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 XX - Why { Structure A } and { Structure B } are Identical In Shape

Summary for the Section :

• In this Section , we shall demonstrate why { Structure A } and { Structure B } are identical in shape ,
  • via this procedure below :
    • FIRST :
      • we set up { Structure A } using { Orthogonal Unit-Vector Basis SET ZERO } ,

    • THEN :
      • we re-interpret the co-ordinates for the now-standing { Structure A }
        • using { Orthogonal Unit-Vector basis SET ALPHA } .

  The results are interesting .

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Recalling the Split for { Structure A } vs. { Structure B } :

• Let us recall that :
  • we split up the 128 Cross-Diagonal Lines of the { 8-Dimension Quasi-Cube }
    • into the 64-Lines { Structure A } and the 64-Lines { Structure B } ;

    Classification	Number of Lines in Class	* * *	Structure A	Structure B
    **************	**************	*	**************	**************
    Class 8-0 / 0-8	1	*	1
    Class 7-1 / 1-7	8	*		8
    Class 6-2 / 2-6	28	*	28
    Class 5-4 / 3-5	56	*		56
    Class 4-4 / 4-4	35	*	35
    **************	**************	*	**************	**************
    Total	128	*	64	64

    so that :

    • for the 128 Position Vectors associated with the 64 Lines in { Structure A } :
      • the [ 8 co-ordinates ] of the 2 Position Vectors for [ Line ZERO in { Class 8-0 / 0-8 } ] will host :
        • { 8 [ +1 ]'s } and { 8 [ -1 ]'s } respectively ;

      • the [ 8 co-ordinates ] of the Position Vectors for the remaining 63 Lines will host :
        • an even number of [ +1 ]'s and a corresponding even number of [ -1 ]'s ;

    • for the 128 Position Vectors associated with the 64 Lines in { Structure B } :
      • the [ 8 co-ordinates ] of the Position Vectors will host :
        • an odd number of [ +1 ]'s and a corresponding odd number of [ -1 ]'s .

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Setting up { Structure A } using { SET ZERO } :

• Let us bring in the { orthogonal unit-vector basis } :
  • { Orthogonal Unit-Vector Basis SET ZERO } .

    Details for the Orthogonal Unit-Vector Basis { SET ZERO }
    Vector I.D.	* *	Co-ordinates based on the Physical Unit Vector
    		PUV-1	PUV-2	PUV-3	PUV-4	PUV-5	PUV-6	PUV-7	PUV-8
    *************	*	******	******	******	******	******	******	******	******
    Vector UVZ-1	*	+1	0	0	0	0	0	0	0
    Vector UVZ-2	*	0	+1	0	0	0	0	0	0
    Vector UVZ-3	*	0	0	+1	0	0	0	0	0
    Vector UVZ-4	*	0	0	0	+1	0	0	0	0
    Vector UVZ-5	*	0	0	0	0	+1	0	0	0
    Vector UVZ-6	*	0	0	0	0	0	+1	0	0
    Vector UVZ-7	*	0	0	0	0	0	0	+1	0
    Vector UVZ-8	*	0	0	0	0	0	0	0	+1

• Let us now construct { Structure A } using :
  • { 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 } .

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Fixing the Structure in { 8-D Space } :

• Once the above-said Structure is completed ,
  • we shall now treat the 64 Lines in the Structure as being fixed-and-invariant in the { 8-Dimension Vector Space } ;

    i.e. :

    • the positioning of the { 128 Position Vectors } associated with the 64 Lines
      • will remain the same at-all-times with respect to 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 } .

  And for our discussion purposes below , we shall now refer this Structure as :

  • the { 64-Lines Structure now-fixed-in-space } .

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Re-interpreting using { SET ALPHA } :

• Let us now bring in the { orthogonal unit-vector basis } ,
  • { Orthogonal Unit-Vector Basis SET ALPHA } .

    Details for the Orthogonal Unit-Vector Basis { SET ALPHA }
    Vector I.D.	* *	Co-ordinates based on the Physical Unit Vector
    		PUV-1	PUV-2	PUV-3	PUV-4	PUV-5	PUV-6	PUV-7	PUV-8
    *************	*	******	******	******	******	******	******	******	******
    Vector UVA-1	*	+1	0	0	0	0	0	0	0
    Vector UVA-2	*	0	+1	0	0	0	0	0	0
    Vector UVA-3	*	0	0	-1	0	0	0	0	0
    Vector UVA-4	*	0	0	0	+1	0	0	0	0
    Vector UVA-5	*	0	0	0	0	+1	0	0	0
    Vector UVA-6	*	0	0	0	0	0	+1	0	0
    Vector UVA-7	*	0	0	0	0	0	0	+1	0
    Vector UVA-8	*	0	0	0	0	0	0	0	+1

• We shall now attempt to interpret :
  • the [ 8 co-ordinates ] for the 128 Position Vectors associated with the { 64-Lines Structure now-fixed-in-space }
    • using { Orthogonal Unit-Vector Basis SET ALPHA } .

