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 XXIV - 2 Identical { 6-D Structures } arising from the { 6-D Quasi-Cube }

Summary for the Section :

• In this Section , we shall split the 32 Cross-Diagonal Lines of the { 6-Dimension Quasi-Cube }
  • into 2 identical Structures , each made-up-of { 16 Cross-Diagonal Lines } .

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Bringing in the { 6-Dimension Quasi-Cube } :

• Let us simply recall here that the { 6-Dimension Quasi-Cube } :
  • has 64 { corners / nodal-points } and 32 Cross-Diagonal Lines .

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Classifying for the 32 [ Cross-Diagonal Lines ] :

• We shall now classify the 32 Cross-Diagonal Lines here into 4 categories ,
  • following closely along-the-line the Classification Scheme for the { 8-Dimension Quasi-Cube } in Section X .

  And the four (4) Categories here are :

  • [ Class 6-0 / 0-6 ] :

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

  • [ Class 5-1 / 1-5 ] :

    • 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 1 [ -1 ] in its 6 co-ordinates ;
      • the Position Vector for the [ 2nd end-point of the Cross-Diagonal Line ]
        • will host 1 [ +1 ] and 5 [ -1 ]'s in its 6 co-ordinates ;

        or vice-versa .

  • [ Class 4-2 / 2-4 ] :

    • 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 ]'s and 2 [ -1 ]'s in its 6 co-ordinates ;
      • the Position Vector for the [ 2nd end-point of the Cross-Diagonal Line ]
        • will host 2 [ +1 ]'s and 4 [ -1 ]'s in its 6 co-ordinates ;

        or vice-versa .

  • [ Class 3-3 / 3-3 ] :

    • For a [ Cross-Diagonal Line ] in this Class :
      • the Position Vector for the [ 1st end-point of the Cross-Diagonal Line ]
        • will host 3 [ +1 ]'s and 3 [ -1 ]'s in its 6 co-ordinates ;
      • the Position Vector for the [ 2nd end-point of the Cross-Diagonal Line ]
        • will host 3 [ +1 ]'s and 3 [ -1 ]'s in its 6 co-ordinates ;

        or vice-versa .

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2 Identical { 16-Lines Structures } :

• Let us now split the 32 Cross-Diagonal Lines into 2 identical { 6-Dimension Structures } ,
  • i.e. : { Structure A } and { Structure B } respectively ;
    • as follows .

    Classification	Number of Lines in Class	* * *	Structure A	Structure B
    **************	**************	*	**************	**************
    Class 6-0 / 0-6	1	*	1
    Class 5-1 / 1-5	6	*		6
    Class 4-2 / 2-4	15	*	15
    Class 3-3 / 3-3	10	*		10
    **************	**************	*	**************	**************
    Total	32	*	16	16

  As such :

  • { Structure A } shall then be made up of the 16 Lines in { Class 6-0 / 0-6 } and { Class 4-2 / 2-4 } ,

    and

  • { Structure B } shall be made up of the 16 Lines in { Class 5-1 / 1-5 } and { Class 3-3 / 3-3 } .

• Let us now list out the 16 Cross-Diagonal Lines for { Structure A } :

