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 XV - A Special Issue on Swapping { 4 Orthogonal-Lines Structures }

Summary for the Section :

• In this Section , we shall take a first look at :
  • the phenomenon of Swapping { 4 Orthogonal-Lines Structures } ,
    • arising from the use of a different set of { 4 orthogonal unit-vectors basis } for our description of the 2 Structures .

  The Swapping Mechanism arising from switching { orthogonal unit-vector basis } as the bases of our description ;

  • will be fully explained later in Section XXI , Section XXII and Section XIII ,
    • when we discuss the swapping of 2 identical { 8-Dimension Structures } under cerain circumstances .

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The Combinatorics for the { Sets of 4 Vectors } :

• For a set of { 4 Lines mutually orthogonal to one-another } :
  • the number of permutations for { order of the 4 lines } is simply [ 24 ] ,
    • i.e. : [ 24 ] = 4 x 3 x 2 x 1 .

  And the number of permutations arising from :

  • the choice of either the { 1st Position Vector } or the { 2nd Position Vector } in each of the 4 Lines
    • to set up the { Set of 4 Vectors }
      • is simply [ 16 ] ,
        • i.e : [ 16 ] = 2 x 2 x 2 x 2 .

• Thus , the number of permutations for setting up the { Set of 4 Vectors }
  • from a set of { 4 Lines mutually orthogonal to one-another } is then :
    • [ 384 ] = 24 x 16 .

  And we simply had to be aware that :

  • the 384 different { Sets of 4 Vectors } do arise from the same { 4 Lines mutually orthogonal to one-another } ,
    • when we do the Tri-Symmetry Analysis .

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But will the results be different :

• But will the results be different :
  • Yes and No :
    • because there is another issue here , namely :
      • the Swapping of { 4 Orthogonal-Lines Structures } under certain special circumstances .

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The Special Issue of Swapping { 4 Orthogonal-Lines Structure } :

• Let us now do an example to illustrate the special issue of Swapping { 4 Orthogonal-Lines Structures } .

  Let us recall that :

  • the { orthogonal unit-vector basis } we originally used consists of the four (4) Unit-Vectors :
    • { Vector UV-1 } ,
    • { Vector UV-2 } ,
    • { Vector UV-3 } , and
    • { Vector UV-4 } .

  And for our discussion to follow , we shall now call this :

  • the { OLD SET of Orthogonal Unit-Vector Basis } .

• Let us now set up a new { orthogonal unit-vector basis } consisting of :
  • { Vector UV-1 } ,
  • { Vector UV-2 } ,
  • { a unit-vector in the opposite direction of [ Vector UV-3 ] } , and
  • { Vector UV-4 } .

  And for our discussion to follow , we shall now call this :

  • the { NEW SET of Orthogonal Unit-Vector Basis } .

• Let us now bring in the 12 Lines in { Set A } , { Set B } and { Set C }
  • based on the { OLD SET of Orthogonal Unit-Vector Basis } .

    Line I.D.	* * *	Position Vector for 1st end-point of the Line					* * *	Position Vector for 2nd end-point of the Line
    	*	Vector I.D.	Co-ordinates				*	Vector I.D.	Co-ordinates
    *********************	*	***********	***	***	***	***	*	***********	***	***	***	***
    Cross-Diagonal Line 1	*	Vector A1	+1	+1	+1	+1	*	Vector A2	-1	-1	-1	-1
    Cross-Diagonal Line 2	*	Vector A3	+1	+1	-1	-1	*	Vector A4	-1	-1	+1	+1
    Cross-Diagonal Line 3	*	Vector A5	+1	-1	-1	+1	*	Vector A6	-1	+1	+1	-1
    Cross-Diagonal Line 4	*	Vector A7	+1	-1	+1	-1	*	Vector A8	-1	+1	-1	+1
    *********************	*	***********	***	***	***	***	*	***********	***	***	***	***
    Cross-Diagonal Line 5	*	Vector B1	+1	+1	+1	-1	*	Vector B2	-1	-1	-1	+1
    Cross-Diagonal Line 6	*	Vector B3	+1	+1	-1	+1	*	Vector B4	-1	-1	+1	-1
    Cross-Diagonal Line 7	*	Vector B5	+1	-1	+1	+1	*	Vector B6	-1	+1	-1	-1
    Cross-Diagonal Line 8	*	Vector B7	-1	+1	+1	+1	*	Vector B8	+1	-1	-1	-1
    *********************	*	***********	***	***	***	***	*	***********	***	***	***	***
    Axial-Direction Line 1	*	Vector C1	+2	0	0	0	*	Vector C2	-2	0	0	0
    Axial-Direction Line 2	*	Vector C3	0	+2	0	0	*	Vector C4	0	-2	0	0
    Axial-Direction Line 3	*	Vector C5	0	0	+2	0	*	Vector C6	0	0	-2	0
    Axial-Direction Line 4	*	Vector C7	0	0	0	+2	*	Vector C8	0	0	0	-2

