On Symmetric Structures within Symmetric Solid Objects

by Frank C. Fung ( 1st published in March, 2010. )

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Section XXXXII - A Special Feature of the [ Quad-Type Structure Coloring Schemes ]

Summary for the Section :

• In this Section , we do the set-up work for our analysis in the next 2 Sections :
  • on the [ 5-color combinatorics ] for the [ Quad-Type Structure Coloring Schemes ] ,
    • the [ ALPHA Series ] vs. the [ OMEGA Series ] .

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The Special Feature :

• Let us now bring-back the Icosahedron for both the [ ALPHA Series ] :

  

  and also the [ OMEGA Series ] :

  

  We noticed here that :

  • the 5 [ pieces ] surrounding each of the 12 [ nodal-points ] are always of-a-different-color ,

    i.e :

    • in the [ circular counterclockwise rotation-order ] for the 5 [ pieces ] surrounding a [ nodal-point ] :
      • each of the [ 5-colors ] appeared once and exactly-once .

• And we shall be taking a closer look at these 12 [ rotation-orders ] in the next 2 Sections .

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Identifying the 12 [ Nodal-Points ] :

• Let us now mark-off the 12 [ nodal-points ] as [ N-1 ] thru [ N-12 ] respectively :
  • as per this set of 3 diagrams below :

  

• And we have also marked-off 5 [ positions ] in the above diagrams , namely :
  • [ Position A ] , [ Position B ] , [ Position C ] , [ Position D ] and [ Position E ] ; respectively .

  And , as a matter of convention for our analysis to follow , we shall :

  • identify the color of the [ piece ] in [ Position A ] as [ Color-1 ] ;
  • identify the color of the [ piece ] in [ Position B ] as [ Color-2 ] ;
  • identify the color of the [ piece ] in [ Position C ] as [ Color-3 ] ;
  • identify the color of the [ piece ] in [ Position D ] as [ Color-4 ] ;
  • identify the color of the [ piece ] in [ Position E ] as [ Color-5 ] .

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5-Colors Combinatorics in general :

• In general , for a [ Circular Counterclockwise Rotation Order ] of 5-colors :
  • each [ rotation-order ] does cover five ( 5) permutations ;

    i.e. : the [ rotation-orders ] :

    • [ 1-2-3-4-5 ] / [ 2-3-4-5-1 ] / [ 3-4-5-1-2 ] / [ 4-5-1-2-3 ] / [ 5-1-2-3-4 ]

      are always considered as the same [ rotation-order ] .

  Since there are 120 permutations of [ rotation-orders ] here ( 120 = 5 x 4 x 3 x 2 x 1 ) :

  • there are then exactly 24 distiguishable [ circular rotation-order ] here for our consideration .

    ( i.e. : 24 = 120 divided by 5 ) .

• We then list-out the 24 [ rotaion-orders ] in this table below for the reader's first perusal :

  1-2-3-4-5	1-2-4-5-3	1-2-5-3-4	1-2-5-4-3	1-2-4-3-5	1-2-3-5-4
  1-3-4-5-2	1-3-5-2-4	1-3-2-4-5	1-3-2-5-4	1-3-5-4-2	1-3-4-2-5
  1-4-5-2-3	1-4-2-3-5	1-4-3-5-2	1-4-3-2-5	1-4-2-5-3	1-4-5-3-2
  1-5-2-3-4	1-5-3-4-2	1-5-4-2-3	1-5-4-3-2	1-5-3-2-4	1-5-2-4-3
  ***********	***********	***********	***********	***********	***********

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go to the next section : Section XXXXIII - 5-Colors Combinatorics for the [ ALPHA Series ]

go to the last section : Section XXXXI - Rotational Orientation for the [ Curved Triangular Pieces ]

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Original dated 2010-3-31