Linking 6-D and 7-D

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 XXX - Linkage System for { 6-D vs. 7-D }

Summary for the Section :

In this Section , we shall again start-off with an { 8-Dimension Vector Space } and :
- go thru the mechanics of how to arrive at both { Structure A } and { Structur B }
  - arising from the { 6-Dimension Quasi-Cube } .

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Setting up the { 8-Dimensional Vector Space } :

Let us set up again , as we have done so earlier in Section XIX ,
- the 8 Physical Unit-Vectors orthogonal to one-another defining the { 8-Dimension Vector Space } ;
  - namely , the 8 physical unit-vectors :
    - { Vector PUV-1 } , { Vector PUV-2 } , { Vector PUV-3 } , { Vector PUV-4 } ,
      { Vector PUV-5 } , { Vector PUV-6 } , { Vector PUV-7 } and { Vector PUV-8 } ;
      
      respectively .

Let us also set up the two (2) { 8-Vectors System }'s for our use here in this Section :

{ SYSTEM ZERO } consisting of the 8 unit-vectors :
- { Vector UVZ-1 } , { Vector UVZ-2 } , { Vector UVZ-3 } , { Vector UVZ-4 } ,
  { Vector UVZ-5 } , { Vector UVZ-6 } , { Vector UVZ-7 } and { Vector UVZ-8 } , respectively ;
  
  as shown on-the-left in the table below .

{ SYSTEM H } consisting of the 8 vectors :

{ Vector H-1 } , { Vector H-2 } , { Vector H-3 } , { Vector H-4 } ,
{ Vector H-5 } , { Vector H-6 } , { Vector H-7 } and { Vector H-8 } , repectively ;

as shown on-the-right in the table below .

Vector I.D.	co-ordinates based on the 8 Physical Unit-Vectors									Vector I.D.	co-ordinates based on the 8 Physical Unit-Vectors
SYSTEM ZERO									* * *	SYSTEM H
Vector I.D.	P U V - 1	P U V - 2	P U V - 3	P U V - 4	P U V - 5	P U V - 6	P U V - 7	P U V - 8		Vector I.D.	P U V - 1	P U V - 2	P U V - 3	P U V - 4	P U V - 5	P U V - 6	P U V - 7	P U V - 8
************	***	***	***	***	***	***	***	***	*	************	***	***	***	***	***	***	***	***
Vector UVZ-1	+1	0	0	0	0	0	0	0	*	Vector H-1	-1	-1	+1	+1	0	0	0	0
Vector UVZ-2	0	+1	0	0	0	0	0	0	*	Vector H-2	-1	+1	-1	+1	0	0	0	0
Vector UVZ-3	0	0	+1	0	0	0	0	0	*	Vector H-3	-1	+1	+1	-1	0	0	0	0
Vector UVZ-4	0	0	0	+1	0	0	0	0	*	Vector H-4	0	0	0	0	-1	+1	+1	-1
Vector UVZ-5	0	0	0	0	+1	0	0	0	*	Vector H-5	0	0	0	0	+1	-1	+1	-1
Vector UVZ-6	0	0	0	0	0	+1	0	0	*	Vector H-6	0	0	0	0	+1	+1	-1	-1
Vector UVZ-7	0	0	0	0	0	0	+1	0	*	Vector H-7	+1	+1	+1	+1	-1	-1	-1	-1
Vector UVZ-8	0	0	0	0	0	0	0	+1	*	Vector H-8	+1	+1	+1	+1	+1	+1	+1	+1

And we take note here that :

the 8 unit-vectors in { SYSTEM ZERO } are orthogonal to one-another ,
- and is therefore a set of { 8 orthogonal unit-vectors basis } for the { 8-Dimension Vector Space } ;
the 8 vectors in { SYSTEM H } are orthogonal to one-another ;

and
- the size of { Vector H-1 / H-2 / H-3 / H-4 / H-5 / H-6 } is [ 2 ] ,
- the size of { Vector H-7 } and { Vector H-8 } is { square-root of [ 8 ] } .

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Choosing { Vector H-8 } as the Line-Of-Concern :

Let us now choose { Vector H-8 } as our Line-Of-Concern for the time-being and :

identify the 28 Lines orthogonal to { Vector H-8 } intersecting one-another at the angle of [ 70.53 degrees ] ,
- using { SYSTEM ZERO } as the core system-of-reference for our identifications ;
  - as shown on-the-left in the table below .

