On Symmetric Structures within Symmetric Solid Objects

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

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Section XXIV - The [ 2-6-6-6-Symmetry ] co-existing within the Icosahedron

Summary for the Section :

• In this Section , we shall take a look at the [ 2-6-6-6-Symmetry ] co-existing within the 20-faced Icosahedron , and in the process :
  • we shall introduce the [ 1-3-6-6-3-1-Split ] on the [ relative positions ] of the 20 pieces of the Icosahedron .

  And the [ 1-3-6-6-3-1 Split ] is useful both here in this Section and in the next Section on [ Quad-Type Structures ] .

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The 4-colors [ 2-6-6-6-Symmetric Configuration ] :

• First , we present this { 4-colors Configuration } of the Icosahedron which exhibits [ Bi-Symmetry ] and triple [ Hexa-Symmetry ] :

  

  Very simply , if we were to toss this Icosahedron into the air and let it land on any one of its 20 faces , then :

  • If the Icosahedron landed on a [ magenta-color ] face :
    • we would not be able to tell which of the two (2) [ magenta-color pieces ] it landed-on .

  • If the Icosahedron landed on a [ orange-color ] face :
    • we would not be able to tell which of the six (6) [ orange-color pieces ] it landed-on .

  • If the Icosahedron landed on a [ light-green-color ] face :
    • we would not be able to tell which of the six (6) [ light-green-color pieces ] it landed-on .

  • If the Icosahedron landed on a [ cornflower-blue-color ] face :
    • we would not be able to tell which of the six (6) [ cornflower-blue-color pieces ] it landed-on .

• And , in order to understand why this [ 4-colors configuration ] here is indeed [ 2-6-6-6-Symmetric ] ,
  • we shall now bring-in the [ 1-3-6-6-3-1 Split ] on the [ relative positions ] of the 20 [ pieces ] ,
    • to start-off the analysis .

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The [ 1-3-6-6-3-1 Split ] on [ relative positions ] :

• For our analysis here on the [ relative positions ] of the 20 [ pieces ] :
  • we shall first select at random any one of of the 20 [ pieces ] as the [ Base Piece ] of our analysis .

  And we shall also immediately identify the [ Opposite Piece ] , being the [ piece ] directly-opposite to the [ Base Piece ] .

• We then present this set of 3 diagrams below :
  • showing the [ Base Piece ] as marked-off in [ magenta-color ] , and
  • showing the [ Opposite Piece ] as marked-off in [ purple-color ] .

  

  • We then construct a straight-line passing-thru the [ centroidal-points ] of the two (2) said-[ pieces ] ,
    • and this then is the [ Axis-Of-Tri-Symmetry ] for our [ 1-3-6-6-3-1 Split ] configuration to follow .

• And for better viewing and easier-to-understand purposes , let us now re-orient the entire Icosahedron , so that :
  • the [ Axis-of-Tri-Symmetry ] is now in a [ quasi-upright ] position ,
    • as per this set of 3 diagrams below .

  

• We shall next identify the 4 layers situated in-between the [ Base Piece ] and the [ Opposite Piece ] , namely :
  • the [ 1st Layer-Of-Three ] ,
  • the [ 1st Layer-Of-Six ] ,
  • the [ Opposite Layer-Of-Six ] , and
  • the [ Opporsite Layer-Of-Three ] .

  We shall now do the following :

  • identify the 3 [ pieces ] adjacent to the [ Base Piece ] as the [ 1st Layer-Of-Three ] ,
  • identify the 3 [ pieces ] adjacent to the [ Opposite Piece ] as the [ Opposite Layer-Of-Three ] ,

  We then have this set of 3 diagrams :

  • showing the 3 [ pieces ] in the [ 1st Layer-Of-Three ] in [ orange-color ] , and
  • showing the 3 [ pieces ] in the [ Opposite Layer-Of-Three ] in [ brown-color ] .

  

  And we simply note here that :

  • For the 3 [ pieces ] in the [ 1st Layer-Of-Three ] :
    • these are [ tri-symmetric ] about the [ Axis-Of-Tri-Symmetry ] , and
    • their [ centriodal-points ] are equi-distant from the [ centroidal-point ] of the [ Base Piece ] .

