On Symmetric Structures within Symmetric Solid Objects

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

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Section XXXXIV - 5-Colors Combinatorics for the [ OMEGA Series ]

Summary for the Section :

• In this Section , we shall take a closer look at the 5-Colors Combinatorics for the [ OMEGA Series ] .

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The 5-Colors Combinatorics :

• Let us first bring back the Icosahedron with the [ OMEGA-Series Quad-Type Structure Coloring Scheme ]:

  

  We then set-up :

  • [ Color-1 ] is [ light-green-color ] ,
  • [ Color-2 ] is [ blue-color ] ,
  • [ Color-3 ] is [ yellow-color ] ,
  • [ Color-4 ] is [ brown-color ] , and
  • [ Color-5 ] is [ red-color ] .

• We then list-out the [ circular counter-clockwise rotation-order ] for the 12 [ nodal-points] :

  Nodal-Point	Rotation-Order		Nodal-Point	Rotation-Order
  N-1	1-2-3-4-5		N-12	1-5-4-3-2
  N-2	1-5-2-4-3		N-11	1-3-4-2-5
  N-3	1-3-5-4-2		N-10	1-2-4-5-3
  N-4	1-5-3-2-4		N-9	1-4-2-3-5
  N-5	1-4-3-5-2		N-8	1-2-5-3-4
  N-6	1-3-2-5-4		N-7	1-4-5-2-3
  **********	**************	***	**********	**************

  For our discussion purposes below :

  • let us call this set of 12 [ rotation-orders ] the [ OSCAR-12 ] .

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The Complimentary Group of 12 :

• Let us also identify the complimentary group of 12 [ rotation-orders ] as the [ OLLIE-12 ] :
  • as shown in [ beige-white-color ] in this table below :

  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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An Interesting Proposal :

• Let us now make this proposal in 2 Stages :

  STAGE ONE :

  • Let us take any of the 12 [ rotation-orders ] in the [ OSCAR-12 ] , or a [ circular-rotationary-variation ] thereof :
    • and slap-it onto the [ Positions A / B / C / D / E ] , and :
      • then expand the 5-colors in accordance to the [ OMEGA-Series Quad-Type Structure Coloring Scheme ] ;

    then :

    • the remaining 11 members of the [ OSCAR-12 ] would show up at the other 11 [ nodal-points ] .

  STAGE TWO :

  • Let us take any of the 12 [ rotation-orders ] in the [ OLLIE-12 ] , or a [ circular-rotationary-variation ] thereof :
    • and slap-it onto the [ Positions A / B / C / D / E ] , and
      • then expand the 5-colors in accordance to the [ OMEGA-Series Quad-Type Structure Coloring Scheme ] ;

    then :

    • the remaining 11 members of the [ OLLIE-12 ] would show up at the other 11 [ nodal-points ] .

• Is this TRUE or NOT ?
  • Again , we shall let the reader call this one !

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A Special Catch here :

• Since the [ OSCAR / OLLIE Separation ] of the 24 [ rotation-orders ] here is exactly the same as the [ ALBERTO / ALBINA Separation ] ,
  • Is there any possibility that we might be looking at a different kind of symmetry here ?

• UH ??

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go to the next section : Section XXXXV - Concluding Remarks

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

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31