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 XXI - Creating and Segregating 256 Sets of { 8-D Orthogonal Unit-Vector Basis }

Summary for the Section :

• In this Section , we shall create 256 sets of { 8-D orthogonal unit-vector basis }
  • and segregate these into two (2) groups :
    • the [ ODD-ODD Series ] containing 128 sets ,

      and

    • the [ EVEN-EVEN Series ] containing the remaining 128 sets .

  We take note here that :

  • the [ 8 x 8 matrix ] associated with an { 8-D orthogonal unit-vector basis }
    • can be used as a { 8 x 8 mapping matrix } for mapping one { 8-D Vector } onto another { 8-D Vector } .

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Characteristics for the 256 sets of { 8-D Orthogonal Unit-Vector Basis } :

• Let us state-out , at this first instance :
  • the characteristics of the 256 sets of { 8-D Orthogonal Unit-Vector Basis } we shall be creating .

  The primary-and-overriding characteristics for the { 8-D orthogonal unit-vector bases } we intend to create are then as follows :

  • For each-and-every set of { 8-D orthogonal unit-vector basis } with [ 8 unit-vectors ] :
    • the { 1st Unit-Vector } shall host :
      • either a [ +1 ] or a [ -1 ] in its [ 1st co-ordinate position ]

        and

      • all the remaining 7 co-ordinates shall be [ 0 ]'s .

    • the { 2nd Unit-Vector } shall host :
      • either a [ +1 ] or a [ -1 ] in its [ 2nd co-ordinate position ]

        and

      • all the remaining 7 co-ordinates shall be [ 0 ]'s .

    • the { 3rd Unit-Vector } shall host :
      • either a [ +1 ] or a [ -1 ] in its [ 3rd co-ordinate position ]

        and

      • all the remaining 7 co-ordinates shall be [ 0 ]'s .

    • the { 4th Unit-Vector } shall host :
      • either a [ +1 ] or a [ -1 ] in its [ 4th co-ordinate position ]

        and

      • all the remaining 7 co-ordinates shall be [ 0 ]'s .

    • the { 5th Unit-Vector } shall host :
      • either a [ +1 ] or a [ -1 ] in its [ 5th co-ordinate position ]

        and

      • all the remaining 7 co-ordinates shall be [ 0 ]'s .

    • the { 6th Unit-Vector } shall host :
      • either a [ +1 ] or a [ -1 ] in its [ 6th co-ordinate position ]

        and

      • and all the remaining 7 co-ordinates shall be [ 0 ]'s .

    • the { 7th Unit-Vector } shall host :
      • either a [ +1 ] or a [ -1 ] in its [ 7th co-ordinate position ]

        and

      • all the remaining 7 co-ordinates shall be [ 0 ]'s .

    • the { 8th Unit-Vector } shall host :
      • either a [ +1 ] or a [ -1 ] in its [ 8th co-ordinate position ]

        and

      • all the remaining 7 co-ordinates shall be [ 0 ]'s .

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Creating the 256 sets of { 8-D Orthogonal Unit-Vector Basis } :

• Since we have a choice of 2 alternatives for each of the { 8 unit vectors } ,
  • i.e. : the choice of either a [ +1 ] or a [ -1 ] ;

  the total number of sets of { 8-D orthogonal unit-vector basis } we shall be creating is then [ 256 ] ;

  • i.e. : [ 256 ] = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 .

• Let us now create the 256 sets of { 8-D orthogonal unit-vector basis } as required .

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{ [ +1 ]-based unit-vector } vs. { [ -1 ]-based unit-vector } :

• For our discussions to follow immediately , we shall simply :
  • refer to a unit-vector with { a [ +1 ] and 7 [ 0] 's } in its [ 8 co-ordinates ] as :
    • a { [ +1 ]-based unit-vector } ;

  • refer to a unit-vector with { a [ -1 ] and 7 [ 0 ]'s } in its [ 8 co-ordinates ] as :
    • a { [ -1 ]-based unit-vector } .

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Segregating the 256 sets of { 8-D orthogonal unit-vector basis } :

• Let us now segregate the 256 sets of { 8-D orthogonal unit-vector basis } into two (2) groups :
  • the { ODD-ODD Series } containing 128 sets ,

    and

  • the { EVEN-EVEN series } containing the remaining 128 sets .

• For the { ODD-ODD Series } ,
  • each set of { 8-D orthogonal unit-vector basis } shall contain :
    • an odd number of { [ +1 ]-based unit-vectors } and correspondingly an odd number of { [ -1 ]-based unit-vectors } ;
  '

• For the { EVEN-EVEN Series } ,
  • each set of { 8-D orthogonal unit-vector basis } shall contain :
    • an even number of { [ +1 ]-based unit-vectors } and correspondingly an even number of { [ -1 ]-based unit-vectors } ;

      noting in particular that :

      • { Orthogonal Unit-Vector SET ZERO } with 8 { [ +1 ]-based unit-vectors }

        and

      • { Orthogonal Unit-Vector SET RICO } with 8 { [ -1 ]-based unit-vectors }
        • are also included in the { EVEN-EVEN Series } to make up the 128 sets .

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Using the { 8 x 8 matrix } as a Mapping Matrix :

• We simply take note here that :
  • the [ 8 x 8 matrix ] associated with a set of { 8-D Orthogonal Unit-Vector Basis }
    • can also be used for mapping [ one vector ] onto [ another vector ] .

  But it is necessary for us to recognize here that :

  • the [ 8 x 8 martix ] associated with any of our 256 sets of { 8-D orthogonal unit-vector basis } above
    • is a { symmetric matrix } with :
      • the diagonal elements hosting only either a [ +1 ] or a [ -1 ]

        and

      • all other elements in the [ 8 x 8 matrix ] being [ 0 ] 's .

  As such , for this special type of [ 8 x 8 matrix ] :

  • the [ inverse ] is its [ transpose ] and is therefore one-and-the-same { 8 x 8 matrix } .

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What's Next :

• With the [ ODD-ODD Series ] and [ EVEN-EVEN Series ] now in-hand ,
  • we are ready to tackle the orientation problem of { Structure A } vs. { Structure B } , next .

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go to the next Section : Section XXII - Orientation Symmetry for { Structure A } vs. { Structure B }

go to the last Section : Section XX - Why { Structure A } and { Structure B } are Identical In Shape

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Original dated 2017-7-20