  • For the 3 [ pieces ] in the [ Opposite Layer-Of-Three ] :
    • these are [ tri-symmetric ] about the [ Axis-Of-Tri-Symmetry ] , and
    • their [ centriodal-points ] are equi-distant from the [ centroidal-point ] of the [ Opposite Piece ] .

  And this arises from the triple-but-equivalent adjacency :

  • of the 3 [ orange-color pieces ] to the one-and-the-same [ Base Piece ] ,
  • of the 3 [ brown-color pieces ] to the one-and-the-same [ Opposite Piece ] .

• Let us now identify the [ 1st Layer-Of-Six ] consisting of :
  • the [ 3 pieces ] as marked-off in [ light-green-color ] and
  • the [ 3 pieces ] as marked-off in [ cornflower-blue-color ] ;

    in the set of 6 diagrams below .

  
  

  • the 6 [ pieces ] are then the 6 [ pieces ] adjacent to the 3 [ orange-color pieces ] .

  Let us now combine the 2 sets of diagrams for a full view of the 6 [ pieces ] in the [ 1st Layer-Of-Six ] :

  

  We simply note here that :

  • the 3 [ light-green-color pieces ] are [ tri-symmetric ] about the [ Axis-Of-Tri-Symmetry ] ,
  • the 3 [ cornflower-blue-color pieces ] are [ tri-symmetric ] about the [ Axis-Of-Tri-Symmetry ] , and
  • the 6 [ centroidal-points ] of the 6 [ pieces ] are equi-distant from the [ centroidal-point ] of the [ Base Piece ] ,

  And to understand why this is so , let us recall 2 things here :

  • First , for any one of the 3 tri-symmetric [ orange-color pieces ] ,
    • the 3 adjacent [ pieces ] are always the [ magenta / light-green / cornflower-blue pieces ] , and :
      • the [ circulra counter-clockwise rotation-order ] here is always [ magenta-lightgreen-cornflowerblue ] .

  

  As such , the 3 adjacent [ pieces ] are always tri-symmetric about the [ orange-color piece ] so that :

  • the [ centroidal-points ] of the [ magenta / light-green / cornflower-blue pieces ] always form an Equilateral Triangle .

  It then follows from the Equilateral Triangle that :

  • the [ centrodial-points ] of the [ light-green / cornflower-blue pieces ] are always equi-distant from the [ centroidal-point ] of the [ magenta-color piece ] .
  • Secondly , the tri-symmetric nature of the 3 [ orange-color pieces ] necessarily dictates :
    • the [ tri-symmetric nature ] of the 3 [ light-green-color pieces ] , and
    • the [ tri-symmetric nature ] of the 3 [ cornflower-blue-color pieces ] ;

      given the fact that :

      • the relative positions of the [ magenta / light-green / cornflower-blue pieces ] about the [ orange-color piece ] are always maintained ;

        i.e. : the [ counter-clockwise circular rotation order ] here is always [ magenta-lightgreen-cornflowerblue ] .

• Let us now identify the [ Opposite Layer-Of-Six ] consisting of :
  • the [ 3 pieces ] as marked-off in [ dark-green-color ] and
  • the [ 3 pieces ] as marked-off in [ cyan-color ] ;

    in the set of 6 diagrams below .

  
  

  • the 6 [ pieces ] are then the 6 [ pieces ] adjacent to the 3 [ brown-color pieces ] .

  Let us now combine the 2 sets of diagrams for a full view of the 6 [ pieces ] in the [ Opposite Layer-Of-Six ] :

  

  We simply note here that :

  • the 3 [ dark-green-color pieces ] are [ tri-symmetric ] about the [ Axis-Of-Tri-Symmetry ] ,
  • the 3 [ cyan-color pieces ] are [ tri-symmetric ] about the [ Axis-Of-Tri-Symmetry ] , and
  • the 6 [ centroidal-points ] of the 6 [ pieces ] are equi-distant from the [ centroidal-point ] of the [ Opposite Piece ] ,

  And to understand why this is so , let us recall 2 things here :

  • First , for any one of the 3 tri-symmetric [ brown-color pieces ] ,
    • the 3 adjacent [ pieces ] are always the [ purple / dark-green / cyan pieces ] , and :
      • the [ circular counter-clockwise rotation-order ] here is always [ purple-darkgreen-cyan ] .

