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 VII - 4 Sets of { 4-Lines Mutually Orthogonal to One-another } in 8-D

Summary for the Section :

• In this Section , we shall :
  • identify a particular set of { 16 Cross-Diagonal Lines } arising from the { 8-Dimension Quasi-Cube }
    • to formulate :
      • 4 sets of { 4-Lines mutually orthogonal to one-another } .

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A Fresh Set of { 4 Lines Mutually Orthogonal to One-another } :

• Let us now bring in this set of four (4) Cross-Diagonal Lines :
  • { Line F-1 / Line F2 / Line F3 / Line F4 } as shown in the table below ;

    and name this as [ Set P ] for our discussion purposes in this Section .

  Line I.D.	* * *	Position Vector for 1st end-point of the Line									* * *	Position Vector for 2nd end-point of the Line
  	*	Vector I.D.	Co-ordinates								*	Vector I.D.	Co-ordinates
  *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
  Line F-1	*	Vector F-1	+1	+1	+1	+1	+1	+1	-1	-1	*	Vector F-2	-1	-1	-1	-1	-1	-1	+1	+1
  Line F-2	*	Vector F-3	+1	+1	+1	+1	-1	-1	+1	+1	*	Vector F-4	-1	-1	-1	-1	+1	+1	-1	-1
  Line F-3	*	Vector F-5	+1	+1	-1	-1	+1	+1	+1	+1	*	Vector F-6	-1	-1	+1	+1	-1	-1	-1	-1
  Line F-4	*	Vector F-7	-1	-1	+1	+1	+1	+1	+1	+1	*	Vector F-8	+1	+1	-1	-1	-1	-1	-1	-1
  *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
  Line Zero	*	Vector Z-1	+1	+1	+1	+1	+1	+1	+1	+1	*	Vector Z-2	-1	-1	-1	-1	-1	-1	-1	-1
  Line D-1	*	Vector D-1	+1	+1	+1	+1	-1	-1	-1	-1	*	Vector D-2	-1	-1	-1	-1	+1	+1	+1	+1
  Line D-18	*	Vector D-35	+1	+1	-1	-1	-1	-1	+1	+1	*	Vector D-36	-1	-1	+1	+1	+1	+1	-1	-1
  Line D-19	*	Vector D-37	+1	+1	-1	-1	+1	+1	-1	-1	*	Vector D-38	-1	-1	+1	+1	-1	-1	+1	+1
  *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
  Line D-26	*	Vector D-51	+1	-1	-1	+1	-1	+1	+1	-1	*	Vector D-52	-1	+1	+1	-1	+1	-1	-1	+1
  Line D-27	*	Vector D-53	+1	-1	-1	+1	+1	-1	-1	+1	*	Vector D-54	-1	+1	+1	-1	-1	+1	+1	-1
  Line D-34	*	Vector D-67	+1	-1	+1	-1	+1	-1	+1	-1	*	Vector D-68	-1	+1	-1	+1	-1	+1	-1	+1
  Line D-35	*	Vector D-69	+1	-1	+1	-1	-1	+1	-1	+1	*	Vector D-70	-1	+1	-1	+1	+1	-1	+1	-1
  *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
  Line D-28	*	Vector D-55	+1	-1	-1	+1	+1	-1	+1	-1	*	Vector D-56	-1	+1	+1	-1	-1	+1	-1	+1
  Line D-29	*	Vector D-57	+1	-1	-1	+1	-1	+1	-1	+1	*	Vector D-58	-1	+1	+1	-1	+1	-1	+1	-1
  Line D-32	*	Vector D-63	+1	-1	+1	-1	-1	+1	+1	-1	*	Vector D-64	-1	+1	-1	+1	+1	-1	-1	+1
  Line D-33	*	Vector D-65	+1	-1	+1	-1	+1	-1	-1	+1	*	Vector D-66	-1	+1	-1	+1	-1	+1	+1	-1

  And we simply make the observation here that :

  • the 4 Lines in { Set P } are orthogonal to one-another .

• And for our discussion purpose below , let us now name :
  • the 4-lines set of { Line Zero / Line D-1 / Line D-18 / Line D-19 } as { Set Q } ;

  • the 4-lines set of { Line D-26 / Line D-27 / Line D-34 / Line D-35 } as { Set R } ;

  • the 4-lines set of { Line D-28 / Line D-29 / Line D-32 / Line D-33 } as { Set S } .

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Making the Observations :

• We now make these observations on the above set of { 16 Cross-Diagonal Lines } arising from our { 8-Dimension Quasi-Cube } .
  • OBSERVATION ONE :
    • Any line in { Set P } is orthogonal to any line in { Set R }

      and

      any line in { Set Q } is orthogonal to any line in { Set R } .

      As such ,

      • { Set P } and { Set Q } reside in the same { 4-Dimension Sub-Space } ,
        • which we shall now identify as the { FIRST 4-Dimension Sub-Space } .

  • OBSERVATION TWO :
    • Any line in { Set R } is orthogonal to any line in { Set P }

      and

      any line in { Set S } is orthogonal to any line in { Set P } .

      As such ,

      • { Set R } and { Set S } reside in the same { 4-Dimension Sub-Space } ,
        • which we shall now identify as the { SECOND 4-Dimension Sub-Space } .

  • OBSERVATION THREE :
    • Any line in { Set P } and any line in { Set Q } will intersect one-another at an angle of [ 60 degrees ]

      and

      any line in { Set R } and any line in { Set S } will intersect one-another at an angle of [ 60 degrees ] .

      As such ,

      • the relative orientation of { Set P } vs. { Set Q } in the { FIRST 4-Dimension Sub-Space }

        and

        the relative orientation of { Set R } vs. { Set S } in the { SECOND 4-Dimension Sub-Space }

        • should be the same .

  But are they really ?

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What's Next :

• Let us bring-in the { 4-Dimension Quasi-Cube } , next :
  • to help us find an answer .

  And the results from the next part , Part III , concentrating on Symmetry in 4-Dimension ,

  • can be intriguing .

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go to the next Section : Section VIII - Setting up the 8 Cross-Diagonal Lines of the { 4-D Quasi-Cube }

go to the last Section : Section VI - The Dot Product Table for the 35 Cross-Diagonal Lines

return to the Symmetry Through-put HomePage

Original dated 2017-7-23