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 V - Classifying the 128 Cross-Diagonal Lines of the { 8-D Quasi-Cube }

Summary for the Section :

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

• And the Classification Scheme here will be used not only in Part II of the paper here :
  • but also in Part IV of the paper ,
    • when we split the 128 Cross-Diagonal Lines of the { 8-Dimension Quasi-Cube } into 2 identical Structures .

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Classifying the 128 [ Cross-Diagonal Lines ] :

• Let us recall that we have 128 Cross-Diagonal Lines in our { 8-Dimension Quasi-Cube } .

  We shall now proceed to divide the 128 Cross-Diagonal Lines into 5 categories ,

  • based on :
    • the number of [ +1 ]'s and [ -1 ]'s in the [ 8 co-ordinates ] of
      • the Position Vector for the 1st end-point of the [ Cross-Diagonal Line ]

      and

    • the corresponding number of [ +1 ]'s and [ -1 ]'s in the [ 8 co-ordinates ] of
      • the Position Vector for the 2nd end-point of the [ Cross-Diagonal Line ] .

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The 5 Categories :

• The five (5) CATEGORIES of Cross-Diagonal Lines are then :

  • [ Class 8-0 / 0-8 ] :

    • For the single [ Cross-Diagonal Line ] in this Class :
      • the Position Vector for the [ 1st end-point of the Cross-Diagonal Line ]
        • will host exactly 8 [ +1 ]'s in its 8 co-ordinates ;

      • the Position Vector for the [ 2nd end-point of the Cross-Diagonal Line ]
        • will host exactly 8 [ -1 ]'s in its 8 co-ordinates ;

        or vice-versa .

  • [ Class 7-1 / 1-7 ] :

    • For a [ Cross-Diagonal Line ] in this Class :
      • the Position Vector for the [ 1st end-point of the Cross-Diagonal Line ]
        • will host 7 [ +1 ]'s and 1 [ -1 ]'s in its 8 co-ordinates ;

      • the Position Vector for the [ 2nd end-point of the Cross-Diagonal Line ]
        • will host 1 [ +1 ]'s and 7 [ -1 ]'s in its 8 co-ordinates ;

        or vice-versa .

  • [ Class 6-2 / 2-6 ] :

    • For a [ Cross-Diagonal Line ] in this Class :
      • the Position Vector for the [ 1st end-point of the Cross-Diagonal Line ]
        • will host 6 [ +1 ]'s and 2 [ -1 ]'s in its 8 co-ordinates ;

      • the Position Vector for the [ 2nd end-point of the Cross-Diagonal Line ]
        • will host 2 [ +1 ]'s and 6 [ -1 ]'s in its 8 co-ordinates ;

        or vice-versa .

  • [ Class 5-3 / 3-5 ] :

    • For a [ Cross-Diagonal Line ] in this Class :
      • the Position Vector for the [ 1st end-point of the Cross-Diagonal Line ]
        • will host 5 [ +1 ]'s and 3 [ -1 ]'s in its 8 co-ordinates ;

      • the Position Vector for the [ 2nd end-point of the Cross-Diagonal Line ]
        • will host 3 [ +1 ]'s and 5 [ -1 ]'s in its 8 co-ordinates ;

        or vice-versa .

  • [ Class 4-4 / 4-4 ] :

    • For a [ Cross-Diagonal Line ] in this Class :
      • the Position Vector for the [ 1st end-point of the Cross-Diagonal Line ]
        • will host 4 [ +1 ] and 4 [ -1 ] in its 8 co-ordinates ;
        • the Position Vector for the [ 2nd end-point of the Cross-Diagonal Line ]
          • will also host 4 [ +1 ] and 4 [ -1 ] in its 8 co-ordinates .

• Let us now take a quick look at some sample Line/s in each of the five (5) Categories ,
  • for a better grasp of the classification system here .

