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 IV - A Listing of the 35 Lines orthogonal to the { Line-Of-Concern }

Summary for the Section :

• In this Section , we shall :
  • do a full listing of the { 35 Lines } orthogonal to the { Line-Of-Concern }

    and

  • make the observation that all 36 Lines here are { Cross-Diagonal Lines } in our { 8-Dimension Quasi-Cube } .

  The full listing of the { 35 Lines } is done here for a specific reason :

  • for cross-checking purposes on the [ Dot Product Table ] for the 35 Lines
    • to be presented later in Section VI .

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A Full Listing of the { 35 Lines } :

• Let us now do a full listing of the 35 [ Type-D Lines ] orthogonal to our [ Line-Of-Concern ] ,
  • taking note here that we have now given our { Line-Of-Concern } a new name :
    • the { Line-Of-Concern } shall now be identified as [ Line Zero ] from here-on-in ,
      • and throughout the remainder of this Part II of the paper .

  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 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-2	*	Vector D-3	+1	+1	+1	-1	-1	-1	-1	+1	*	Vector D-4	-1	-1	-1	+1	+1	+1	+1	-1
  Line D-3	*	Vector D-5	+1	+1	+1	-1	-1	-1	+1	-1	*	Vector D-6	-1	-1	-1	+1	+1	+1	-1	+1
  Line D-4	*	Vector D-7	+1	+1	+1	-1	-1	+1	-1	-1	*	Vector D-8	-1	-1	-1	+1	+1	-1	+1	+1
  Line D-5	*	Vector D-9	+1	+1	+1	-1	+1	-1	-1	-1	*	Vector D-10	-1	-1	-1	+1	-1	+1	+1	+1
  Line D-6	*	Vector D-11	+1	+1	-1	+1	-1	-1	-1	+1	*	Vector D-12	-1	-1	+1	-1	+1	+1	+1	-1
  Line D-7	*	Vector D-13	+1	+1	-1	+1	-1	-1	+1	-1	*	Vector D-14	-1	-1	+1	-1	+1	+1	-1	+1
  Line D-8	*	Vector D-15	+1	+1	-1	+1	-1	+1	-1	-1	*	Vector D-16	-1	-1	+1	-1	+1	-1	+1	+1
  Line D-9	*	Vector D-17	+1	+1	-1	+1	+1	-1	-1	-1	*	Vector D-18	-1	-1	+1	-1	-1	+1	+1	+1
  Line D-10	*	Vector D-19	+1	-1	+1	+1	-1	-1	-1	+1	*	Vector D-20	-1	+1	-1	-1	+1	+1	+1	-1
  Line D-11	*	Vector D-21	+1	-1	+1	+1	-1	-1	+1	-1	*	Vector D-22	-1	+1	-1	-1	+1	+1	-1	+1
  Line D-12	*	Vector D-23	+1	-1	+1	+1	-1	+1	-1	-1	*	Vector D-24	-1	+1	-1	-1	+1	-1	+1	+1
  Line D-13	*	Vector D-25	+1	-1	+1	+1	+1	-1	-1	-1	*	Vector D-26	-1	+1	-1	-1	-1	+1	+1	+1
  Line D-14	*	Vector D-27	+1	-1	-1	-1	+1	+1	+1	-1	*	Vector D-28	-1	+1	+1	+1	-1	-1	-1	+1
  Line D-15	*	Vector D-29	+1	-1	-1	-1	+1	+1	-1	+1	*	Vector VD30	-1	+1	+1	+1	-1	-1	+1	-1
  Line D-16	*	Vector D-31	+1	-1	-1	-1	+1	-1	+1	+1	*	Vector D-32	-1	+1	+1	+1	-1	+1	-1	-1
  Line D-17	*	Vector D-33	+1	-1	-1	-1	-1	+1	+1	+1	*	Vector D-34	-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-20	*	Vector D-39	+1	+1	-1	-1	-1	+1	+1	-1	*	Vector D-40	-1	-1	+1	+1	+1	-1	-1	+1
  Line D-21	*	Vector D-41	+1	+1	-1	-1	+1	-1	-1	+1	*	Vector D-42	-1	-1	+1	+1	-1	+1	+1	-1
  Line D-22	*	Vector D-43	+1	+1	-1	-1	+1	-1	+1	-1	*	Vector D-44	-1	-1	+1	+1	-1	+1	-1	+1
  Line D-23	*	Vector D-45	+1	+1	-1	-1	-1	+1	-1	+1	*	Vector D-46	-1	-1	+1	+1	+1	-1	+1	-1
  Line D-24	*	Vector D-47	+1	-1	-1	+1	-1	-1	+1	+1	*	Vector D-48	-1	+1	+1	-1	+1	+1	-1	-1
  Line D-25	*	Vector D-49	+1	-1	-1	+1	+1	+1	-1	-1	*	Vector D-50	-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-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-30	*	Vector D-59	+1	-1	+1	-1	-1	-1	+1	+1	*	Vector D-60	-1	+1	-1	+1	+1	+1	-1	-1
  Line D-31	*	Vector D-61	+1	-1	+1	-1	+1	+1	-1	-1	*	Vector D-62	-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
  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

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Making the Observation :

• We now make the observation that :
  • all 36 Lines above are { Cross-Diagonal Lines } in our { 8-Dimension Quasi-Cube }
    • being characterised by :
      • the [ 8 co-ordinates ] for both :
        • the { Position Vector for the 1st end-point of the Line }

          and

        • the { Position Vector for the 2nd end-point of the Line }

        will host :

        • only [ +1 ]'s and [ -1 ]'s and no other value .

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The Intrigue here :

• Let us recall that we arrived at the { 35 Lines } here via :
  • choosing [ Line Zero ] as the { line-of-concern }

    and

  • using an extension of the { N+1 Symmetric Lines in a N-Dimension Space } procedure .

  And we have demonstrated here that :

  • [ Line Zero ] and the { 35 Lines } above all belong to :
    • the set of 128 Cross-Diagonal Lines of the { 8-Dimension Quasi-Cube } .

• It would seem that we now have a crossing-point for :
  • the { N+1 Symmetric Lines in a N-Dimension Space } procedure and its extentions

    and

  • the { N-Dimension Quasi-Cube } expansion procedure
    • for identifying Symmetric Lines in { higher-Dimension Vector Space }'s .

  It is therefore our prerogative in this Part II of the paper :

  • to find out more about these { 36 Lines } within the framework of the { 8-Dimension Quasi-Cube } .

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What's Next :

• In the next Section , we shall classify the 128 Cross-Diagonal Lines of the { 8-Dimension Quasi-Cube }
  • into 5 distinct and mutually-exclusive categories .

  And it is necessary to do the classifications now , because :

  • these classifications will greatly facilitate our discussions in our analysis to follow ,
    • both here in Part II and later in Part IV of this paper .

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go to the next Section : Section V - Classifying the 128 Cross-Diagonal Lines of the { 8-D Quasi-Cube }

go to the last Section : Section III - Recalling the 28 Symmetric Lines in a { 7-Dimension Vector Spacce }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23