  16 Cross-Diagonal Lines in { Structure A }	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 6-0 / 0-6	*	Vector Z-1	+1	+1	+1	+1	+1	+1	*	Vector Z-2	-1	-1	-1	-1	-1	-1
  **************	**************	*	***********	***	***	***	***	***	***	*	***********	***	***	***	***	***	***
  Line Q-1	Class 4-2 / 2-4	*	Vector Q-1	+1	+1	+1	+1	-1	-1	*	Vector Q-2	-1	-1	-1	-1	+1	+1
  Line Q-2	Class 4-2 / 2-4	*	Vector Q-3	+1	+1	+1	-1	-1	+1	*	Vector Q-4	-1	-1	-1	+1	+1	-1
  Line Q-3	Class 4-2 / 2-4	*	Vector Q-5	+1	+1	-1	-1	+1	+1	*	Vector Q-6	-1	-1	+1	+1	-1	-1
  Line Q-4	Class 4-2 / 2-4	*	Vector Q-7	+1	-1	-1	+1	+1	+1	*	Vector Q-8	-1	+1	+1	-1	-1	-1
  Line Q-5	Class 4-2 / 2-4	*	Vector Q-9	-1	-1	+1	+1	+1	+1	*	Vector Q-10	+1	+1	-1	-1	-1	-1
  Line Q-6	Class 4-2 / 2-4	*	Vector Q-11	+1	+1	+1	-1	+1	-1	*	Vector Q-12	-1	-1	-1	+1	-1	+1
  Line Q-7	Class 4-2 / 2-4	*	Vector Q-13	+1	+1	-1	+1	-1	+1	*	Vector Q-14	-1	-1	+1	-1	+1	-1
  Line Q-8	Class 4-2 / 2-4	*	Vector Q-15	+1	-1	+1	-1	+1	+1	*	Vector Q-16	-1	+1	-1	+1	-1	-1
  Line Q-9	Class 4-2 / 2-4	*	Vector Q-17	-1	+1	-1	+1	+1	+1	*	Vector Q-18	+1	-1	+1	-1	-1	-1
  Line Q-10	Class 4-2 / 2-4	*	Vector Q-19	+1	+1	-1	+1	+1	-1	*	Vector Q-20	-1	-1	+1	-1	-1	+1
  Line Q-11	Class 4-2 / 2-4	*	Vector Q-21	+1	-1	+1	+1	-1	+1	*	Vector Q-22	-1	+1	-1	-1	+1	-1
  Line Q-12	Class 4-2 / 2-4	*	Vector Q-23	-1	+1	+1	-1	+1	+1	*	Vector Q-24	+1	-1	-1	+1	-1	-1
  Line Q-13	Class 4-2 / 2-4	*	Vector Q-25	+1	-1	+1	+1	+1	-1	*	Vector Q-26	-1	+1	-1	-1	-1	+1
  Line Q-14	Class 4-2 / 2-4	*	Vector Q-27	-1	+1	+1	+1	-1	+1	*	Vector Q-28	+1	-1	-1	-1	+1	-1
  Line Q-15	Class 4-2 / 2-4	*	Vector Q-29	-1	+1	+1	+1	+1	-1	*	Vector Q-30	+1	-1	-1	-1	-1	+1

• Let us now also list out the 16 Cross-Diagonal Lines for { Structure B } :

