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 VI - The Dot Product Table for the 35 Cross-Diagonal Lines

Summary for the Section :

• In this Section , we shall present :
  • the { Dot Product Table } for the 35 Cross-Diagonal Lines .

  Interestingly , we were able to identify :

  • two (2) sets of { 4 mutually orthogonal Cross-Diagonal Lines }
    • residing in the same { 4-Dimension Sub-Space } .

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The Dot Product Table for the 35 Cross-Diagonal Lines :

• Let us now bring-in the { Dot Product Table } for the 35 Cross-Diagonal Lines :
  • marking-off with an [ O ] whenever a pair of lines intersect one-another at the angle of [ 90 degrees ] , and

  • marking-off with an [ X ] whenever a pair of lines intersect one-another at the angle of [ 60 degrees ] .

  	D 1	* * *	D 2	D 3	D 4	D 5	D 6	D 7	D 8	D 9		D 1 0	D 1 1	D 1 2	D 1 3	D 1 4	D 1 5	D 1 6	D 1 7	* * *	D 1 8	D 1 9	D 2 0	D 2 1	D 2 2	D 2 3		D 2 4	D 2 5	D 2 6	D 2 7	D 2 8	D 2 9		D 3 0	D 3 1	D 3 2	D 3 3	D 3 4	D 3 5	* * *
  D1		*	X	X	X	X	X	X	X	X		X	X	X	X	X	X	X	X	*	O	O	O	O	O	O		O	O	O	O	O	O		O	O	O	O	O	O	*
  ****	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	*	**	**	**	**	**	**	*	**	**	**	**	**	**	*
  D2	X	*		X	X	X	X	O	O	O		X	O	O	O	X	O	O	O	*	X	O	O	X	O	X		O	X	X	O	X	O		X	O	O	X	O	X	*
  D3	X	*	X		X	X	O	X	O	O		O	X	O	O	O	X	O	O	*	X	O	X	O	X	O		O	X	O	X	O	X		X	O	X	O	X	O	*
  D4	X	*	X	X		X	O	O	X	O		O	O	X	O	O	O	X	O	*	O	X	X	O	O	X		X	O	O	X	X	O		O	X	X	O	O	X	*
  D5	X	*	X	X	X		O	O	O	X		O	O	O	X	O	O	O	X	*	O	X	O	X	X	O		X	O	X	O	O	X		O	X	O	X	X	O	*
  D6	X	*	X	O	O	O		X	X	X		X	O	O	O	X	O	O	O	*	X	O	O	X	O	X		X	O	O	X	O	X		O	X	X	O	X	O	*
  D7	X	*	O	X	O	O	X		X	X		O	X	O	O	O	X	O	O	*	X	O	X	O	X	O		X	O	X	O	X	O		O	X	O	X	O	X	*
  D8	X	*	O	O	X	O	X	X		X		O	O	X	O	O	O	X	O	*	O	X	X	O	O	X		O	X	X	O	O	X		X	O	O	X	X	O	*
  D-9	X	*	O	O	O	X	X	X	X			O	O	O	X	O	O	O	X	*	O	X	O	X	X	O		O	X	O	X	X	O		X	O	X	O	O	X	*
  		*																		*																					*
  D10	X	*	X	O	O	O	X	O	O	O			X	X	X	X	O	O	O	*	O	X	X	O	X	O		X	O	O	X	O	X		X	O	O	X	O	X	*
  D11	X	*	O	X	O	O	O	X	O	O		X		X	X	O	X	O	O	*	O	X	O	X	O	X		X	O	X	O	X	O		X	O	X	O	X	O	*
  D12	X	*	O	O	X	O	O	O	X	O		X	X		X	O	O	X	O	*	X	O	O	X	X	O		O	X	X	O	O	X		O	X	X	O	O	X	*
  D13	X	*	O	O	O	X	O	O	O	X		X	X	X		O	O	O	X	*	X	O	X	O	O	X		O	X	O	X	X	O		O	X	O	X	X	O	*