  Cross-Diagonal Line I.D.	Class	* * *	Position Vector for 1st end-point of the Cross-Diagonal Line									* * *	Position Vector for 2nd end-point of the Cross-Diagonal Line
  		*	Vector I.D.	Co-ordinates								*	Vector I.D.	Co-ordinates
  **************	**************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
  Line Zero	Class 8-0 / 0-8	*	Vector Z-1	+1	+1	+1	+1	+1	+1	+1	+1	*	Vector Z-2	-1	-1	-1	-1	-1	-1	-1	-1
  **************	**************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
  Sample Line H	Class 7-1 / 1-7	*	Vector H-1	+1	+1	+1	+1	+1	+1	+1	-1	*	Vector H-2	-1	-1	-1	-1	-1	-1	-1	+1
  Sample Line J	Class 7-1 / 1-7	*	Vector J-1	+1	+1	+1	-1	+1	+1	+1	+1	*	Vector J-2	-1	-1	-1	+1	-1	-1	-1	-1
  **************	**************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
  Sample Line L	Class 6-2 / 2-6	*	Vector L-1	+1	+1	+1	+1	+1	+1	-1	-1	*	Vector L-2	-1	-1	-1	-1	-1	-1	+1	+1
  Sample Line M	Class 6-2 / 2-6	*	Vector M-1	+1	+1	+1	-1	+1	+1	+1	-1	*	Vector M-2	-1	-1	-1	+1	-1	-1	-1	+1
  Sample Line N	Class 6-2 / 2-6	*	Vector N-1	+1	+1	-1	-1	+1	+1	+1	+1	*	Vector N-2	-1	-1	+1	+1	-1	-1	-1	-1
  **************	**************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
  Sample Line P	Class 5-3 / 3-5	*	Vector P-1	+1	+1	+1	+1	+1	-1	-1	-1	*	Vector P-2	-1	-1	-1	-1	-1	+1	+1	+1
  Sample Line Q	Class 5-3 / 3-5	*	Vector Q-1	+1	+1	-1	+1	+1	-1	-1	+1	*	Vector Q-2	-1	-1	+1	-1	-1	+1	+1	-1
  Sample Line R	Class 5-3 / 3-5	*	Vector R-1	+1	-1	-1	+1	+1	-1	+1	+1	*	Vector R-2	-1	+1	+1	-1	-1	+1	-1	-1
  Sample Line S	Class 5-3 / 3-5	*	Vector S-1	-1	-1	-1	+1	+1	+1	+1	+1	*	Vector R-2	+1	+1	+1	-1	-1	-1	-1	-1
  **************	**************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
  Line D-1	Class 4-4 / 4-4	*	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	Class 4-4 / 4-4	*	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	Class 4-4 / 4-4	*	Vector D-37	+1	+1	-1	-1	+1	+1	-1	-1	*	Vector D-38	-1	-1	+1	+1	-1	-1	+1	+1

  And we take note here that :

  • Line Zero / Line D-1 / Line D-18 / Line D-19
    • are orthogonal to one-another ;

    so that :

    • we do now entertain the possibility of identifying :
      • a set of { 4 Cross-Diagonal Lines mutually orthogonal to one-another } ,
        • and thereby defining a { 4-Dimension Sub-Space } .

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Line-Count for the 5 categories :

• Let us now take a look at :
  • the number of Cross-Diagonal Lines in each of the 5 Categories .

    Classification	Number of Lines in Class	* * *	Comment
    **************	**************	*	************************************
    Class 8-0 / 0-8	1	*
    Class 7-1 / 1-7	8	*	8 = 8 / 1
    Class 6-2 / 2-6	28	*	28 = 8 x 7 / 2 x 1
    Class 5-4 / 3-5	56	*	56 = 8 x 7 x 6 / 3 x 2 x 1
    Class 4-4 / 4-4	35	*	35 = one-half of [ 8 x 7 x 6 x 5 / 4 x 3 x 2 x 1 ]
    **************	**************	*	************************************
    Total	128	*

  And we simply take note here that :

  • [ Line Zero ] is the one-and-only line in { Class 8-0 / 0-8 } , and

  • the 35 Lines from the last Section belong to { Class 4-4 / 4-4 } .

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What's Next :

• We shall next set up the { Dot Product Table for the 35 Cross-Diagonal Lines }
  • to determine which pairs-of-lines are orthogonal to one-another .

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

go to the last Section : Section IV - A Listing of the 35 Lines orthogonal to the { Line-Of-Concern }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23