		U V Z - 1	U V Z - 2	U V Z - 3	U V Z - 4	U V Z - 5	U V Z - 6	U V Z - 7	U V Z - 8	*	U V Z - 1	U V Z - 2	U V Z - 3	U V Z - 4	U V Z - 5	U V Z - 6	U V Z - 7	U V Z - 8		H - 1	H - 2	H - 3	H - 4	H - 5	H - 6	H - 7	H - 8	*	H - 1	H - 2	H - 3	H - 4	H - 5	H - 6	H - 7	H - 8
Line I.D. for 28 Lines orthogonal to Vector H-8	* *	SYSTEM ZERO																	* *	SYSTEM H																	* *
		Co-ordinates for 1st Position Vector								*	Co-ordinates for 2nd Position Vector									Co-ordinates for 1st Position Vector								*	Co-ordinates for 2nd Position Vector
***********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*
Line P-1	*	-3	+1	+1	+1	+1	+1	+1	-3	*	+3	-1	-1	-1	-1	-1	-1	+3	*	+1	+1	+1	+1	+1	+1	0	0	*	-1	-1	-1	-1	-1	-1	0	0	*
Line P-2	*	-3	+1	+1	+1	+1	+1	-3	+1	*	+3	-1	-1	-1	-1	-1	+3	-1	*	+1	+1	+1	-1	-1	+1	0	0	*	-1	-1	-1	+1	+1	-1	0	0	*
Line P-3	*	-3	+1	+1	+1	+1	-3	+1	+1	*	+3	-1	-1	-1	-1	+3	-1	-1	*	+1	+1	+1	-1	+1	-1	0	0	*	-1	-1	-1	+1	-1	+1	0	0	*
Line P-4	*	-3	+1	+1	+1	-3	+1	+1	+1	*	+3	-1	-1	-1	+3	-1	-1	-1	*	+1	+1	+1	+1	-1	-1	0	0	*	-1	-1	-1	-1	+1	+1	0	0	*
Line P-5	*	+1	-3	+1	+1	+1	+1	+1	-3	*	-1	+3	-1	-1	-1	-1	-1	+3	*	+1	-1	-1	+1	+1	+1	0	0	*	-1	+1	+1	-1	-1	-1	0	0	*
Line P-6	*	+1	-3	+1	+1	+1	+1	-3	+1	*	-1	+3	-1	-1	-1	-1	+3	-1	*	+1	-1	-1	-1	-1	+1	0	0	*	-1	+1	+1	+1	+1	-1	0	0	*
Line P-7	*	+1	-3	+1	+1	+1	-3	+1	+1	*	-1	+3	-1	-1	-1	+3	-1	-1	*	+1	-1	-1	-1	+1	-1	0	0	*	-1	+1	+1	+1	-1	+1	0	0	*
Line P-8	*	+1	-3	+1	+1	-3	+1	+1	+1	*	-1	+3	-1	-1	+3	-1	-1	-1	*	+1	-1	-1	+1	-1	-1	0	0	*	-1	+1	+1	-1	+1	+1	0	0	*
Line P-9	*	+1	+1	-3	+1	+1	+1	+1	-3	*	-1	-1	+3	-1	-1	-1	-1	+3	*	-1	+1	-1	+1	+1	+1	0	0	*	+1	-1	+1	-1	-1	-1	0	0	*
Line P-10	*	+1	+1	-3	+1	+1	+1	-3	+1	*	-1	-1	+3	-1	-1	-1	+3	-1	*	-1	+1	-1	-1	-1	+1	0	0	*	+1	-1	+1	+1	+1	-1	0	0	*
Line P-11	*	+1	+1	-3	+1	+1	-3	+1	+1	*	-1	-1	+3	-1	-1	+3	-1	-1	*	-1	+1	-1	-1	+1	-1	0	0	*	+1	-1	+1	+1	-1	+1	0	0	*
Line P-12	*	+1	+1	-3	+1	-3	+1	+1	+1	*	-1	-1	+3	-1	+3	-1	-1	-1	*	-1	+1	-1	+1	-1	-1	0	0	*	+1	-1	+1	-1	+1	+1	0	0	*
Line P-13	*	+1	+1	+1	-3	+1	+1	+1	-3	*	-1	-1	-1	+3	-1	-1	-1	+3	*	-1	-1	+1	+1	+1	+1	0	0	*	+1	+1	-1	-1	-1	-1	0	0	*
Line P-14	*	+1	+1	+1	-3	+1	+1	-3	+1	*	-1	-1	-1	+3	-1	-1	+3	-1	*	-1	-1	+1	-1	-1	+1	0	0	*	+1	+1	-1	+1	+1	-1	0	0	*
Line P-15	*	+1	+1	+1	-3	+1	-3	+1	+1	*	-1	-1	-1	+3	-1	+3	-1	-1	*	-1	-1	+1	-1	+1	-1	0	0	*	+1	+1	-1	+1	-1	+1	0	0	*
Line P-16	*	+1	+1	+1	-3	-3	+1	+1	+1	*	-1	-1	-1	+3	+3	-1	-1	-1	*	-1	-1	+1	+1	-1	-1	0	0	*	+1	+1	-1	-1	+1	+1	0	0	*
***********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*
Line P-17	*	-3	-3	+1	+1	+1	+1	+1	+1	*	+3	+3	-1	-1	-1	-1	-1	-1	*	+2	0	0	0	0	0	-1	0	*	-2	0	0	0	0	0	+1	0	*
Line P-18	*	+1	+1	-3	-3	+1	+1	+1	+1	*	-1	-1	+3	+3	-1	-1	-1	-1	*	-2	0	0	0	0	0	-1	0	*	+2	0	0	0	0	0	+1	0	*
Line P-19	*	-3	+1	-3	+1	+1	+1	+1	+1	*	+3	-1	+3	-1	-1	-1	-1	-1	*	0	+2	0	0	0	0	-1	0	*	0	-2	0	0	0	0	+1	0	*
Line P-20	*	+1	-3	+1	-3	+1	+1	+1	+1	*	-1	+3	-1	+3	-1	-1	-1	-1	*	0	-2	0	0	0	0	-1	0	*	0	+2	0	0	0	0	+1	0	*
Line P-21	*	-3	+1	+1	-3	+1	+1	+1	+1	*	+3	-1	-1	+3	-1	-1	-1	-1	*	0	0	+2	0	0	0	-1	0	*	0	0	-2	0	0	0	+1	0	*
Line P-22	*	+1	-3	-3	+1	+1	+1	+1	+1	*	-1	+3	+3	-1	-1	-1	-1	-1	*	0	0	-2	0	0	0	-1	0	*	0	0	+2	0	0	0	+1	0	*
Line P-23	*	+1	+1	+1	+1	+1	-3	-3	+1	*	-1	-1	-1	-1	-1	+3	+3	-1	*	0	0	0	-2	0	0	+1	0	*	0	0	0	+2	0	0	-1	0	*
Line P-24	*	+1	+1	+1	+1	-3	+1	+1	-3	*	-1	-1	-1	-1	+3	-1	-1	+3	*	0	0	0	+2	0	0	+1	0	*	0	0	0	-2	0	0	-1	0	*
Line P-25	*	+1	+1	+1	+1	-3	+1	-3	+1	*	-1	-1	-1	-1	+3	-1	+3	-1	*	0	0	0	0	-2	0	+1	0	*	0	0	0	0	+2	0	-1	0	*
Line P-26	*	+1	+1	+1	+1	+1	-3	+1	-3	*	-1	-1	-1	-1	-1	+3	-1	+3	*	0	0	0	0	+2	0	+1	0	*	0	0	0	0	-2	0	-1	0	*
Line P-27	*	+1	+1	+1	+1	-3	-3	+1	+1	*	-1	-1	-1	-1	+3	+3	-1	-1	*	0	0	0	0	0	-2	+1	0	*	0	0	0	0	0	+2	-1	0	*
Line P-28	*	+1	+1	+1	+1	+1	+1	-3	-3	*	-1	-1	-1	-1	-1	-1	+3	+3	*	0	0	0	0	0	+2	+1	0	*	0	0	0	0	0	-2	-1	0	*

