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 XI - Reconciling the { 12 Lines vs. 8 Lines } Difference

Summary for the Section :

• In this Section , we shall reconcile :
  • the { 12 Lines vs. 8 Lines } Difference .

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Recalling the 8 Cross-Diagonal Lines of the { 4-D Quasi-Cube } :

• Let us now bring-in again the 8 Cross-Diagonal Lines from Section VIII ,
  • and add to it 4 more lines :
    • 4 Lines in the 4 Axial-Directions for the { 4-Dimension Vector Space } ;

    as shown below :

    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

  For the purpose of facilitating our discussions to follow :

  • let us re-group the 12 Lines above into 3 sets of [ 4 lines ] each :
    • { Set A } consisting of :
      • { Cross-Diagonal 1 / Cross-Diagonal Line 2 / Cross-Diagonal Line 3 / Cross-Diagonal Line 4 } ;

    • { Set B } consisting of :
      • { Cross-Diagonal 5 / Cross-Diagonal Line 6 / Cross-Diagonal Line 7 / Cross-Diagonal Line 8 } ;

    • { Set C } consisting of :
      • { Axial-Direction Line 1 / Axial-Direction Line 2 / Axial-Direction Line 3 / Axial-Direction Line 4 } .

• Let us now take a look at some special features for the { 12 Lines } :
  • FEATURE ONE :
    • { Set A } , { Set B } and { Set C } do all qualify as :
      • a set of { 4 lines mutually orthogonal to one-another } .

  • FEATURE TWO :
    • any line in either { Set A } , { Set B } or { Set C }

      and

      any line from a different set

      • will intersect one-another at an angle of [ 60 degrees ] .

• Isn't that wonderful ,
  • same features for { Set A / B / C } here and { Set D / E / F } from the last Section .

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Listing out the { 12 Lines vs. 12 Lines } :

• Before leaving this Section , let us simply list-out :
  • the { 12 Lines above } and the { 12 Symmetric Lines } from the last Section
    • in a single table , as shown below .

    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
    *********************	*	***********	***	***	***	***	*	***********	***	***	***	***
    Symmetric Line 1	*	Vector D1	+1	+1	0	0	*	Vector D2	-1	-1	0	0
    Symmetric Line 2	*	Vector D3	+1	-1	0	0	*	Vector D4	-1	+1	0	0
    Symmetric Line 3	*	Vector D5	0	0	-1	+1	*	Vector D6	0	0	+1	-1
    Symmetric Line 4	*	Vector D7	0	0	+1	+1	*	Vector D8	0	0	-1	-1
    *********************	*	***********	***	***	***	***	*	***********	***	***	***	***
    Symmetric Line 5	*	Vector E1	+1	0	0	-1	*	Vector E2	-1	0	0	+1
    Symmetric Line 6	*	Vector E3	+1	0	0	+1	*	Vector E4	-1	0	0	-1
    Symmetric Line 7	*	Vector E5	0	+1	+1	0	*	Vector E6	0	-1	-1	0
    Symmetric Line 8	*	Vector E7	0	+1	-1	0	*	Vector E8	0	-1	+1	0
    *********************	*	***********	***	***	***	***	*	***********	***	***	***	***
    Symmetric Line 9	*	Vector F1	+1	0	-1	0	*	Vector F2	-1	0	+1	0
    Symmetric Line 10	*	Vector F3	+1	0	+1	0	*	Vector F4	-1	0	-1	0
    Symmetric Line 11	*	Vector F5	0	+1	0	+1	*	Vector F6	0	-1	0	-1
    Symmetric Line 12	*	Vector F7	0	+1	0	-1	*	Vector F8	0	-1	0	+1

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What's Next :

• It is our view here that :
  • { Set A } , { Set B } and { Set C } may be Tri-Symmetric , and

  • { Set D } , { Set E } and { Set F } may also be Tri-Symmetric .

  In the next Section ,

  • we shall do the preliminary preparartion work for our Analysis on Tri-Symmetry :
    • on { Set A / Set B / Set C } and { Set D / Set E / Set F } respectively in the 2 Sections to follow thereafter .

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go to the next Section : Section XII - Setting up for an Analysis on Tri-Symmetry

go to the last Section : Section X - 12 Symmetric Lines in a { 4-D Vector Space }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23