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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Appendix A - Further Thoughts on Symmetric Lines in 6-D and 7-D

Summary for the Appendix :

• In this Appendix ,
  • we shall provide further thoughts on Symmetric Lines in 6-D and 7-D .

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Setting up the { 8-Dimensional Vector Space } :

• Let us first set up 8 unit-vectors defining a { 8-Dimension Vector Space }
  • namely , the 8 unit-vectors :
    • { Vector J-1 } , { Vector J-2 } , { Vector J-3 } , { Vector J-4 } ,
      { Vector J-5 } , { Vector J-6 } , { Vector J-7 } and { Vector J-8 } ;

      respectively .

  Let us now reserve the first 6 unit-vectors for the purpose of constructing a { 6-Dimension Quasi-Cube } ,

  • i.e. the 6 vectors :
    • { Vectors J-1 / J-2 / J-3 / J-4 / J-5 / J-6 } .

• And we take note here that :
  • { VectorJ-7 } and { Vector J-8 } do form a 2-Dimesion Plane .

  Let us now create 2 Vectors residing in this 2-Dimension Plane ,

  • namely { Vector X-9 } and { Vector Y-9 } respectively ,
    • so that :
      • both Vectors are of size { Square-root of [ 2 ] } ;

    but without stating :

    • the exact oriention of either vector ,

    or for that matter :

    • whether or not { Vector X-9 } and { Vector Y-9 } are one-and-the-same vector ,
      • in which case the Dot Product of the 2 Vectors is [ +2 ] ;

    • whether or not { Vector X-9 } and { Vector Y-9 } point in exact-opposite directions ,
      • in which case the Dot Product of the 2 Vectors is [ -2 ] ;

    • whether or not { Vector X-9 } and { Vector Y-9 } are orthogonal ,
      • in which case the Dot Product of the 2 Vectors is [ 0 ] ;

    • whether or not { Vector X-9 } and { Vector Y-9 } intersect at any other specific angle ,
      • in which case the Dot Product of the 2 Vectors is any value from [ -2 ] to [ +2 ] .

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Constructing { Structure A } / { Structure B } / { Structure X } / { Structure Y } :

• Let us now contruct the two (2) { 6-Dimension 16-Lines Structures } ,
  • { Structure A } and { Structure B } respectively ,

    same as in Section XXIV before .