Let us now convert ,
- for each of the 28 Lines in the table above :
  - the co-ordinates for the { 1st Position Vector } and the { 2nd Position Vector } under { SYSTEM ZERO }
    
    into
  - co-ordinates for the same { 1st Position Vector } and the same { 2nd Position Vector } under { SYSTEM H } ;
    - as shown on-the-right in the table above .
And we take note here that :
- the 8 co-ordinates under { SYSTEM H } on-the-right is simply given by :
  - the [ Dot Product ] of the { Position Vector in-question } and { Vector H-1 } ,
  - the [ Dot Product ] of the { Position Vector in-question } and { Vector H-2 } ,
  - the [ Dot Product ] of the { Position Vector in-question } and { Vector H-3 } ,
  - the [ Dot Product ] of the { Position Vector in-question } and { Vector H-4 } ,
  - the [ Dot Product ] of the { Position Vector in-question } and { Vector H-5 } ,
  - the [ Dot Product ] of the { Position Vector in-question } and { Vector H-6 } ,
  - the [ Dot Product ] of the { Position Vector in-question } and { Vector H-7 } ,
  - the [ Dot Product ] of the { Position Vector in-question } and { Vector H-8 } ;
    
    respectively .
And the only observation we wish to make at this point is that :
- the { 16 Lines from [ Line P-1 ] thru [ Line P-16 ] } are orthogonal to both { Vector H-7 } and { Vector H-8 }
  - and therefore reside in a { 6-Dimension Sub-Space } .

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Choosing { Vector H-7 } as the Line-Of-Concern :

Let us now choose { Vector H-7 } as our Line-Of-Concern for the time-being and :

identify the 28 Lines orthogonal to { Vector H-7 } intersecting one-another at the angle of [ 70.53 degrees ] ,
- using { SYSTEM ZERO } as the core system-of-reference for our identifications ;
  - as shown on-the-left in the table below .

