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

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

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Section XXXI - The [ 6-elements loops ] for the Single Rolls

Summary for the Section :

• In this Section , we shall be looking at the [ 6-rolls sequences ] for the Half-on-Half rotations of the [ 8-Colors Cube ] :
  • but involving [ 1-notch rotations ] and [ 3-notch rotations ] only .

  And to the extent that :

  • a [ 3-notch ( 270 degrees ) counter-clockwise rotation ] and a [ 1-notch ( 90-degrees ) clockwise rotation ]
    • will always yield the same results ;

    we shall now , for our analysis purposes here , treat the [ 3-notch counter-clockwise rotation ] as a [ single roll ] :

    • i.e. : the equivalent of a single [ 1-notch rotation in the clockwise direction ] .

• And our approach here is to first present four (4) different Scenarios , for illustration purposes ,
  • ahead of drawing the final conclusions on the [ 6-rolls sequences ] .

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Scenario One :

• For Scenario One , the 6 rotations here are , in sequence :
  • a [ 1-notch +X Axis rotation ] ,
  • a [ 1-notch +Y Axis rotation ] ,
  • a [ 1-notch +X Axis rotation ] ,
  • a [ 1-notch +Y Axis rotation ] ,
  • a [ 1-notch +X Axis rotation ] , and
  • a [ 1-notch +Y Axis rotation ] .

  We then have this set of diagrams for the said rotations :

  

  

  Question :

  • Do we get back to the [ Starting Position ] after these 6 rolls ?

  And the answer here is [ NO ] .

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Scenario Two :

• For Scenario Two , the 6 rotations here are , in sequence :
  • a [ 1-notch +X Axis rotation ] ,
  • a [ 3-notch +Y Axis rotation ] ,
  • a [ 1-notch +X Axis rotation ] ,
  • a [ 3-notch +Y Axis rotation ] ,
  • a [ 1-notch +X Axis rotation ] , and
  • a [ 3-notch +Y Axis rotation ] .

  We then have this set of diagrams for the said rotations :

  

  

  Question :

  • Do we get back to the [ Starting Position ] after these 6 rolls ?

  And the answer here is [ YES ] .

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Scenario Three :

• For Scenario Three , the 6 rotations here are , in sequence :
  • a [ 3-notch +X Axis rotation ] ,
  • a [ 1-notch +Y Axis rotation ] ,
  • a [ 3-notch +X Axis rotation ] ,
  • a [ 1-notch +Y Axis rotation ] ,
  • a [ 3-notch +X Axis rotation ] , and
  • a [ 1-notch +Y Axis rotation ] .

  We then have this set of diagrams for the said rotations :

  

  

  Question :

  • Do we get back to the [ Starting Position ] after these 6 rolls ?

  And the answer here is [ YES ] .

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Scenario Four :

• For Scenario Four , the 6 rotations here are , in sequence :
  • a [ 3-notch +X Axis rotation ] ,
  • a [ 3-notch +Y Axis rotation ] ,
  • a [ 3-notch +X Axis rotation ] ,
  • a [ 3-notch +Y Axis rotation ] ,
  • a [ 3-notch +X Axis rotation ] , and
  • a [ 3-notch +Y Axis rotation ] .

  We then have this set of diagrams for the said rotations :

  

  

  Question :

  • Do we get back to the [ Starting Position ] after these 6 rolls ?

  And the answer here is [ NO ] .

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Summarizing the 4 Scenarios :

• Based on the results of the above four (4) scenearios ,
  • we then have this proposal below for identifying the [ 6-elements loops ] for the Single Rolls .

  And the 2 steps procedure for the said proposal is as follows :

  • STEP ONE :
    • Select any two (2) orthogonal axis and label these as the [ 1st-Axis ] and the [ 2nd-Axis ] .

  • STEP TWO :
    • Use either of the following [ 6-rolls sequence ] as the rotation-sequence for the [ 6-elements loops ] :
      • Sequence One :
        • a [ 1-notch rotation in the 1st-Axis Axial-Direction ] ,
        • a [ 3-notch rotation in the 2nd-Axis Axial-Direction ] ,
        • a [ 1-notch rotation in the 1st-Axis Axial-Direction ] ,
        • a [ 3-notch rotation in the 2nd-Axis Axial-Direction ] ,
        • a [ 1-notch rotation in the 1st-Axis Axial-Direction ] ,
        • a [ 3-notch rotation in the 2nd-Axis Axial-Direction ] .

      • Sequence Two :
        • a [ 3-notch rotation in the 1st-Axis Axial-Direction ] ,
        • a [ 1-notch rotation in the 2nd-Axis Axial-Direction ] ,
        • a [ 3-notch rotation in the 1st-Axis Axial-Direction ] ,
        • a [ 1-notch rotation in the 2nd-Axis Axial-Direction ] ,
        • a [ 3-notch rotation in the 1st-Axis Axial-Direction ] ,
        • a [ 1-notch rotation in the 2nd-Axis Axial-Direction ] .

  And the 6-elements [ rotation sequence/s ] above will always get us back to the original [ Starting Position ] .

• Wasn't that simple !

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A Quick Comment on [ 6-elements loops ] :

• Let us take a quick look at this [ 6-elements loops ] ,
  • where [ Annapurna ] and [ K-5 ] occupy directly-opposite positions in the loop .

  

• Let us suppose for the moment that we are at [ Annapurna ] ,
  • and we want to do some minor changes / adjustments / mutations .

  But the conditions at [ Annapurna ] are not condusive to these minor changes ,

  • whereas the conditions at [ K-5 ] are ripe for these minor changes .

  So we need to go to [ K-5 ] and come back .

• If we go-to [ K-5 ] , brake-and-stop , and reverse-and-come-back ,
  • this might be high-energy-consuming maneuver .

  But if we use the [ 6-elements loop ] to get-there-and-come-back ,

  • this might be a smoother affair and possibly requires less-energy .

• The key point here is this :
  • When we do our routing procedure going thru a maze-of-path-options ,
    • we should always be on-the-look-out for alteranative routes .

  And [ loops ] are always an alternative .

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go to the next section : Section XXXII - Revisiting Mirror Images

go to the last section : Section XXX - Revisiting the Concept of [ QI ] and [ OU ]

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

Original dated 2011-3-06