structure as being fixed-and-invariant in the { 8-Dimension Vector Space } ,
  • and again name this the { 64-Lines Structure now-fixed-in-space } .

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Interpreting using different { 8-D orthogaonal unit-vector basis } :

• Let us now attempt to re-interpret the co-ordinates for the { 64-Lines Structure now-fixed-in-space } ,
  • using any one of the 128 { 8-D Orthogonal Unit-Vector Basis in the ODD-ODD Series } .

  Interesting enough , the [ 8 co-ordinates ] for 128 Position Vectors for the { 64-Lines Structure now-fixed-in-space }

  • will now fall into the { Class 7-1 / 1-7 } / { Class 5-3 / 3-5 } categories ;
    • by the same reasoning as before in Section XX ,

      i.e. :

      • for { either 1 , 3 , 5 or 7 of the co-ordinate position/s } of the [ 8 co-ordinate positions ] of the { 128 Position Vectors } here ,
        • the co-ordinate will either :
          • change from a [ +1 ] to a [ -1 ] ,

            or alternately

          • change from a [ -1 ] to a [ +1 ] ;

          thereby :

          • making the original Position Vector in { Class 8-0 / 0-8 } or { Class 6-2 / 2-6 } or { Class 4-4 / 4-4}
            • to fall into the classification of either { class 7-1 / 1-7 } or { Class 5-3 / 3-5 } ,
              • after the change .

• Let us also attempt to re-interpret the co-ordinates for the { 64-Lines Structure now-fixed-in-space } ,
  • using any one of the 128 { 8-D Orthogonal Unit-Vector Basis in the EVEN-EVEN Series } .

  And in this case , the [ 8 co-ordinates ] for 128 Position Vectors for the { 64-Lines Structure now-fixed-in-space }

  • will now fall into the { Class 8-0 / 0-8 } / { Class 6-2 / 2-6 } / { Class 4-4 / 4-4 } categories ;
    • again by the same type of reasoning as before in Section XX

      i.e. :

      • for { either 0 , 2 , 4 , 6 or 8 of the co-ordinate position/s } of the [ 8 co-ordinate positions ] of the { 128 Position Vectors } here ,
        • the co-ordinate will either :
          • change from a [ +1 ] to a [ -1 ] ,

            or alternately

          • change from a [ -1 ] to a [ +1 ] ;

          thereby :

          • making the original Position Vector in { Class 8-0 / 0-8 } or { Class 6-2 / 2-6 } or { Class 4-4 / 4-4 }
            • to fall into the classification of either { Class 8-0 / 0-8 } , { Class 6-2 / 2-6 } or { Class 4-4 / 4-4 } ,
              • after the change .

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An Alternate Approach to creating the { 64-Lines Structure now-fixed-in-space } :

• Let us now create an alternate { 64-Lines Structure now-fixed-in-space } using instead :
  • the { 8-D Orthogonal Unit-Vector Basis SET ZERO }

    and

  • the co-ordinates of the 128 Position Vectors for the 64 Lines in :
    • { Class 7-1 / 1-7 } and { Class 5-3 / 3-5 } .

  And this is { Structure B } .

• Under such circumstances ,
  • When we attempt to re-interpret the co-ordinates for the new { 64-Lines Structure now-fixed-in-space } ,
    • using any one of the 128 { 8-D Orthogonal Unit-Vector Basis in the ODD-ODD Series } ;
      • the [ 8 co-ordinates ] for 128 Position Vectors for the new { 64-Lines Structure now-fixed-in-space }
        • will now fall into the { Class 8-0 / 0-8 } / { Class 6-2 / 2-6 } / { Class 4-4 / 4-4 } categories ;
          • by the same type of reasoning as above .

  • When we attempt to re-interpret the co-ordinates for the new { 64-Lines Structure now-fixed-in-space } ,
    • using any one of the 128 { 8-D Orthogonal Unit-Vector Basis in the EVEN-EVEN series } ;
      • the [ 8 co-ordinates ] for 128 Position Vectors for the new { 64-Lines Structure now-fixed-in-space }
        • will now fall into the { Class 7-1 / 1-7 } / { Class 5-3 / 3-5 } categories ;
          • by the same type of reasoning as above .

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Summarizing on Orientation :

• In the 2 operations above , we did create two (2) structures :
  • { Structure A } via the use of :
    • { 8-D Orthogonal Unit-Vector Basis SET ZERO }

      and

    • the co-ordinates of the 128 Position Vectors for the 64 Lines in { Class 8-0 / 0-8 } , { Class 6-2 / 2-6 } and { Class 4-4 / 4-4 } ;

  • { Structure B } via the use of :
    • { 8-D Orthogonal Unit-Vector Basis SET ZERO }

      and

    • the co-ordinates of the 128 Position Vectors for the 64 Lines in { Class 7-1 / 1-7 } and { Class 5-3 / 3-5 } .

  And :

  • When we use any one of the 128 { 8-D Orthogonal Unit-Vectors Basis in the ODD-ODD series } to interpret the 2 Structures ,
    • the 2 Structures swap positions ;

      i.e. :

      • { Structure A } is now { Structure B } and { Structure B } is now { Structure A } ;

  • When we use any one of the 128 { 8-D Orthogonal Unit-Vectors Basis in the EVEN-EVEN series } to interpret the 2 Structures ,
    • the 2 Structures retained their positions ;

      i.e. :

      • { Structure A } is still { Structure A } and { Structure B } is still { Structure B } .

• From a orientation stand-point ,
  • the 2 Structures are considered to be oriented symmetrically ;

  because :

  • we can always start with { Structure A } and obtain { Structure B } via :
    • re-interpreting { Structure A } using any one of the 128 { 8-D Orthogonal Unit-Vector Basis in the ODD-ODD Series } ;

    and

  • we can also start with { Structure B } and obtain { Structure A } via :
    • re-interpreting { Structure B } using any one of the 128 { 8-D Orthogonal Unit-Vector Basis in the ODD-ODD Series } .

  But we should be aware at the same time that :

  • both { Structure A } and { Structure B } may be considered :
    • 64-symnnetric on a Line-On-Line bases ,

      or for that matter :

    • 128-symmetric on a Vector-On-Vector bases .

  So , the orientation symmetry here can be multiplex in nature .

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The Swapping Mechanism for swapping 2 identical Structures :

• The Swapping Mechanism we have just demonstrated above is then :
  • the same type of Swapping Mechanism for Swapping 2 { 4 Orthogonal-Lines Structures } in 4-Dimension ,
    • as discussed back in Section XV .

  A quick reminder here :

  • One should be aware that :
    • this type of Swapping 2 identical Structures is also in-full-force ,
      • when we split the 32 Cross-Diagonal Lines of { 6-Dimension Quasi-Cube }
        • into 2 identical { 16 Lines Structures } ;

        later in Section XXIV , in Part V of this paper .

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What's Next :

• We shall look at the mechanics for the use of [ 8 x 8 Mappig Matrices ] ,
  • on { Structure A } and { Structure B } , next .

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go to the next Section : Section XXIII - Using [ 8 x 8 Mapping Matrices ] on { Structure A } and { Structure B }

go to the last Section : Section XXI - Creating and Segregating 256 Sets of { 8-D Orthogonal Unit-Vector Basis }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23