		U V Z - 1	U V Z - 2	U V Z - 3	U V Z - 4	U V Z - 5	U V Z - 6	U V Z - 7	U V Z - 8	*	U V Z - 1	U V Z - 2	U V Z - 3	U V Z - 4	U V Z - 5	U V Z - 6	U V Z - 7	U V Z - 8		H - 1	H - 2	H - 3	H - 4	H - 5	H - 6	H - 7	H - 8	*	H - 1	H - 2	H - 3	H - 4	H - 5	H - 6	H - 7	H - 8
Line I.D. for 28 Lines orthogonal to Vector H-7	* *	SYSTEM ZERO																	* *	SYSTEM H																	* *
		Co-ordinates for 1st Position Vector								*	Co-ordinates for 2nd Position Vector									Co-ordinates for 1st Position Vector								*	Co-ordinates for 2nd Position Vector
***********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*
Line Q-1	*	-3	+1	+1	+1	-1	-1	-1	+3	*	+3	-1	-1	-1	+1	+1	+1	-3	*	+1	+1	+1	-1	-1	-1	0	0	*	-1	-1	-1	+1	+1	+1	0	0	*
Line Q-2	*	-3	+1	+1	+1	-1	-1	+3	-1	*	+3	-1	-1	-1	+1	+1	-3	+1	*	+1	+1	+1	+1	+1	-1	0	0	*	-1	-1	-1	-1	-1	+1	0	0	*
Line Q-3	*	-3	+1	+1	+1	-1	+3	-1	-1	*	+3	-1	-1	-1	+1	-3	+1	+1	*	+1	+1	+1	+1	-1	+1	0	0	*	-1	-1	-1	-1	+1	-1	0	0	*
Line Q-4	*	-3	+1	+1	+1	+3	-1	-1	-1	*	+3	-1	-1	-1	-3	+1	+1	+1	*	+1	+1	+1	-1	+1	+1	0	0	*	-1	-1	-1	+1	-1	-1	0	0	*
Line Q-5	*	+1	-3	+1	+1	-1	-1	-1	+3	*	-1	+3	-1	-1	+1	+1	+1	-3	*	+1	-1	-1	-1	-1	-1	0	0	*	-1	+1	+1	+1	+1	+1	0	0	*
Line Q-6	*	+1	-3	+1	+1	-1	-1	+3	-1	*	-1	+3	-1	-1	+1	+1	-3	+1	*	+1	-1	-1	+1	+1	-1	0	0	*	-1	+1	+1	-1	-1	+1	0	0	*
Line Q-7	*	+1	-3	+1	+1	-1	+3	-1	-1	*	-1	+3	-1	-1	+1	-3	+1	+1	*	+1	-1	-1	+1	-1	+1	0	0	*	-1	+1	+1	-1	+1	-1	0	0	*
Line Q-8	*	+1	-3	+1	+1	+3	-1	-1	-1	*	-1	+3	-1	-1	-3	+1	+1	+1	*	+1	-1	-1	-1	+1	+1	0	0	*	-1	+1	+1	+1	-1	-1	0	0	*
Line Q-9	*	+1	+1	-3	+1	-1	-1	-1	+3	*	-1	-1	+3	-1	+1	+1	+1	-3	*	-1	+1	-1	-1	-1	-1	0	0	*	+1	-1	+1	+1	+1	+1	0	0	*
Line Q-10	*	+1	+1	-3	+1	-1	-1	+3	-1	*	-1	-1	+3	-1	+1	+1	-3	+1	*	-1	+1	-1	+1	+1	-1	0	0	*	+1	-1	+1	-1	-1	+1	0	0	*
Line Q-11	*	+1	+1	-3	+1	-1	+3	-1	-1	*	-1	-1	+3	-1	+1	-3	+1	+1	*	-1	+1	-1	+1	-1	+1	0	0	*	+1	-1	+1	-1	+1	-1	0	0	*
