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. )

Top Of Page

Section XXIII - Using [ 8 x 8 Mapping Matrices ] on { Structure A } and { Structure B }

Summary for the Section :

• In this Section , we shall demonstrate :
  • how to construct { Structure A } and { Structure B } via the use of :
    • [ 8 x 8 mapping matrices ] associated with the 256 sets of { 8-D Orthogonal Unit-Vector Basis }
      • we have set up back in Section XXI .

Using { Orthogonal Unit-Vector Basis Sets } as Mapping Matrices :

• We simply that note here that :
  • the [ 8 x 8 matrix ] associated with a set of { 8-D Orthogonal Unit-Vector Basis }
    • can also be used as mapping matrix for mapping one vector onto another vector .

  Let us bring-in { Orthogonal Unit-Vector Basis SET ZERO } and { Orthogonal Unit-Vector Basis SET ALPHA }

  • for a quick review :

    Details for the Orthogonal Unit-Vector Basis { SET ZERO }
    Vector I.D.	* *	Co-ordinates based on the Physical Unit Vector
    		PUV-1	PUV-2	PUV-3	PUV-4	PUV-5	PUV-6	PUV-7	PUV-8
    *************	*	******	******	******	******	******	******	******	******
    Vector UVZ-1	*	+1	0	0	0	0	0	0	0
    Vector UVZ-2	*	0	+1	0	0	0	0	0	0
    Vector UVZ-3	*	0	0	+1	0	0	0	0	0
    Vector UVZ-4	*	0	0	0	+1	0	0	0	0
    Vector UVZ-5	*	0	0	0	0	+1	0	0	0
    Vector UVZ-6	*	0	0	0	0	0	+1	0	0
    Vector UVZ-7	*	0	0	0	0	0	0	+1	0
    Vector UVZ-8	*	0	0	0	0	0	0	0	+1

    Details for the Orthogonal Unit-Vector Basis { SET ALPHA }
    Vector I.D.	* *	Co-ordinates based on the Physical Unit Vector
    		PUV-1	PUV-2	PUV-3	PUV-4	PUV-5	PUV-6	PUV-7	PUV-8
    *************	*	******	******	******	******	******	******	******	******
    Vector UVA-1	*	+1	0	0	0	0	0	0	0
    Vector UVA-2	*	0	+1	0	0	0	0	0	0
    Vector UVA-3	*	0	0	-1	0	0	0	0	0
    Vector UVA-4	*	0	0	0	+1	0	0	0	0
    Vector UVA-5	*	0	0	0	0	+1	0	0	0
    Vector UVA-6	*	0	0	0	0	0	+1	0	0
    Vector UVA-7	*	0	0	0	0	0	0	+1	0
    Vector UVA-8	*	0	0	0	0	0	0	0	+1

• We take note here that :
  • the [ 8 x 8 matrix ] associated with a set of { 8-D Orthogonal Unit-Vector Basis } is a { diagonal matrix }
    • with :
      • only [ +1 ]'s or [ -1 ]'s as the diagonal elements of the Matrix ,

        and

      • [ 0 ]'s for all other elements of the Matrix .

  As such :

  • the [ inverse ] of the Matrix is its [ transpose ] ;

    and therefore :

    • the [ inverse ] of [ 8 x 8 Mapping Matrix ] is the [ 8 x 8 Mapping Matrix ] itself .

• Thus ,
  • when we use a [ 8 X 8 matrix ] to map the { First Vector } onto the { Second Vector } ,
    • we can always use the same [ 8 x 8 matrix ] to map the { Second Vector } onto the { First Vector } .

go to Top Of Page

A Quick Example on the Mapping Mechnism :

• Let us bring-in these four (4) Vectors below :
  • based on the 8 Physical Unit-Vectors defining the { 8-Dimension Vector Space } ,

    i.e. :

    • { Vector PUV-1 } , { Vector PUV-2 } , { Vector PUV-3 } , { Vector PUV-4 } ,
      { Vector PUV-5 } , { Vector PUV-6 } , { Vector PUV-7 } , { Vector PUV-8 } .