  Structure A																			*	Structure B
  Line I.D.	*	Co-ordinates for 1st Position Vector								*	Co-ordinates for 2nd Position Vector									Line I.D.	*	Co-ordinates for 1st Position Vector								*	Co-ordinates for 2nd Position Vector
  		J - 1	J - 2	J - 3	J - 4	J - 5	J - 6	X - 9	Y - 9		J - 1	J - 2	J - 3	J - 4	J - 5	J - 6	X - 9	Y - 9				J - 1	J - 2	J - 3	J - 4	J - 5	J - 6	X - 9	Y - 9		J - 1	J - 2	J - 3	J - 4	J - 5	J - 6	X - 9	Y - 9
  ********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**
  P-1	*	+1	+1	+1	+1	+1	+1	0	0	*	-1	-1	-1	-1	-1	-1	0	0	*	Q-1	*	+1	+1	+1	-1	-1	-1	0	0	*	-1	-1	-1	+1	+1	+1	0	0
  P-2	*	+1	+1	+1	-1	-1	+1	0	0	*	-1	-1	-1	+1	+1	-1	0	0	*	Q-2	*	+1	+1	+1	+1	+1	-1	0	0	*	-1	-1	-1	-1	-1	+1	0	0
  P-3	*	+1	+1	+1	-1	+1	-1	0	0	*	-1	-1	-1	+1	-1	+1	0	0	*	Q-3	*	+1	+1	+1	+1	-1	+1	0	0	*	-1	-1	-1	-1	+1	-1	0	0
  P-4	*	+1	+1	+1	+1	-1	-1	0	0	*	-1	-1	-1	-1	+1	+1	0	0	*	Q-4	*	+1	+1	+1	-1	+1	+1	0	0	*	-1	-1	-1	+1	-1	-1	0	0
  P-5	*	+1	-1	-1	+1	+1	+1	0	0	*	-1	+1	+1	-1	-1	-1	0	0	*	Q-5	*	+1	-1	-1	-1	-1	-1	0	0	*	-1	+1	+1	+1	+1	+1	0	0
  P-6	*	+1	-1	-1	-1	-1	+1	0	0	*	-1	+1	+1	+1	+1	-1	0	0	*	Q-6	*	+1	-1	-1	+1	+1	-1	0	0	*	-1	+1	+1	-1	-1	+1	0	0
  P-7	*	+1	-1	-1	-1	+1	-1	0	0	*	-1	+1	+1	+1	-1	+1	0	0	*	Q-7	*	+1	-1	-1	+1	-1	+1	0	0	*	-1	+1	+1	-1	+1	-1	0	0
  P-8	*	+1	-1	-1	+1	-1	-1	0	0	*	-1	+1	+1	-1	+1	+1	0	0	*	Q-8	*	+1	-1	-1	-1	+1	+1	0	0	*	-1	+1	+1	+1	-1	-1	0	0
  P-9	*	-1	+1	-1	+1	+1	+1	0	0	*	+1	-1	+1	-1	-1	-1	0	0	*	Q-9	*	-1	+1	-1	-1	-1	-1	0	0	*	+1	-1	+1	+1	+1	+1	0	0
  P-10	*	-1	+1	-1	-1	-1	+1	0	0	*	+1	-1	+1	+1	+1	-1	0	0	*	Q-10	*	-1	+1	-1	+1	+1	-1	0	0	*	+1	-1	+1	-1	-1	+1	0	0
  P-11	*	-1	+1	-1	-1	+1	-1	0	0	*	+1	-1	+1	+1	-1	+1	0	0	*	Q-11	*	-1	+1	-1	+1	-1	+1	0	0	*	+1	-1	+1	-1	+1	-1	0	0
  P-12	*	-1	+1	-1	+1	-1	-1	0	0	*	+1	-1	+1	-1	+1	+1	0	0	*	Q-12	*	-1	+1	-1	-1	+1	+1	0	0	*	+1	-1	+1	+1	-1	-1	0	0
  P-13	*	-1	-1	+1	+1	+1	+1	0	0	*	+1	+1	-1	-1	-1	-1	0	0	*	Q-13	*	-1	-1	+1	-1	-1	-1	0	0	*	+1	+1	-1	+1	+1	+1	0	0
  P-14	*	-1	-1	+1	-1	-1	+1	0	0	*	+1	+1	-1	+1	+1	-1	0	0	*	Q-14	*	-1	-1	+1	+1	+1	-1	0	0	*	+1	+1	-1	-1	-1	+1	0	0
  P-15	*	-1	-1	+1	-1	+1	-1	0	0	*	+1	+1	-1	+1	-1	+1	0	0	*	Q-15	*	-1	-1	+1	+1	-1	+1	0	0	*	+1	+1	-1	-1	+1	-1	0	0
  P-16	*	-1	-1	+1	+1	-1	-1	0	0	*	+1	+1	-1	-1	+1	+1	0	0	*	Q-16	*	-1	-1	+1	-1	+1	+1	0	0	*	+1	+1	-1	+1	-1	-1	0	0

• Let us also construct { Structure X } and { Structure Y } ,
  • as shown below .