  16 Cross-Diagonal Lines in { Structure B }	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 P-1	Class 5-1 / 1-5	*	Vector P-1	+1	+1	+1	+1	+1	-1	*	Vector P-2	-1	-1	-1	-1	-1	+1
  Line P-2	Class 5-1 / 1-5	*	Vector P-3	+1	+1	+1	+1	-1	+1	*	Vector P-4	-1	-1	-1	-1	+1	-1
  Line P-3	Class 5-1 / 1-5	*	Vector P-5	+1	+1	+1	-1	+1	+1	*	Vector P-6	-1	-1	-1	+1	-1	-1
  Line P-4	Class 5-1 / 1-5	*	Vector P-7	+1	+1	-1	+1	+1	+1	*	Vector P-8	-1	-1	-1	+1	-1	-1
  Line P-5	Class 5-1 / 1-5	*	Vector P-9	+1	-1	+1	+1	+1	+1	*	Vector P-10	-1	+1	-1	-1	-1	-1
  Line P-6	Class 5-1 / 1-5	*	Vector P-11	-1	+1	+1	+1	+1	+1	*	Vector P-12	+1	-1	-1	-1	-1	-1
  **************	**************	*	***********	***	***	***	***	***	***	*	***********	***	***	***	***	***	***
  Line R-1	Class 3-3 / 3-3	*	Vector R-1	+1	+1	+1	-1	-1	-1	*	Vector R-2	-1	-1	-1	+1	+1	+1
  Line R-2	Class 3-3 / 3-3	*	Vector R-3	+1	+1	-1	+1	-1	-1	*	Vector R-4	-1	-1	+1	-1	+1	+1
  Line R-3	Class 3-3 / 3-3	*	Vector R-5	+1	-1	+1	+1	-1	-1	*	Vector R-6	-1	+1	-1	-1	+1	+1
  Line R-4	Class 3-3 / 3-3	*	Vector R-7	-1	+1	+1	+1	-1	-1	*	Vector R-8	+1	-1	-1	-1	+1	+1
  Line R-5	Class 3-3 / 3-3	*	Vector R-9	+1	+1	-1	-1	+1	-1	*	Vector R-10	-1	-1	+1	+1	-1	+1
  Line R-6	Class 3-3 / 3-3	*	Vector R-11	-1	-1	+1	+1	+1	-1	*	Vector R-12	+1	+1	-1	-1	-1	+1
  Line R-7	Class 3-3 / 3-3	*	Vector R-13	+1	-1	-1	+1	+1	-1	*	Vector R-14	-1	+1	+1	-1	-1	+1
  Line R-8	Class 3-3 / 3-3	*	Vector R-15	-1	+1	+1	-1	+1	-1	*	Vector R-16	+1	-1	-1	+1	-1	+1
  Line R-9	Class 3-3 / 3-3	*	Vector R-17	-1	+1	-1	+1	+1	-1	*	Vector R-18	+1	-1	+1	-1	-1	+1
  Line R-10	Class 3-3 / 3-3	*	Vector R-19	+1	-1	+1	-1	+1	-1	*	Vector R-20	-1	+1	-1	+1	-1	+1

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Observations on the [ Angle of Intersection ] :

• Let us now make these observations below on the [ angle of intersection ] :

  OBSERVATION ONE :

  • The 16 Lines in { Structure A } will intersect one-another at an angle of [ 70.53 degrees ] ,
    • i.e. :
      • [ 70.53 degrees ] = arccos [ 1 / 3 ] .

  OBSERVATION TWO :

  • The 16 Lines in { Structure B } will intersect one-another at an angle of [ 70.53 degrees ] ,
    • i.e. :
      • [ 70.53 degrees ] = arccos [ 1 / 3 ] .

  OBSERVATION THREE :

  • Any Line in { Structure A } will intersect :
    • 10 of the 16 Lines in { Structure B } at an angle of [ 90.00 degrees ] ,

      and

    • the remaining 6 of the 16 Lines in { Structure B } at an angle of [ 48.19 degrees ] ,

      i.e. :

      • [ 90.00 degrees ] = arccos [ 0 / 3 ] ,

      • [ 48.19 degrees ] = arccos [ 2 / 3 ] .

  OBSERVATION FOUR :

  • Any Line in { Structure B } will intersect :
    • 10 of the 16 Lines in { Structure A } at an angle of [ 90.00 degrees ] ,

      and

    • the remaining 6 of the 16 Lines in { Structure A } at an angle of [ 48.19 degrees ] ,

      i.e. :

      • [ 90.00 degrees ] = arccos [ 0 / 3 ] ,

      • [ 48.19 degrees ] = arccos [ 2 / 3 ] .

• And we take note here , of course , that :
  • the { 28 Symmetric Lines in 7-Dimension } also intersect one-another at an angle of [ 70.53 degrees ] .

    so that :

    • the { 16 Symmetric Lines in 6-Dimension } here may possibly be a sub-set of the said { 28 Lines in 7-D } .

  But more on this later in Section XXX .

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What's Next :

• Based on Observation Three and Observation Four above , we now do have :
  • 10 Lines residing in a { 5-Dimension Sub-Space } intersecting one-another at an angle of [ 70.53 degrees ] .

  Let us take a closer look at these [ 10 lines ] , next .

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go to the next Section : Section XXV - Identifying { 10 Symmetric Lines in 5-D }

go to the last Section : Section XXIII - Using [ 8 x 8 Mapping Matrices ] on { Structure A } and { Structure B }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23