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 XXVIII - Orthogonal Lines arising from the { 6-D Quasi-Cube }

Summary for the Section :

• In this Section , we shall identify , within the { 6-D Quasi-Cube } framework :
  • lines in the { 16-Lines Structure A } and lines in the { 16-Lines Structure B }
    • that are orthogonal to one-another ;

    in preparation of identifying matching pairs of cubes residing in a { 3-D sub-space } and its Kernal .

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

• Let us bring-in again the 32 Cross-Diagonal Lines of the { 6-Dimension Quasi-Cube } ,
  • split into the 2 identical structures , { Structure A } and { Structure B } .

  Structure A with 16 Cross-Diagonal Lines																	* *	Structure B with 16 Cross-Diagonal Lines
  16 C-D Lines in STR A	* * *	Position Vector for 1st end-point of the Cross-Diagonal Line							* * *	Position Vector for 2nd end-point of the Cross-Diagonal Line							* * *	16 C-D Lines in STR B	* * *	Position Vector for 1st end-point of the Cross-Diagonal Line							* * *	Position Vector for 2nd end-point of the Cross-Diagonal Line
  	*	Vctr I.D.	Co-ordinates						*	Vctr I.D.	Co-ordinates								*	Vctr I.D.	Co-ordinates						*	Vctr I.D.	Co-ordinates
  ********	*	****	**	**	**	**	**	**	*	****	**	**	**	**	**	**	*	********	*	****	**	**	**	**	**	**	*	****	**	**	**	**	**	**
  Line A1	*	J-1	+1	+1	+1	+1	+1	+1	*	J-2	-1	-1	-1	-1	-1	-1	*	Line B1	*	K-1	+1	-1	+1	-1	+1	-1	*	K-2	-1	+1	-1	+1	-1	+1
  Line A2	*	J-3	-1	+1	+1	+1	+1	-1	*	J-4	+1	-1	-1	-1	-1	+1	*	Line B2	*	K-3	+1	+1	-1	+1	-1	-1	*	K-4	-1	-1	+1	-1	+1	+1
  Line A3	*	J-5	+1	-1	+1	+1	-1	+1	*	J-6	-1	+1	-1	-1	+1	-1	*	Line B3	*	K-5	+1	+1	+1	-1	-1	-1	*	K-6	-1	-1	-1	+1	+1	+1
  Line A4	*	J-7	+1	+1	-1	-1	+1	+1	*	J-8	-1	-1	+1	+1	-1	-1	*	Line B4	*	K-7	+1	-1	-1	+1	+1	-1	*	K-8	-1	+1	+1	-1	-1	+1
  Line A5	*	J-9	+1	+1	-1	+1	+1	-1	*	J-10	-1	-1	+1	-1	-1	+1	*	Line B5	*	K-9	+1	-1	-1	-1	+1	+1	*	K-10	-1	+1	+1	+1	-1	-1
  Line A6	*	J-11	+1	+1	-1	+1	-1	+1	*	J-12	-1	-1	+1	-1	+1	-1	*	Line B6	*	K-11	-1	+1	+1	+1	+1	+1	*	K-12	+1	-1	-1	-1	-1	-1
  Line A7	*	J-13	+1	+1	+1	-1	-1	+1	*	J-14	-1	-1	-1	+1	+1	-1	*	Line B7	*	K-13	+1	-1	+1	+1	-1	-1	*	K-14	-1	+1	-1	-1	+1	+1
  Line A8	*	J-15	+1	+1	+1	-1	+1	-1	*	J-16	-1	-1	-1	+1	-1	+1	*	Line B8	*	K-15	+1	-1	+1	+1	+1	+1	*	K-16	-1	+1	-1	-1	-1	-1
  Line A9	*	J-17	-1	+1	-1	+1	+1	+1	*	J-18	+1	-1	+1	-1	-1	-1	*	Line B9	*	K-17	+1	+1	+1	+1	-1	+1	*	K-18	-1	-1	-1	-1	+1	-1
  Line A10	*	J-19	+1	-1	-1	+1	+1	+1	*	J-20	-1	+1	+1	-1	-1	-1	*	Line B10	*	K-19	+1	+1	-1	-1	+1	-1	*	K-20	-1	-1	+1	+1	-1	+1
  Line A11	*	J-21	+1	-1	+1	-1	+1	+1	*	J-22	-1	+1	-1	+1	-1	-1	*	Line B11	*	K-21	+1	+1	+1	+1	+1	-1	*	K-22	-1	-1	-1	-1	-1	+1
  Line A12	*	J-23	-1	+1	+1	-1	+1	+1	*	J-24	+1	-1	-1	+1	-1	-1	*	Line B12	*	K-23	+1	+1	-1	-1	-1	+1	*	K-24	-1	-1	+1	+1	+1	-1
  Line A13	*	J-25	+1	+1	+1	+1	-1	-1	*	J-26	-1	-1	-1	-1	+1	+1	*	Line B13	*	K-25	+1	-1	-1	+1	-1	+1	*	K-26	-1	+1	+1	-1	+1	-1
  Line A14	*	J-27	+1	-1	+1	+1	+1	-1	*	J-28	-1	+1	-1	-1	-1	+1	*	Line B14	*	K-27	+1	+1	-1	+1	+1	+1	*	K-28	-1	-1	+1	-1	-1	-1
  Line A15	*	J-29	-1	+1	+1	+1	-1	+1	*	J-30	+1	-1	-1	-1	+1	-1	*	Line B15	*	K-29	+1	+1	+1	-1	+1	+1	*	K-30	-1	-1	-1	+1	-1	-1
  Line A16	*	J-31	-1	-1	+1	+1	+1	+1	*	J-32	+1	+1	-1	-1	-1	-1	*	Line B16	*	K-31	+1	-1	+1	-1	-1	+1	*	K-32	-1	-1	+1	-1	-1	-1