  Structure X																			*	Structure Y
  Line I.D.	*	Co-ordinates for 1st Position Vector								*	Co-ordinates for 2nd Position Vector									Line I.D.	*	Co-ordinates for 1st Position Vector								*	Co-ordinates for 2nd Position Vector
  		J - 1	J - 2	J - 3	J - 4	J - 5	J - 6	X - 9	Y - 9		J - 1	J - 2	J - 3	J - 4	J - 5	J - 6	X - 9	Y - 9				J - 1	J - 2	J - 3	J - 4	J - 5	J - 6	X - 9	Y - 9		J - 1	J - 2	J - 3	J - 4	J - 5	J - 6	X - 9	Y - 9
  ********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**
  X-17	*	+2	0	0	0	0	0	+1	0	*	-2	0	0	0	0	0	-1	0	*	Y-17	*	+2	0	0	0	0	0	0	+1	*	-2	0	0	0	0	0	0	-1
  X-18	*	-2	0	0	0	0	0	+1	0	*	+2	0	0	0	0	0	-1	0	*	Y-18	*	-2	0	0	0	0	0	0	+1	*	+2	0	0	0	0	0	0	-1
  X-19	*	0	+2	0	0	0	0	+1	0	*	0	-2	0	0	0	0	-1	0	*	Y-19	*	0	+2	0	0	0	0	0	+1	*	0	-2	0	0	0	0	0	-1
  X-20	*	0	-2	0	0	0	0	+1	0	*	0	+2	0	0	0	0	-1	0	*	Y-20	*	0	-2	0	0	0	0	0	+1	*	0	+2	0	0	0	0	0	-1
  X-21	*	0	0	+2	0	0	0	+1	0	*	0	0	-2	0	0	0	-1	0	*	Y-21	*	0	0	+2	0	0	0	0	+1	*	0	0	-2	0	0	0	0	-1
  X-22	*	0	0	-2	0	0	0	+1	0	*	0	0	+2	0	0	0	-1	0	*	Y-22	*	0	0	-2	0	0	0	0	+1	*	0	0	+2	0	0	0	0	-1
  X-23	*	0	0	0	+2	0	0	+1	0	*	0	0	0	-2	0	0	-1	0	*	Y-23	*	0	0	0	+2	0	0	0	+1	*	0	0	0	-2	0	0	0	-1
  X-24	*	0	0	0	-2	0	0	+1	0	*	0	0	0	+2	0	0	-1	0	*	Y-24	*	0	0	0	-2	0	0	0	+1	*	0	0	0	+2	0	0	0	-1
  X-25	*	0	0	0	0	+2	0	+1	0	*	0	0	0	0	-2	0	-1	0	*	Y-25	*	0	0	0	0	+2	0	0	+1	*	0	0	0	0	-2	0	0	-1
  X-26	*	0	0	0	0	-2	0	+1	0	*	0	0	0	0	+2	0	-1	0	*	Y-26	*	0	0	0	0	-2	0	0	+1	*	0	0	0	0	+2	0	0	-1
  X-27	*	0	0	0	0	0	+2	+1	0	*	0	0	0	0	0	-2	-1	0	*	Y-27	*	0	0	0	0	0	+2	0	+1	*	0	0	0	0	0	-2	0	-1
  X-28	*	0	0	0	0	0	-2	+1	0	*	0	0	0	0	0	+2	-1	0	*	Y-28	*	0	0	0	0	0	-2	0	+1	*	0	0	0	0	0	+2	0	-1

• We take note here that :
  • the 16 lines in { Structure A } intersects one-another at the angle of [ 70.53 degrees ] ;
  • the 16 lines in { Structure B } intersects one-another at the angle of [ 70.53 degrees ] ;
  • the 12 lines in { Structure X } intersects one-another at the angle of [ 70.53 degrees ] ;
  • the 12 lines in { Structure Y } intersects one-another at the angle of [ 70.53 degrees ] .

  Further ,

  • any line in { Structure X } will intersect any line in { Structure A } or { Struture B }
    • at the angle of [ 70.53 degrees ] ;

  • any line in { Structure Y } will intersect any line in { Structure A } or { Struture B }
    • at the angle of [ 70.53 degrees ] .

• Thus , we can construct the { 28 Lines in 7-D } via :
  • { Structure A } plus { Structure X } ,
  • { Structure A } plus { Structure Y } ,
  • { Structure B } plus { Structure X } ,
  • { Structure B } plus { Structure Y } .

  Isn't that interesting !

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Original dated 2017-7-23