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So what is the Difference here :

• Let us make a quick comparison of the 8 unit-vectors in { SET ALPHA } vs. { SET ZERO } :
  • { Vector UVA-1 } and { Vector UVZ-1 } are the same ;
  • { Vector UVA-2 } and { Vector UVZ-2 } are the same ;

  • { Vector UVA-3 } and { Vector UVZ-3 }are not the same ,
    • { Vector UVZ-3 } points in one direction and { Vector UVA-3 } points in the opposite direction .

  • { Vector UVA-4 } and { Vector UVZ-4 } are the same ;
  • { Vector UVA-5 } and { Vector UVZ-5 } are the same ;
  • { Vector UVA-6 } and { Vector UVZ-6 } are the same ;
  • { Vector UVA-7 } and { Vector UVZ-7 } are the same ;
  • { Vector UVA-8 } and { Vector UVZ-8 } are the same .

  Thus , for the 128 Position Vectors associated with the { 64-Lines Structure now-fixed-in-space } :

  • the original [ 8 co-odinates ] using { Orthogonal Unit-Vector Basis SET ZERO }
    • shall now make a change at the { 3rd co-ordinate position } , changing :
      • either :
        • from a [ +1 ] to a [ -1 ] ,

        or alternately :

        • from a [ -1 ] to a [ + 1 ] ,

        as the case may be ;

      when interpreted using the { Orthogonal Unit-Vector Basis SET ALPHA } .

• So , what is the net result here ?

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The Net Result :

• The net result is rather straight-forward :
  • The new [ 8 co-ordinates ] for the 128 Position Vectors associated with the { 64-Lines Structure now-fixed-in-space }
    • will now host an odd number of [ +1 ]'s and a corresponding odd number of [ -1 ]'s ,
      • under the new interpretation using { Orthogonal Unit-Vector Basis SET ALPHA } .

• And because ,
  • the 64 lines in the { 64-Lines Structure now-fixed-in-space } are distinct separate and unique ,

    and

  • the newly interpreted [ 8 co-ordinates ] for the 128 Position Vectors have remained in the { Cross-Diagonal Line format } ,
    • i.e. :
      • hosting only either a [ +1 ] or a [ -1 ] is all 8 co-ordinates ;

  we have no alternative other than to conclude that :

  • the newly interpreted [ 8 co-ordinates ] for each pair of Position Vectors for the 64 Lines must match :
    • the 64 Lines in { Class 7-1 / 1-7 } and { Class 5-3 / 3-5 } ;
      • on a { line-for-line } basis.

  And the 64 Lines in { Class 7-1 / 1-7 } and { Class 5-3 / 3-5 } is { Structure B } .

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Summarizing :

• Let us summarize our findings here as follows :
  • First , we created the { 64-Lines Structure now-fixed-in-space } ,
    • using { Orthogonal Unit-Vector Basis SET ZERO }

      and

    • { Structure A }'s { Class 8-0 / 0-8 }-{ Class 6-2 / 2-6 }-{ Class 4-4 / 4-4 } co-ordinates .

  • By default ,
    • the { 64-Lines Structure now-fixed-in-space } is { Structure A }
      • when interpreted using { Orthogonal Unit-Vector Basis SET ZERO } .

  • At the same time ,
    • the { 64-Lines Structure now-fixed-in-space } is { Structure B }
      • when interpreted using { Orthogonal Unit-Vector Basis SET ALPHA } .

• This then proves that :
  • the two (2) 64-lines structures , { Structure A } and { Structure B } are identical in shape .

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What's Next :

• We also need to look at orientations .

  And in order to do so ,

  • we must first create and segregate 256 sets of { orthogonal unit-vector basis } , next .

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

go to the last Section : Section XIX - Setting up 4 Sets of { 8-D Orthogonal Unit-Vector Basis } for our analysis

return to the Symmetry Through-put HomePage

Original dated 2017-7-23