On 8-Colors Combinatorics and [ Quasi 24-Symmetry ]

by Frank C. Fung ( 1st published in March, 2011. )

Top Of Page

Section XVI - Setting up the [ 8-Colors Cube ]

Summary for the Section :

• In this Section , we shall set up the [ 8-Colors Cube ] for our use here in Part V and beyond ,
  • with the key feature here being that :
    • one-half of the [ 8-Colors Cube ] may be rotated against the other-half .

go to Top Of Page

Setting up the Cube :

• Let us first set-up :
  • a set of three (3) orthogonal axis defining a 3-Dimensional Space , i.e. :
    • the [ X-Axis ] , the [ Y-Axis ] and the [ Z-Axis ] respectively .

  

• Next , let us also set up a cube , so that :
  • the co-ordinates of its 8 corners are :
    • ( +1, +1, +1 ) / ( +1, +1, -1 ) / ( +1, -1, +1 ) / ( +1, -1, -1 ) /
      ( -1, +1, +1 ) / ( -1, +1, -1 ) / ( -1, -1, +1 ) / ( -1, -1, -1 ) .

      respectively .

go to Top Of Page

Setting up the [ 8-Colors Cube ] :

• Let us now divide the above-said cube into 8 equal-eighths , via :
  • the use of the [ X-Y Plane ] , the [ Y-Z Plane ] and the [ Z-X Plane ] as the dividing-planes ,

  

  We can then :

  • color each of the 8 [ eighths ] a different color to arrive at the [ 8-Colors Cube ] ,
    • as shown in the diagram on-the-right above .

  And the 8 colors we shall be using throughout are then the colors :

  • Red / Cyan / Purple / Magenta / Green / Yellow / Orange / Blue .

go to Top Of Page

The Rotational Feature of the [ 8-Colors Cube ] :

• Let us assume throughout that :
  • We can physically rotate one-half of the cube against the other-half ,
    • thru the angles of [ 90 degrees ] , [ 180 degrees ] , [ 270 degrees ] and [ 360 degrees ] ;

    so that :

    • the resultant object will always retain the shape of a cube ,
      • and thus always a [ colored-cube ] with 8 colors .

go to Top Of Page

Defining the [ Types of Rotation ] :

• For the sake of clarity and consistency , we shall now list-out :
  • the convention for labelling / identifying the different [ types of half-on-half rotation ] .

  First ,

  • we shall define :
    • a [ 1-notch rotation ] as a counter-clockwise rotation of [ 90 degrees ] ,

    • a [ 2-notch rotation ] as a counter-clockwise rotation of [ 180 degrees ] ,

    • a [ 3-notch rotation ] as a counter-clockwise rotation of [ 270 degrees ] , and

    • a [ 4-notch rotation ] as a counter-clockwise rotation of [ 360 degrees ] .

  Secondly ,

  • there are exactly six (6) [ axial-directions ] that we can select as the axis-of-rotation for our counter-clockwise rotations ,
    • and these 6 [ axial directions ] are :
      • the [ +X Axial-Direction ] ,
      • the [ +Y Axial-Direction ] ,
      • the [ +Z Axial-Direction ] ,
      • the [ -X Axial-Direction ] ,
      • the [ -Y Axial-Direction ] , and
      • the [ -Z Axial-Direction ] ;

        respectively .

• And for the sake of clarity and precision :
  • we shall now present the rotational-mechanics for a [ 1-notch rotation ] in each of the 6 [ Axial-Directions ] ,
    • in the 6 sets of diagrams to follow .

  First , for a [ 1-notch +X Axis Rotation ] , we have :

  

  Secondly , for a [ 1-notch +Y Axis Rotation ] , we have :

  

  Thirdly , for a [ 1-notch +Z Axis Rotation ] , we have :

  

  Fourthly , for a [ 1-notch -X Axis Rotation ] , we have :

  

  Fifthly, for a [ 1-notch -Y Axis Rotation ] , we have :

  

  Sixth-and-finally , for a [ 1-notch -Z Axis Rotation ] , we have :

  

• Rather simple and straight-forward up to this point .

go to Top Of Page

go to the next section : Section XVII - The Global Set of 40,320 [ 8-colors configurations ]

go to the last section : Section XIV - Structural Considerations in Quasi Symmetry

return to the 8-Colors Combinatorics / Quasi 24-Symmetry HomePage

Original dated 2011-3-06