  Cross-Diagonal Line I.D.	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
  				P U V - 1	P U V - 2	P U V - 3	P U V - 4	P U V - 5	P U V - 6	P U V - 7	P U V - 8			P U V - 1	P U V - 2	P U V - 3	P U V - 4	P U V - 5	P U V - 6	P U V - 7	P U V - 8
  *************	*************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***	*
  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	*
  Line G	Class 7-1 / 1-7	*	Vector G-1	+1	+1	-1	+1	+1	+1	+1	+1	*	Vector G-2	-1	-1	+1	-1	-1	-1	-1	-1	*

• We see here that :
  • we can map { Vector Z-1 } onto { Vector G-1 } via using :
    • the [ 8 x 8 matrix ] associated with the { 8-D Orthogonal Unit-Vector Basis SET ALPHA } ;

  • we can map { Vector Z-1 } onto { Vector G-2 } via using :
    • the [ 8 x 8 matrix ] associated with the { 8-D Orthogonal Unit-Vector Basis SET OMEGA } ;

  • we can map { Vector Z-1 } onto { Vector Z-2 } via using :
    • the [ 8 x 8 matrix ] associated with the { 8-D Orthogonal Unit-Vector Basis SET RICO } ;

  • we can map { Vector Z-1 } onto itself via using :
    • the [ 8 x 8 matrix ] associated with the { 8-D Orthogonal Unit-Vector Basis SET ZERO } .

• And for the reader's quick reference , we bring-in again :
  • { 8-D Orthogonal Unit-Vector Basis SET OMEGA } and

  • { 8-D Orthogonal Unit-Vector Basis SET RICO } ;
    • as shown below .

    Details for the Orthogonal Unit-Vector Basis { SET OMEGA }
    Vector I.D.	* *	Co-ordinates based on the Physical Unit Vector
    		PUV-1	PUV-2	PUV-3	PUV-4	PUV-5	PUV-6	PUV-7	PUV-8
    *************	*	******	******	******	******	******	******	******	******
    Vector UVO-1	*	-1	0	0	0	0	0	0	0
    Vector UVO-2	*	0	-1	0	0	0	0	0	0
    Vector UVO-3	*	0	0	+1	0	0	0	0	0
    Vector UVA-4	*	0	0	0	-1	0	0	0	0
    Vector UVO-5	*	0	0	0	0	-1	0	0	0
    Vector UVO-6	*	0	0	0	0	0	-1	0	0
    Vector UVO-7	*	0	0	0	0	0	0	-1	0
    Vector UVO-8	*	0	0	0	0	0	0	0	-1

    Details for the Orthogonal Unit-Vector Basis { SET RICO }
    Vector I.D.	* *	Co-ordinates based on the Physical Unit Vector
    		PUV-1	PUV-2	PUV-3	PUV-4	PUV-5	PUV-6	PUV-7	PUV-8
    *************	*	******	******	******	******	******	******	******	******
    Vector UVR-1	*	-1	0	0	0	0	0	0	0
    Vector UVR-2	*	0	-1	0	0	0	0	0	0
    Vector UVR-3	*	0	0	-1	0	0	0	0	0
    Vector UVR-4	*	0	0	0	-1	0	0	0	0
    Vector UVR-5	*	0	0	0	0	-1	0	0	0
    Vector UVR-6	*	0	0	0	0	0	-1	0	0
    Vector UVR-7	*	0	0	0	0	0	0	-1	0
    Vector UVR-8	*	0	0	0	0	0	0	0	-1

go to Top Of Page

The 256 sets of { 8-D Orthogonal Unit-Vector Basis } :

• Let us recall that we did set up , in Section XXI ,
  • 256 sets of { 8-D Orthogonal Unit-Vector Basis } .

  Interesting enough , we can now actually start-off with a single vector ,

  • namely { Vector Z-1 } with the co-ordinates :
    • { +1, +1, +1, +1 , +1, +1, +1, +1 }

    and

    • construct { Structure A } via :
      • mapping { Vector Z-1 } onto the 128 Position Vectors for the 64 Lines of { Structure A }

        using :

        • the [ 8 x 8 matrices ] associated with the 128 { 8-D Orthogonal Unit-Vector Basis in the EVEN-EVEN Series } ;

    and also

    • construct { Structure B } via :
      • mapping { Vector Z-1 } onto the 128 Position Vectors for the 64 Lines of { Structure B }

        using :

        • the [ 8 x 8 matrices ] associated with the 128 { 8-D Orthogonal Unit-Vector Basis in the ODD-ODD Series } .

go to Top Of Page

What is the significance here :

• We take note here that :
  • { Cross-Product of 2 Vectors } in 3-Dimension is a common tool used in identifying [ rotation-type symmetry ] .

  The same might not be available in 4-Dimension and higher-Dimension's .

• Thus ,
  • [ mapping matrix ] may be a useful tool to fill in the gap .

go to Top Of Page

What's Next :

• Let us move on to Part V of this paper on Symmetry Through-put linking 3-4-5-6-7 Dimension's , next .

go to Top Of Page

go to the next Section : Section XXIV - 2 Identical { 6-D Strutures } arising from the { 6-D Quasi-Cube }

go to the last Section : Section XXII - Orientation Symmetry for { Structure A } vs. { Structure B }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23