• Let us re-do the co-ordinates for the 12 Lines above :
  • using the { NEW SET of Orthogonal Unit-Vector Basis } as our bases for the description ,

    via :

    • changing the [ 3rd co-ordinate of the Position Vector ] :
      • from a [ +1 ] to a [ -1 ] , or
      • from a [ -1 ] to a [ +1 ] , or
      • from a [ +2 ] to a [ -2 ] , or
      • from a [ -2 ] to a [ +2 ] ;

      or

    • letting the [ 3rd co-ordinate of the Position Vector ] :
      • remain at [ 0 ] if this is the case .

  We then have this table below :

  • showing the resultant Position Vectors after the change .

    Line I.D.	* * *	Position Vector for 1st end-point of the Line					* * *	Position Vector for 2nd end-point of the Line
    	*	Vector I.D.	Co-ordinates				*	Vector I.D.	Co-ordinates
    *********************	*	***********	***	***	***	***	*	***********	***	***	***	***
    Cross-Diagonal Line 1	*	Vector B3	+1	+1	-1	+1	*	Vector B4	-1	-1	+1	-1
    Cross-Diagonal Line 2	*	Vector B1	+1	+1	+1	-1	*	Vector B2	-1	-1	-1	+1
    Cross-Diagonal Line 3	*	Vector B5	+1	-1	+1	+1	*	Vector B6	-1	+1	-1	-1
    Cross-Diagonal Line 4	*	Vector B8	+1	-1	-1	-1	*	Vector B7	-1	+1	+1	+1
    *********************	*	***********	***	***	***	***	*	***********	***	***	***	***
    Cross-Diagonal Line 5	*	Vector A3	+1	+1	-1	-1	*	Vector A4	-1	-1	+1	+1
    Cross-Diagonal Line 6	*	Vector A1	+1	+1	+1	+1	*	Vector A2	-1	-1	-1	-1
    Cross-Diagonal Line 7	*	Vector A5	+1	-1	-1	+1	*	Vector A6	-1	+1	+1	-1
    Cross-Diagonal Line 8	*	Vector A8	-1	+1	-1	+1	*	Vector A7	+1	-1	+1	-1
    *********************	*	***********	***	***	***	***	*	***********	***	***	***	***
    Axial-Direction Line 1	*	Vector C1	+2	0	0	0	*	Vector C2	-2	0	0	0
    Axial-Direction Line 2	*	Vector C3	0	+2	0	0	*	Vector C4	0	-2	0	0
    Axial-Direction Line 3	*	Vector C5	0	0	-2	0	*	Vector C6	0	0	+2	0
    Axial-Direction Line 4	*	Vector C7	0	0	0	+2	*	Vector C8	0	0	0	-2

• Let us summarize the results after the change here as follows :
  • When we use the { NEW SET of Orthogonal-Unit Vector Basis } to describe the 12 Lines :
    • the original { Cross-Diagonal Line 1 } , { Cross-Diagonal Line 2 } , { Cross-Diagonal Line 3 } , and { Cross-Diagonal Line 4 }
      • will now spot the 8 { B-Series Vectors } as their position-vectors ;

    • the original { Cross-Diagonal Line 5 } , { Cross-Diagonal Line 6 } , { Cross-Diagonal Line 7 } , and { Cross-Diagonal Line 8 }
      • will now spot the 8 { A-Series Vectors } as their position-vectors ;

    • the original { Axial-Direction Line 1 } , { Axial-Direction Line 2 } , { Axial-Direction Line 3 } , and { Axial-Direction Line 4 }
      • will now spot the same 8 { C-Series Vectors } as their position-vectors .

  In essense ,

  • [ Set A ] and [ Set B ] have swapped positions ,
    • while [ Set C ] retained the 4 orthogonal Axial-Directions .

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The Nature of Swapping Positions :

• This type of Swapping Phenomenon arises from the use of a different set of { orthogonal unit-vector basis }
  • for the description of the 2 Structures-in-question ;

  and

  • swapping may occur and swapping may not occur ,
    • depending on characteristics of the { orthogonal unit-vector basis } in use .

  We shall have a full explanation on this type of Swapping Phenomenon later in Part IV ,

  • when we look at the orientations of 2 identical { 8-Dimension Structures }
    • in Section XXI , Section XXII , and Section XIII .

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What's Next :

• Let us do the 96-triangles { 4-Dimension Quasi-Octahedron } , next .

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go to the next Section : Section XVI - The Octo-Symmetric Nature of the { 4-D Quasi-Octahedron }

go to the last Section : Section XIV - The Tri-Symmetry Analysis for { Set D / E / F }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23