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 XXVI - Identifying { 6 Symmetric Lines in 4-D }

Summary for the Section :

• In this Section , we shall identify :
  • 6 Symmetric Lines residing in a { 4-Dimension Vector Space } ,
    • intersecting one-another at the angle of [ 70.53 degrees ] .

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Selecting the 2 { Anchor Lines } :

• Let us now select a pair of { Anchor Lines } from { Structure B } ,
  • for the purpose of finding out which 6 Lines in { Structure A } are orthogonal to the 2 { Anchor Lines } ,
    • and would therefore reside in a { 4-Dimension Sub-Space } .

  And for the sake of convenience ,

  • let us now select { Line R-1 } and { Line R-10 } in { Structure B } as our { 2 Anchor Lines } ,
    • for our analysis to follow .

  2 Anchor Lines from { 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 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-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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Creating 2 Orthogonal Lines from the 2 Anchore Lines :

• We take note here that :
  • the 2 Anchor Lines from { Structure B } do intersect one-another at the angle of [ 70.53 degrees ] ,
    • and would therefore reside in a 2-Dimension Plane .

  For our analytic convenience , let us now create :

  • 2 orthogonal lines in the same plane as the { 2 Anchor Lines } ,
    • namely { Line T-1 }and { Line T-2 } respectively ;
      • as shown in this table below .

  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 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-10	Class 3-3 / 3-3	*	Vector R-19	+1	-1	+1	-1	+1	-1	*	Vector R-20	-1	+1	-1	+1	-1	+1
  **************	**************	*	***********	***	***	***	***	***	***	*	***********	***	***	***	***	***	***
  Line T-1	R-1 plus R-10	*	Vector T-1	+2	0	+2	-2	0	-2	*	Vector T2	-2	0	-2	+2	0	+2
  Line T-2	R-1 minus R-10	*	Vector T-3	0	+2	0	0	-2	0	*	Vector T-4	0	-2	0	0	+2	0

  We then have :

  • [ Vector T-1 ] = [ Vector R-1 ] plus [ Vector R-19 ]
  • [ Vector T-2 ] = [ Vector R-2 ] plus [ Vector R-20 ]
  • [ Vector T-3 ] = [ Vector R-1 ] minus [ Vector R-19 ]
  • [ Vector T-4 ] = [ Vector R-2 ] minus [ Vector R-20 ]

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Identifying the { 6 Symmetric Lines in 4-D } :

• Let us now identify the { 6 Symmetric Lines in 4-D } ,
  • as shown in this table below .

  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 T-1	R-1 plus R-10	*	Vector T-1	+2	0	+2	-2	0	-2	*	Vector T2	-2	0	-2	+2	0	+2
  Line T-2	R-1 minus R-10	*	Vector T-3	0	+2	0	0	-2	0	*	Vector T-4	0	-2	0	0	+2	0
  **************	**************	*	***********	***	***	***	***	***	***	*	***********	***	***	***	***	***	***
  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-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-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-15	Class 4-2 / 2-4	*	Vector Q-29	-1	+1	+1	+1	+1	-1	*	Vector Q-30	+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-10	Class 4-2 / 2-4	*	Vector Q-19	+1	+1	-1	+1	+1	-1	*	Vector Q-20	-1	-1	+1	-1	-1	+1

• Let us now also bring back the { 10 Symmetric Lines in 5-D } ,
  • for a quick comparison as to which lines did not qualify ,
    • as marked-off in white-color below .

  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 T-1	R-1 plus R-10	*	Vector T-1	+2	0	+2	-2	0	-2	*	Vector T2	-2	0	-2	+2	0	+2
  Line T-2	R-1 minus R-10	*	Vector T-3	0	+2	0	0	-2	0	*	Vector T-4	0	-2	0	0	+2	0
  **************	**************	*	***********	***	***	***	***	***	***	*	***********	***	***	***	***	***	***
  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-15	Class 4-2 / 2-4	*	Vector Q-29	-1	+1	+1	+1	+1	-1	*	Vector Q-30	+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-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-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-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-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-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-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-3	Class 4-2 / 2-4	*	Vector Q-5	+1	+1	-1	-1	+1	+1	*	Vector Q-6	-1	-1	+1	+1	-1	-1

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The number of choices for the { 2nd Anchor Line } :

• We take note here that :
  • once we have chosen a { Line in Structure B } as the { 1st Anchor Line } ,
    • for the purpose of identifying the { 10 Symmetric Lines in 5-D } ;

    we still have a choice of [ 15 ] for the selection of the { 2nd Anchor Line } ,

    i.e.:

    • the choice of any of the 15 remaining lines in { Structure B } .

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What's Next :

• YES , let us do the { 4 Symmetric Lines in 3-D } , next .

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

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

return to the Symmetry Through-put HomePage

Original dated 2017-7-23