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 X - 12 Symmetric Lines in a { 4-D Vector Space }

Summary for the Section :

• In this Section , we shall take a look at :
  • the 12 Symmetric Lines in a { 4-Dimension Vector Space } .

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Recalling the 12 Lines for the 24 Squares :

• Let us now bring-in again :
  • the 12 Lines from the last Section , Section IX ,

    i.e. :

    • the 12 lines joining the center-points of each pair of opposite squares
      • found within the { 24-Squares Structure } arising from the { 4-Dimension Quasi-Cube } .

    Symmetric Line I.D.	* * *	Position Vector for 1st end-point of the Symmetric Line					* * *	Position Vector for 2nd end-point of the Symmetric Line
    	*	Vector I.D.	Co-ordinates				*	Vector I.D.	Co-ordinates
    *********************	*	***********	***	***	***	***	*	***********	***	***	***	***
    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

  For the purpose of facilitating our discussions to follow :

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

    • { Set E } consisting of :
      • { Symmetric Line 5 / Symmetric Line 6 / Symmetric Line 7 / Symmetric Line 8 } ;

    • { Set F } consisting of :
      • { Symmetric Line 9 / Symmetric Line 10 / Symmetric Line 11 / Symmetric Line 12 } .

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

  • FEATURE TWO :
    • any line in either { Set D } , { Set E } or { Set F }

      and

      any line from a different set

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

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Something sounds familiar :

• The two (2) Features above do sound familiar .

  As a matter of fact , we have encounter the same kind of features :

  • for the 8 Cross-Diagonal Lines of the { 4-Dimension Quasi-Cube } ,
    • as discussed back in Section VIII .

• But why { 12 Lines } here and only { 8 Lines } before ?

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What's Next :

• Let us try to reconcile the { 12 Lines vs. 8 Lines } difference , next .

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go to the next Section : Section XI - Reconciling the { 12 Lines vs. 8 Lines } Difference

go to the last Section : Section IX - 24 Squares arising from the { 4-D Quasi-Cube }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23