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 XXV - Identifying { 10 Symmetric Lines in 5-D }

Summary for the Section :

• In this Section , we shall identify :
  • the { 10 Symmetric Lines } residing in a { 5-Dimension Vector Space } ,
    • intersecting one-another at the angle of [ 70.53 degrees ] .

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Selecting an { Anchor Line } :

• Let us now select an { Anchor Line } from { Structure B } ,
  • for the purpose of finding out which 10 Lines in { Structure A } are orthogonal to the { Anchor Line } ,
    • and would therefore reside in a { 5-Dimension Sub-Space } .

  And for the sake of convenience ,

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

  Anchor Line 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

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Identifying { 10 Symmetric Lines in 5-D } :

• Let us now identify the { 10 Symmetric Lines in 5-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
  **************	**************	*	***********	***	***	***	***	***	***	*	***********	***	***	***	***	***	***
  Anchor 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 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

• Let us now also bring back the 16 Lines in { Strcuture A }
  • for a quick comparison as to which Lines did not qualify ,
    • as marked-off in white-color below .

  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 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 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

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The number of choices for the { Anchor Line } :

• We take note here that :
  • the number of choices for the { Anchor Line } is simply [ 16 ] ,
    i.e. :
    • any of the 16 Lines in { Structure B } can be selected as the { Anchor Line } .

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What's Next :

• YES , let us do the { 6 Symmetric Lines in 4-D } , next .

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go to the next Section : Section XXVI - Identifying { 6 Symmetric Lines in 4-D }

go to the last Section : Section XXIV - 2 Identical { 6-D Structures } arising from the { 6-D Quasi-Cube }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23