  D14	X	*	X	O	O	O	X	O	O	O		X	O	O	O		X	X	X	*	O	X	X	O	X	O		O	X	X	O	X	O		O	X	X	O	X	O	*
  D15	X	*	O	X	O	O	O	X	O	O		O	X	O	O	X		X	X	*	O	X	O	X	O	X		O	X	O	X	O	X		O	X	O	X	O	X	*
  D16	X	*	O	O	X	O	O	O	X	O		O	O	X	O	X	X		X	*	X	O	O	X	X	O		X	O	O	X	X	O		X	O	O	X	X	O	*
  D17	X	*	O	O	O	X	O	O	O	X		O	O	O	X	X	X	X		*	X	O	X	O	O	X		X	O	X	O	O	X		X	O	X	O	O	X	*
  		*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	*	**	**	**	**	**	**	*	**	**	**	**	**	**	*
  D18	O	*	X	X	O	O	X	X	O	O		O	O	X	X	O	O	X	X	*		O	X	X	X	X		X	X	O	O	O	O		X	X	O	O	O	O	*
  D19	O	*	O	O	X	X	O	O	X	X		X	X	O	O	X	X	O	O	*	O		X	X	X	X		X	X	O	O	O	O		X	X	O	O	O	O	*
  D20	O	*	O	X	X	O	O	X	X	O		X	O	O	X	X	O	O	X	*	X	X		O	X	X		O	O	X	X	O	O		O	O	X	X	O	O	*
  D21	O	*	X	O	O	X	X	O	O	X		O	X	X	O	O	X	X	O	*	X	X	O		X	X		O	O	X	X	O	O		O	O	X	X	O	O	*
  D22	O	*	O	X	O	X	O	X	O	X		X	O	X	O	X	O	X	O	*	X	X	X	X		O		O	O	O	O	X	X		O	O	O	O	X	X	*
  D23	O	*	X	O	X	O	X	O	X	O		O	X	O	X	O	X	O	X	*	X	X	X	X	O			O	O	O	O	X	X		O	O	O	O	X	X	*
  		*																		*																					*
  D24	O	*	X	X	O	O	O	O	X	X		O	O	X	X	X	X	O	O	*	X	X	O	O	O	O			O	X	X	X	X		X	X	O	O	O	O	*
  D25	O	*	O	O	X	X	X	X	O	O		X	X	O	O	O	O	X	X	*	X	X	O	O	O	O		O		X	X	X	X		X	X	O	O	O	O	*
  D26	O	*	X	O	O	X	O	X	X	O		O	X	X	O	X	O	O	X	*	O	O	X	X	O	O		X	X		O	X	X		O	O	X	X	O	O	*
  D27	O	*	O	X	X	O	X	O	O	X		X	O	O	X	O	X	X	O	*	O	O	X	X	O	O		X	X	O		X	X		O	O	X	X	O	O	*
  D28	O	*	X	O	X	O	O	X	O	X		O	X	O	X	X	O	X	O	*	O	O	O	O	X	X		X	X	X	X		O		O	O	O	O	X	X	*
  D29	O	*	O	X	O	X	X	O	X	O		X	O	X	O	O	X	O	X	*	O	O	O	O	X	X		X	X	X	X	O			O	O	O	O	X	X	*
  		*																		*																					*
  D30	O	*	X	X	O	O	O	O	X	X		X	X	O	O	O	O	X	X	*	X	X	O	O	O	O		X	X	O	O	O	O			O	X	X	X	X	*
  D31	O	*	O	O	X	X	X	X	O	O		O	O	X	X	X	X	O	O	*	X	X	O	O	O	O		X	X	O	O	O	O		O		X	X	X	X	*
  D32	O	*	O	X	X	O	X	O	O	X		O	X	X	O	X	O	O	X	*	O	O	X	X	O	O		O	O	X	X	O	O		X	X		O	X	X	*
  D33	O	*	X	O	O	X	O	X	X	O		X	O	O	X	O	X	X	O	*	O	O	X	X	O	O		O	O	X	X	O	O		X	X	O		X	X	*
  D34	O	*	O	X	O	X	X	O	X	O		O	X	O	X	X	O	X	O	*	O	O	O	O	X	X		O	O	O	O	X	X		X	X	X	X		O	*
  D35	O	*	X	O	X	O	O	X	O	X		X	O	X	O	O	X	O	X	*	O	O	O	O	X	X		O	O	O	O	X	X		X	X	X	X	O		*
  		*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	*	**	**	**	**	**	**	*	**	**	**	**	**	**	*
  			O = orthogonal									X = not orthogonal ( angle of intersection is always 60 degrees )