• Let us recall that :
  • any line in { Structure A } will intersect any line in { Structure B } at :
    • either the angle of [ 90.00 degrees ] or the angle of [ 48.19 degrees ] ;
      • i.e. :
        • [ 90.00 degrees ] = arccos [ 0 / 3 ] ,

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

    and vice-versa .

• Let us now identify as to :
  • which lines in { Structure A } are orthogonal to which lines in { Strcture B } ,
    • as marked-off with an [ X ] in the table below .

    16 Lines in Structure B	* * *	16 Lines in Structure A
    		A 1	A 2	A 3	A 4	* *	A 5	A 6	A 7	A 8	* *	A 9	A 10	A 11	A 12	* *	A 13	A 14	A 15	A 16
    ********	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***
    B1	*	X	X	X	X	*	X		X		*		X		X	*	X			X
    B2	*	X	X	X	X	*			X	X	*	X	X			*		X	X
    B3	*	X	X	X	X	*	X	X			*			X	X	*		X	X
    B4	*	X	X	X	X	*		X		X	*	X		X		*	X			X
    ********	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***
    B5	*	X		X		*	X	X	X	X	*	X			X	*		X		X
    B6	*			X	X	*	X	X	X	X	*		X	X		*	X	X
    B7	*	X	X			*	X	X	X	X	*		X	X		*			X	X
    B8	*		X		X	*	X	X	X	X	*	X			X	*	X		X
    ********	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***
    B9	*		X		X	*	X			X	*	X	X	X	X	*		X		X
    B10	*	X	X			*		X	X		*	X	X	X	X	*	X	X
    B11	*			X	X	*		X	X		*	X	X	X	X	*			X	X
    B12	*	X		X		*	X			X	*	X	X	X	X	*	X		X
    ********	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***
    B13	*	X			X	*	X		X		*	X		X		*	X	X	X	X
    B14	*		X	X		*			X	X	*			X	X	*	X	X	X	X
    B15	*		X	X		*	X	X			*	X	X			*	X	X	X	X
    B16	*	X			X	*		X		X	*		X		X	*	X	X	X	X

• We notice here that :
  • { Lines A1 / A2 / A3 / A4 } are othogonal to { Lines B1 / B2 / B3 / B4 } .

  And this can happen when :

  • { Lines A1 / A2 / A3 / A4 } are in the formation of the { 4 Cross-Diagonal Lines of a cube }

    and

    { Lines B1 / B2 / B3 / B4 } are also in the formation of the { 4 Cross-Diagonal Lines of a cube } ;

    so that :

    • the 2 sets of { 4 Lines } will reside in a { 3-D Sub-Space } and its Kernal .

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What's Next :

• Let us identify the 60 pairs of matching cubes , next .

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go to the next Section : Section XXIX - Matching Cubes residing in a { 3-D Sub-Space } and its Kernal

go to the last Section : Section XXVII - Identifying { 4 Symmetric Lines in 3-D }

return to the Symmetry Through-Put HomePage

Original dated 2017-7-20