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

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

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Section XXXII - Revisting Mirror Images

Summary for the Section :

• In this Section , we shall looking into [ Mirror Image Pairs ] , for the [ 24 QSM-Configurations ] .

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Recalling the [ 24 QSM-Configurations ] :

• Let us bring-back , at this point , the [ 24 QSM-Configurations --- Himalaya Series ] amd the [ 24 QSM-Configurations --- Canadian Rockies Series ] :

  

  • Do we seee any mirror images here .
    • Not really .

• Very simply , the orientations are not right for mirror images ,
  • and we need to redraw these diagrams in the [ isometric view ] orientation .

  

  And we see here that the following pairs are the [ mirror-image pairs ] :

  • in the 1st row : [ A - F ] , [ B - E ] and [ C - D ] ,
  • in the 2nd row : [ G - M ] , [ H - L ] and [ J - K ] ,
  • in the 3rd row : [ N - T ] , [ P - S ] and [ Q - R ] ,
  • in the 4th row : [ U - Z ] , [ V - Y ] and [ W - X ] .

  Wasn't that simple !

  • But there is a special [ catch ] here , as detailed below .

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The special [ catch ] on Mirror Images :

• For illustration purposes , let us conduct the following experiment :

  Step One :

  • We paint a very large [ 2-D Image ] on a flat playing-field on the ground that can be seen from the air ,
    • as per the diagram on-the-left , below .

  

  Step Two :

  • We invite the [ R-S-T Sky-Diving Team ] consisting of Rex , Sid and Tom to do a sky diving maneuver ,

    namely , holding-hands-together in the formation of a circle while sky-diving ,

    • as per the digram in-the-middle, above .

• Let us suppose that 3 sky-divers in the circular-formation are now directly above the [ 2-D Image ] ,
  • as per the diagram on-the-right above .

  And we ask Rex , what is the [ mirror image ] of the painted-diagram you see on the ground ?

  • And Rex would answer with the diagram on-the-left , below .

  

  And we ask Sid , what is the [ mirror image ] of the painted-diagram you see on the ground ?

  • And Sid would answer with the diagram in-the-middle , above .

  And we ask Tom , what is the [ mirror image ] of the painted-diagram you see on the ground ?

  • And Tom would answer with the diagram on-the-right , above .

• So who is right and who is wrong ?
  • Good question !

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The [ Mirror Image ] of [ A / B / C / D / E / F ] :

• Based on the above , let us now bring-in this set of 6 diagrams ,
  • for the [ mirror images ] for [ A ] , [ B ] , [ C ] , [ D ] , [ E ] , and [ F ] .

  
  

  Looks vaguely familiar ?

  Let us now also bring-in this set of 6 diagrams , from Section XXX when we looked at [ Bi-Polar Pairs ] :

  
  

• And any resemblence here is , of-course , co-incidental , in the sense that :
  • In Mirror Images :
    • the pieces on the Center-Line do not move , and

    • each pair of [ left-piece / right-piece ] exchange places .

  • In [ Bi-Polar Pairs ] :
    • an even number of pieces do not move , and

    • one or more [ pairs of pieces ] exchange positions .

    ( [ 3-pieces rotary-exchanges ] and [ 4-pieces rotary-exchange ] are always ruled-out in a [ Bi-Polar situation ] ,

    • and this is because :
      • neither of these will permit you get back to the original-poistion-of-the-pieces on the reverse-link ) .

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go to the next section : Section XXXIII - The Tetrahedron and the Full-Body-Rolls

go to the last section : Section XXXI - The [ 6-elements loops ] for the Single Rolls

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

Original dated 2011-3-06