Line Q-12	*	+1	+1	-3	+1	+3	-1	-1	-1	*	-1	-1	+3	-1	-3	+1	+1	+1	*	-1	+1	-1	-1	+1	+1	0	0	*	+1	-1	+1	+1	-1	-1	0	0	*
Line Q-13	*	+1	+1	+1	-3	-1	-1	-1	+3	*	-1	-1	-1	+3	+1	+1	+1	-3	*	-1	-1	+1	-1	-1	-1	0	0	*	+1	+1	-1	+1	+1	+1	0	0	*
Line Q-14	*	+1	+1	+1	-3	-1	-1	+3	-1	*	-1	-1	-1	+3	+1	+1	-3	+1	*	-1	-1	+1	+1	+1	-1	0	0	*	+1	+1	-1	-1	-1	+1	0	0	*
Line Q-15	*	+1	+1	+1	-3	-1	+3	-1	-1	*	-1	-1	-1	+3	+1	-3	+1	+1	*	-1	-1	+1	+1	-1	+1	0	0	*	+1	+1	-1	-1	+1	-1	0	0	*
Line Q-16	*	+1	+1	+1	-3	+3	-1	-1	-1	*	-1	-1	-1	+3	-3	+1	+1	+1	*	-1	-1	+1	-1	+1	+1	0	0	*	+1	+1	-1	+1	-1	-1	0	0	*
***********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*
Line Q-17	*	-3	-3	+1	+1	-1	-1	-1	-1	*	+3	+3	-1	-1	+1	+1	+1	+1	*	+2	0	0	0	0	0	0	-1	*	-2	0	0	0	0	0	0	+1	*
Line Q-18	*	+1	+1	-3	-3	-1	-1	-1	-1	*	-1	-1	+3	+3	+1	+1	+1	+1	*	-2	0	0	0	0	0	0	-1	*	+2	0	0	0	0	0	0	+1	*
Line Q-19	*	-3	+1	-3	+1	-1	-1	-1	-1	*	+3	-1	+3	-1	+1	+1	+1	+1	*	0	+2	0	0	0	0	0	-1	*	0	-2	0	0	0	0	0	+1	*
Line Q-20	*	+1	-3	+1	-3	-1	-1	-1	-1	*	-1	+3	-1	+3	+1	+1	+1	+1	*	0	-2	0	0	0	0	0	-1	*	0	+2	0	0	0	0	0	+1	*
Line Q-21	*	-3	+1	+1	-3	-1	-1	-1	-1	*	+3	-1	-1	+3	+1	+1	+1	+1	*	0	0	+2	0	0	0	0	-1	*	0	0	-2	0	0	0	0	+1	*
Line Q-22	*	+1	-3	-3	+1	-1	-1	-1	-1	*	-1	+3	+3	-1	+1	+1	+1	+1	*	0	0	-2	0	0	0	0	-1	*	0	0	+2	0	0	0	0	+1	*
Line Q-23	*	-1	-1	-1	-1	+1	-3	-3	+1	*	+1	+1	+1	+1	-1	+3	+3	-1	*	0	0	0	-2	0	0	0	-1	*	0	0	0	+2	0	0	0	+1	*
Line Q-24	*	-1	-1	-1	-1	-3	+1	+1	-3	*	+1	+1	+1	+1	+3	-1	-1	+3	*	0	0	0	+2	0	0	0	-1	*	0	0	0	-2	0	0	0	+1	*
Line Q-25	*	-1	-1	-1	-1	-3	+1	-3	+1	*	+1	+1	+1	+1	+3	-1	+3	-1	*	0	0	0	0	-2	0	0	-1	*	0	0	0	0	+2	0	0	+1	*
Line Q-26	*	-1	-1	-1	-1	+1	-3	+1	-3	*	+1	+1	+1	+1	-1	+3	-1	+3	*	0	0	0	0	+2	0	0	-1	*	0	0	0	0	-2	0	0	+1	*
Line Q-27	*	-1	-1	-1	-1	-3	-3	+1	+1	*	+1	+1	+1	+1	+3	+3	-1	-1	*	0	0	0	0	0	-2	0	-1	*	0	0	0	0	0	+2	0	+1	*
Line Q-28	*	-1	-1	-1	-1	+1	+1	-3	-3	*	+1	+1	+1	+1	-1	-1	+3	+3	*	0	0	0	0	0	+2	0	-1	*	0	0	0	0	0	-2	0	+1	*

