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

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

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Section XVII - The Global Set of 40,320 [ 8-colors configurations ]

Summary for the Section :

• In this Section , we shall set up the global set of 40,320 [ 8-colors configurations ] , and :
  • identify a particular configuration , [ Configuration Annapurna ] , as the starting-point of our analysis .

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The 40,320 permutations of [ 8-colors configurations ] :

• Let us first take note here that :
  • the [ 8-Colors Cube ] is actually made-up-of 8 differently-colored pieces ,
    • with each piece occupying one of the 8 neatly-arranged [ fixed-positions ] .

  

  And each [ fixed-position ] must always spot a differently-colored piece .

• As such :
  • there are a total of 40,320 possible [ 8-colors configurations ] here for our consideration .

    ( 40,320 = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 )

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Selecting a starting-point for our analysis :

• For our analysis on the combinatorics of the [ 8-Colors Cube ] , in this Section and beyond :
  • it is necessary for us to select one of the 40,320 [ 8-colors configurations ] as our starting-point / base-point .

  And we shall select , at this first-instance , this configuration below :

  • 
    • as our starting-point / base-point .

• And we shall name this configuration [ Configuration Annapurna ] :
  • for our easy reference throughout the remainder of this paper .

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The First Question on the 40,320 permutations :

• Our First Question here on the 40,320 permutations is simply this :
  • If we were to start with [ Configuration Annapurna ] and permit an unlimited number of :
    • [ 1 / 2 / 3 / 4-notch rotations ] using the [ +X / +Y / +Z / -X / -Y / -Z Axis ] as the axis-of-rotation ,

    would be be able to reach each-and-every one of the global set of 40,320 [ 8-colors configurations ] ?

  And the answer here is YES .

  ( The analysis here is done thru a computer programming procedure . )

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The Second Question on the 40,320 permutations :

• Our Second Question here is this :
  • If we were to start with [ Configuration Annapurna ] and permit an unlimited number of :
    • [ 2-notch rotations ] using the [ +X / +Y / +Z / -X / -Y / -Z Axis ] as the axis-of-rotation ,

    would be be able to reach each-and-every one of the global set of [ 40,320 configurations ] ?

  And the answer here is NO .

• In fact :
  • we would only reach a set of 96 [ 8-colors configurations ] :
    • [ Configuration Annapurna ] including ;

    if we were to permit only [ 2-notch rotations ] in all six (6) [ Axial-Directions ] .

  ( Again , the analysis here is done thru a computer programming procedure . )

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go to the next section : Section XVIII - The Reduced Set of 5,040 [ 8-colors configurations ]

go to the last section : Section XVI - Setting up the [ 8-Colors Cube ]

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

original dated 2011-3-06