• We simply take note here that :
  • each of the 35 Lines will :
    • intersect 16 of the remaining 34 Lines at an angle of [ 60 degrees ] ,

      and

    • intersect 18 of the remaining 34 Lines at an angle of [ 90 degrees ] .

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Setting up the First { 4-Dimension Sub-Space } :

• Based on the above , let us now identify and set up :
  • four (4) Cross-Diagonal Lines that are orthogonal to one-another ,
    • within the { 128 Cross-Diagonal Lines of the 8-Dimension Quasi-Cube } framework .

  The { 4 Lines } are then :

  • { Line Zero / Line D-1 / Line D-18 / Line D-19 } ;
    • and they together will define our First { 4-Dimension Sub-Space } .

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Setting up the Second { 4-Dimension Sub-Space } :

• Let us now identify the 8 Lines in the { Dot Product Table } above
  • orthogonal to { Line Zero / Line D-1 / Line D-18 / Line D-19 } .

  And these 8 Lines are then :

  • { Line D-26 / Line D-27 / Line D-28 / Line D-29 / Line D-32 / Line D-33 / Line D-34 / Line D-35 }

    respectively .

• Let us now take a look at :
  • the { Dot Product Table } for these 8 Lines .

    	* * *	D 2 6	D 2 7	D 3 4	D 3 5		D 2 8	D 2 9	D 3 2	D 3 3	* * *
    ****	*	**	**	**	**	*	**	**	**	**	*
    D26	*		O	O	O		X	X	X	X	*
    D27	*	O		O	O		X	X	X	X	*
    D34	*	O	O		O		X	X	X	X	*
    D35	*	O	O	O			X	X	X	X	*
    	*										*
    D28	*	X	X	X	X			O	O	O	*
    D29	*	X	X	X	X		O		O	O	*
    D32	*	X	X	X	X		O	O		O	*
    D33	*	X	X	X	X		O	O	O		*
    ****	*	***	***	***	***	*	***	***	***	***	*
    	O = orthogonal						X = not orthogonal

• And based on this { Dot Product Table } above , we can now identify :
  • { Line D-26 / Line D-27 / Line D-34 / Line D-35 } as one set of { 4 Lines mutually orthogonal to one-another } ,

    and

  • { Line D-28 / Line D-29 / Line D-32 / Line D-33 } as another set of { 4 Lines mutually orthogonal to one-another } .

  And we take note here that both of the 2 sets of { 4 Lines mutually orthogonal to one-another }

  • do reside in the same { 4-Dimension Sub-Space } ,
    • being the kernal to the First { 4-Dimension Sub-Space } above .

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What's Next :

• The nagging question here is of course :
  • Since we have 2 sets of { 4 Lines mutually orthogonal to one-another }
    • in the Kernal { 4-Dimension Sub-Space } ;

    there should also be 2 sets of { 4 Lines mutually orthogonal to one-another }

    • in the First { 4-Dimension Sub-Space } .

  Let us find out if this is so , next .

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go to the next section : Section VII - Four (4) Sets of { 4-Lines Mutually Orthogonal to One-another } in 8-D

go to the last section : Section V - Classifying the 128 Cross-Diagonal Lines in our { 8-Dimension Quasi-Cube }

return to the Symmetry Through-put Homepage

Original dated 2017-7-23