Let us now convert ,
- for each of the 28 Lines in the table above :
  - the co-ordinates for the { 1st Position Vector } and the { 2nd Position Vector } under { SYSTEM ZERO }
    
    into
  - co-ordinates for the same { 1st Position Vector } and the same { 2nd Position Vector } under { SYSTEM H } ;
    - as shown on-the-right in the table above .
And the only observation we wish to make at this point is that :
- the { 16 Lines from [ Line Q-1 ] thru [ Line Q-16 ] } are orthogonal to both { Vector H-7 } and { Vector H-8 }
  - and therefore reside in a { 6-Dimension Sub-Space } .

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Summarizing :

Let us quickly summarize our findings here :
- Using { Vector H-8 } as our Line-Of-Concern for the time-being ,
  - we were able to identify a set 16 Lines intersecting one-another at the angle of [ 70.53 degrees ]
    - residing in a { 6-Dimension Sub-Space } orthogonal to both { Vector H-7 } and { Vector H-8 } ;
- Using { Vector H-7 } as our Line-Of-Concern for the time-being ,
  - we were able to identify another set of 16 Lines intersecting one-another at the angle of [ 70.53 degrees ]
    - residing in the same { 6-Dimension Sub-Space } orthogonal to both { Vector H-7 } and { Vector H-8 } .
But the key observation we wish to make here is that :
- IF :
  - we take { Vector H-1 / Vector H-2 / Vector H-3 / Vector H-4 / Vector H-5 / Vector H-6 }
    - as the 6 Axial-Directions for the { 6 Dimension Sub-Space } ;
  THEN :
  - the 32 Lines consisting of :
    - the { 16 Lines from [ Line P-1 ] thru [ Line P-16 ] } and the { 16 Lines from [ Line Q-1 ] thru [ Line Q-16 ] }
      - would become the 32 Cross-Diagonal Lines of a { 6-Dimension Quasi-Cube } ;
    and
  - the { 16 Lines from [ Line P-1 ] thru [ Line P-16 ] } would be in the formation of { Structure A } , and
    
    the { 16 Lines from [ Line Q-1 ] thru [ Line Q-16 ] } would be in the formation of { Structure B } ;
    - with { Structure A } and { Structure B } here being the 2 identical { 16-Lines Structures } we did come up with ,
      - when did our specific split-up of the { 6-Dimension Quasi-Cube } back in Section XXIV .
Isn't that interesting !

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What's Next :

This then close out Part V of this paper ,
- but we do provide further thoughts on { Symmetric Lines in 6-D and 7-D }
  - in Appendix A for interested readers .

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. )

Top Of Page

Section XXX - Linkage System for { 6-D vs. 7-D }

Summary for the Section :

go to Top Of Page

Setting up the { 8-Dimensional Vector Space } :

go to Top Of Page

Choosing { Vector H-8 } as the Line-Of-Concern :

go to Top Of Page

Choosing { Vector H-7 } as the Line-Of-Concern :

go to Top Of Page

Summarizing :

go to Top Of Page

What's Next :

go to Top Of Page

go to the next Section : Section XXXI - Concluding Remarks

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

return to the Symmetry Through-put HomePage

Original dated 2017-7-23

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. )

Top Of Page

Section XXX - Linkage System for { 6-D vs. 7-D }

Summary for the Section :

go to Top Of Page

Setting up the { 8-Dimensional Vector Space } :

go to Top Of Page

Choosing { Vector H-8 } as the Line-Of-Concern :

go to Top Of Page

Choosing { Vector H-7 } as the Line-Of-Concern :

go to Top Of Page

Summarizing :

go to Top Of Page

What's Next :

go to Top Of Page

go to the next Section : Section XXXI - Concluding Remarks

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

return to the Symmetry Through-put HomePage

Original dated 2017-7-23

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