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 II - Defining the { 8-Dimension Quasi-Cube }

Summary for the Section :

• In this Section , we shall give a formal definition of :
  • the { 8-Dimension Quasi-Cube } that we shall be using throughout the entire paper .

  And this is done for the sake of consistency throughout .

go to Top Of Page

Recalling the Cube :

• Let us recall that a Cube , residing in { 3-Dimension Vector Space } , has :
  • 8 [ corners / nodal-points ]

    and

  • 4 [ Cross-Diagonal Lines ] .

  

  And the [ 3 co-ordinates ] for the 8 { corners / nodal-points } can be expressed in this manner :

  Cross-Diagonal Line I.D.	* * *	1st end-point ( a nodal-point ) of the Cross-Diagonal Line				* * *	2nd end-point ( a nodal-point ) of the Cross-Diagonal Line
  	*	Nodal-Point I.D.	Co-ordinates			*	Nodal-Point I.D.	Co-ordinates
  *********************	*	****************	*****	*****	*****	*	****************	*****	*****	*****
  Cross-Diagonal Line 1	*	1st Nodal-Point	+1	+1	+1	*	2nd Nodal-Point	-1	-1	-1
  Cross-Diagonal Line 2	*	3rd Nodal-Point	+1	+1	-1	*	4th Nodal-Point	-1	-1	+1
  Cross-Diagonal Line 3	*	5th Nodal-Point	+1	-1	+1	*	6th Nodal-Point	-1	+1	-1
  Cross-Diagonal Line 4	*	7th Nodal-Point	+1	-1	-1	*	8th Nodal-Point	-1	+1	+1

  We take note here that :

  • the [ 3 co-ordinates ] for each of the 8 { corners / nodal-points } will always :
    • host either a [ +1 ] or a [ -1 ] ;

    and

  • each of the 4 [ Cross-Diagonal Lines ] joins a pair of opposite { corners / nodal-points }
    • whose [ position vectors ] point in the exact opposite directions .

go to Top Of Page

Defining the { 8-D Quasi-Cube } :

• Let us now define the { 8-Dimension Quasi-Cube } for our use in this paper .

  DEFINITION :

  • For each-and-every { corner / nodal-point } of our { 8-Dimension Quasi-Cube } :
    • the [ 8 co-ordinates ] for the said { corner / nodal-point } will always
      • host either a [ +1 ] or a [ -1 ] ,

        and

        no other value is allowed .

  • each-and-every [ Cross-Diagonal Line ] then joins a pair of opposite { corners / nodal-points }
    • whose [ position vectors ] point in the exact opposite directions .

• Let us now take a quick look at several sample [ Cross-Diagonal Lines ] and the associated pair of [ corners / nodal-points ] :

  Cross-Diagonal Line I.D.	* * *	1st end-point ( a nodal-point ) of the Cross-Diagonal Line									* * *	2nd end-point ( a nodal-point ) of the Cross-Diagonal Line
  	*	Nodal-Point I.D.	Co-ordinates								*	Nodal-Point I.D.	Co-ordinates
  *********************	*	***************	***	***	***	***	***	***	***	***	*	***************	***	***	***	***	***	***	***	***
  Sample Line A	*	Nodal-Point A-1	+1	+1	+1	+1	+1	+1	+1	+1	*	Nodal-Point A-2	-1	-1	-1	-1	-1	-1	-1	-1
  Sample Line D	*	Nodal-Point D-1	+1	+1	+1	+1	-1	-1	-1	-1	*	Nodal-Point D-2	-1	-1	-1	-1	+1	+1	+1	+1
  Sample Line Q	*	Nodal-Point Q-1	+1	-1	+1	+1	-1	+1	+1	-1	*	Nodal-Point Q-2	-1	+1	-1	-1	+1	-1	-1	+1
  Sample Line W	*	Nodal-Point W-1	+1	-1	+1	-1	+1	-1	+1	-1	*	Nodal-Point W-2	-1	+1	-1	+1	-1	+1	-1	+1

• Since we have exactly 2 choices for each of the [ 8 co-ordinates ] for a { corner / nodal-point } ,
  • i.e. :
    • the choice of either a [ +1 ] or a [ -1 ] ;

  the [ total number of corners / nodal-points ] for our { 8-Dimension Quasi-Cube } is then [ 256 ] ,

  • i.e. :
    • [ 256 ] = [ 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 ] or [ 2 raised to the 8th-power ] .

  And the [ total number of Cross-Diagonal Lines ] for our { 8-Dimension Quasi-Cube } is then [ 128 ] ,

  • i.e. :
    • [ 128 ] = one-half of [ 256 ] .

go to Top Of Page

What's Next :

• Let us now proceed to Part II of this paper :
  • to find out more about the different sets of { mutually orthogonal [ Cross-Diagonal Lines ] }
    • to be found within our { 8-Dimension Quasi-Cube } .

go to Top Of Page

go to the next Section : Section III - Recalling the 28 Symmetric Lines in a { 7-Dimension Vector Space }

go to the last Section : Section I - Introduction

return to the Symmetry Through-put HomePage

Original dated 2017-7-23