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 XVIII - 2 Identical Structures arising from the { 8-D Quasi-Cube }

Summary for the Section :

• In this Section , we shall split the 128 Cross-Diagonal Lines of the { 8-Dimension Quasi-Cube }
  • into 2 identical Structures , each made-up-of { 64 Cross-Diagonal Lines } .

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Recalling the Classifications for the 128 [ Cross-Diagonal Lines ] :

• Let us recall that we have 128 [ Cross-Diagonal Lines ] in our { 8-Dimension Quasi-Cube } and :
  • we have classified these into 5 categories / classes .

    ( see Section V for the definition details ) .

    Classification	Number of Lines in Class	* * *	Structure A	Structure B
    **************	**************	*	**************	**************
    Class 8-0 / 0-8	1	*	1
    Class 7-1 / 1-7	8	*		8
    Class 6-2 / 2-6	28	*	28
    Class 5-3 / 3-5	56	*		56
    Class 4-4 / 4-4	35	*	35
    **************	**************	*	**************	**************
    Total	128	*	64	64

• Let us now construct 2 Structures :
  • { Structure A } being made up of :
    • the 64 Cross-Diagonal Lines in { Classs 8-0 / 0-8 } , { Class 6-2 / 2-6 } and { Class 4-4 / 4-4 } ;

  • { Structure B } being made up of :
    • the 64 Cross-Diagonal Lines in { Classs 7-1 / 1-7 } and { Class 5-3 / 3-5 } ;

    as shown in the table immediately above , on the right-hand side .

  This is an { EVEN vs. ODD } type Split :

  • for { Structure A } :
    • the two (2) Position Vectors for the single line in { Class 8-0 / 0-8 } is characterised by :
      • the 1st Position Vector having { 8 [ +1 ]'s } in its [ 8 co-ordinates ] ,

        and

      • the 2nd Position Vector having { 8 [ -1 ]'s } in its [ 8 co-ordinates ] ;

    • the two (2) Position Vectors for the remaining 63 Lines in { Class 6-2 / 2-6 } and { Class 4-4 / 4-4 } will have :
      • an { even number of [ +1 ]'s } and correspondingly an { even number of [ -1 ] } in its [ 8 co-ordinates ] .

  whereas ,

  • for { Structure B } :
    • the two (2) Position Vectors for each of the 64 Lines in { Class 7-1 / 1-7 } and { Class 5-3 / 3-5 } will have :
      • an { odd number of [ +1 ] 's } and correspondingly an { odd number of [ -1 ] } in its [ 8 co-ordinates ] .

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A First Look at the 2 Structures :

• Let us take a first look at these 2 structures by :
  • first bringing in some sample Lines in { Structure A } ,

  Sample 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 Zero	Class 8-0 / 0-8	*	Vector Z-1	+1	+1	+1	+1	+1	+1	+1	+1	*	Vector Z-2	-1	-1	-1	-1	-1	-1	-1	-1
  Sample Line L	Class 6-2 / 2-6	*	Vector L-1	+1	+1	+1	+1	+1	+1	-1	-1	*	Vector L-2	-1	-1	-1	-1	-1	-1	+1	+1
  Sample Line M	Class 6-2 / 2-6	*	Vector M-1	+1	+1	+1	-1	+1	+1	+1	-1	*	Vector M-2	-1	-1	-1	+1	-1	-1	-1	+1
  Sample Line N	Class 6-2 / 2-6	*	Vector N-1	+1	+1	-1	-1	+1	+1	+1	+1	*	Vector N-2	-1	-1	+1	+1	-1	-1	-1	-1
  Line D-1	Class 4-4 / 4-4	*	Vector D-1	+1	+1	+1	+1	-1	-1	-1	-1	*	Vector D-2	-1	-1	-1	-1	+1	+1	+1	+1
  Line D-18	Class 4-4 / 4-4	*	Vector D-35	+1	+1	-1	-1	-1	-1	+1	+1	*	Vector D-36	-1	-1	+1	+1	+1	+1	-1	-1
  Line D-19	Class 4-4 / 4-4	*	Vector D-37	+1	+1	-1	-1	+1	+1	-1	-1	*	Vector D-38	-1	-1	+1	+1	-1	-1	+1	+1

  • and secondly bringing in some sample Lines in { Structure B } :

  Sample 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
  **************	**************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
  Sample Line H	Class 7-1 / 1-7	*	Vector H-1	+1	+1	+1	+1	+1	+1	+1	-1	*	Vector H-2	-1	-1	-1	-1	-1	-1	-1	+1
  Sample Line J	Class 7-1 / 1-7	*	Vector J-1	+1	+1	+1	-1	+1	+1	+1	+1	*	Vector J-2	-1	-1	-1	+1	-1	-1	-1	-1
  Sample Line P	Class 5-3 / 3-5	*	Vector P-1	+1	+1	+1	+1	+1	-1	-1	-1	*	Vector P-2	-1	-1	-1	-1	-1	+1	+1	+1
  Sample Line Q	Class 5-3 / 3-5	*	Vector Q-1	+1	+1	-1	+1	+1	-1	-1	+1	*	Vector Q-2	-1	-1	+1	-1	-1	+1	+1	-1
  Sample Line R	Class 5-3 / 3-5	*	Vector R-1	+1	-1	-1	+1	+1	-1	+1	+1	*	Vector R-2	-1	+1	+1	-1	-1	+1	-1	-1
  Sample Line S	Class 5-3 / 3-5	*	Vector S-1	-1	-1	-1	+1	+1	+1	+1	+1	*	Vector R-2	+1	+1	+1	-1	-1	-1	-1	-1

• Let us now make these preliminary statements to be confirmed in later Sections ,
  • for a first feel on the 2 Structures .

  STATEMENT ONE :

  • Any pair of Lines in { Structure A } will intersect one-another :
    • either at an angle of [ 60 degrees ] or [ 90 degrees ] ;

      i.e. :

      • arccos [ 2 / 4 ] = 60 degrees ,
      • arccos [ 0 / 4 ] = 90 degrees .

  STATEMENT TWO :

  • Any pair of Lines in { Structure B } will intersect one-another :
    • either at an angle of [ 60 degrees ] or [ 90 degrees ] ;

      i.e. :

      • arccos [ 2 / 4 ] = 60 degrees ,
      • arccos [ 0 / 4 ] = 90 degrees .

  STATEMENT THREE :

  • Any Line in { Structure A } will intersect any Line in { Structure B } :
    • either at an angle of [ 41.41 degrees ] or [ 75.52 degrees ] ;

      i.e. :

      • arccos [ 3 / 4 ] = 41.41 degrees ,
      • arccos [ 1 / 4 ] = 75.52 degrees .

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The Questions needing an Answer :

• We now have three (3) questions that need to be answered :
  • Question One :
    • Are the 2 { 64-Lines Structures } identical in shape ?

  • Question Two :
    • Are the orientations of the 2 Structures symmetrical ?
      • and if so , how so ?

  • Question Three :
    • Will we be able to confirm our 3 statements just-made above ?
      • And what is the significance here ?

  TOUGH QUESTIONS at first glance , of course .

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What's Next :

• We shall be attempting to answer the 3 questions above , but first :
  • let us set up the four (4) sets of { 8-D orthogonal unit-vector basis } necessary for our analysis , next .

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go to the next Section : Section XIX - Setting up 4 Sets of { 8-D Orthogonal Unit-Vector Basis } for our Analysis

go to the last Section : Section XVII - Structural Analysis for the { 4-D Quasi-Octahedron }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23