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 VIII - Setting up the 8 Cross-Diagonal Lines of the { 4-D Quasi-Cube }

Summary for the Section :

• In this Part III of the paper ,
  • we shall be concentrating our effort on { Symmetry in 4-Dimension } .

  And in this Section , we shall set up :

  • the { 4-Dimension Unit-Vector Basis } defining the { 4-Dimension Vector Space } and correspondingly :
    • the { 4-Dimension Quasi-Cube } ;

    for our use throughout this Part III of the paper .

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Setting up the { 4-Dimension Vector Space } :

• Let us now set up four (4) unit-vectors in a { 4-Dimension Vector Space } ,
  • so that :
    • the 4 unit-vectors are orthogonal to one-another

      and therefore :

      • form a set { 4-Dimension Unit-Vector Basis } .

  For identification purposes , we shall name these 4 Unit-Vectors :

  • { Vector UV-1 } ,
  • { Vector UV-2 } ,
  • { Vector UV-3 } and
  • { Vector UV-4 } ;

    respectively .

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Setting up the { 4-Dimension Quasi-Cube } :

• We take note here that :
  • the { 4-Dimension Quasi-Cube } will have :
    • 16 corners / nodal-points and 8 Cross-Diagonal Lines .

    Cross-Diagonal Line I.D.	* * *	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
    *********************	*	***********	***	***	***	***	*	***********	***	***	***	***
    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

• We simply make these observations here :
  • { Line 1 / Line 2 / Line 3 / Line 4 } do form the first set of { 4 Lines mutually orthogonal to one-another } ;

  • { Line 5 / Line 6 / Line 7 / Line 8 } do form the second set of { 4 Lines mutually orthogonal to one-another } ; and

  • any line in the { first set } will intersect any line in the { second set } at an angle of [ 60 degrees ] .

  And this does confirm our findings on the { 4-Dimension Sub-Space } in Part II of this paper .

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What's Next :

• We shall look at , next :
  • the 24 squares arising from the { 16 corners / nodal-points } of the { 4-Dimension Quasi-Cube } .

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

go to the last Section : Section VII - 4 Sets of { 4 Lines Mutually Orthogonal to One-Another }

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Original dated 2017-7-23