  

  As such , the 3 adjacent [ pieces ] are always tri-symmetric about the [ brown-color piece ] so that :

  • the [ centroidal-points ] of the [ purple / dark-green / cyan pieces ] always form an Equilateral Triangle .

  It then follows from the Equilateral Triangle that :

  • the [ centrodial-points ] of the [ dark-green / cyan pieces ] are always equi-distant from the [ centroidal-point ] of the [ purple-color piece ] .
  • Secondly , the tri-symmetric nature of the 3 [ brown-color pieces ] necessarily dictates :
    • the [ tri-symmetric nature ] of the 3 [ dark-green-color pieces ] , and
    • the [ tri-symmetric nature ] of the 3 [ cyan-color pieces ] ;

      given the fact that :

      • the relative positions of the [ purple / dark-green / cyan pieces ] about the [ brown-color piece ] are always maintained ;

        i.e. : the [ counter-clockwise circular rotation order ] here is always [ purple-darkgreen-cyan ] .

• This then completes the [ 1-3-6-6-3-1 Split ] of the 20 [ pieces ] comprising of :
  • the [ Base Piece ] ,
  • the [ 1st Layer-Of-Three ] ,
  • the [ 1st Layer-Of-Six ] ,
  • the [ Opposite Layer-Of-Six ] ,
  • the [ Opposite Layer-Of-Three ] , and
  • the [ Opposite Piece ] .

  

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Flipping-Over the Icosahedron :

• We are now ready to do the Flip-Over Operation on the [ 1-3-6-6-3-1 Split ] Icosahedron .

  And we shall do so now , such that :

  • the [ Opposite Piece ] now becomes the [ Base Piece ] , and
  • the [ Base Piece ] now becomes the [ Opposite Piece ] .

  And by the Flipping Over process , we have also :

  • exchanged the positions of the [ 1st Layer-Of-Three ] and the [ Opposite Layer-Of-Three ] ; and
  • excnanged the positions of the [ 1st Layer-Of-Six ] and the [ Opposite Layer-Of-Six ] .

  The situation is therefore ripe for the [ doubling-up on the Tri-Symmetries ] .

• Let us just take note here the [ Axis-Of-Tri-Symmetry ] have remained looking-the-same on the Flip-Over .

  This is because the [ Axis-Of-Tri-Symmetry ] is a straight-line passing thru 2 [ nodal-points ] , namely :

  • the [ centroid ] of the [ Base Piece ] and the [ centroid ] of the [ Opposite Piece ] .

  While the exact positions of these 2 [ nodal-points ] are exchanged during the Flip-Over ,

  • the straight-line passsing thru the 2 [ nodal-points ] is also flipped-over ,

    but the [ axial-direction ] and the [ position-of-the-line ] itself remained in-tact / looking-the-same .

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Merging Pieces for the [ 2-6-6-6-Symmetry ] :

• Let us now merge the appropriate [ pieces ] to-the-same-color to create the [ bi-symmetry ] and the triple [ hexa-symmetries ] :
  • First , we merge the [ Base Piece ] and the [ Opposite Piece ] into-the-same-color to create the [ bi-symmetry ] ,
    • and we shall use the [ magenta-color ] for these 2 [ pieces ] ;

  • Secondly , we merge the 3 [ pieces ] in the [ 1st Layer-Of-Three ] and the 3 [ pieces ] in the [ Opposite Layer-Of-Three ] into-the-same-color ,
    • to create the 1st [ hexa-symmetry ] , and we shall use the [ orange-color ] for these 6 [ pieces ] ;

  • And for the 12 [ pieces ] in the [ 1st Layer-Of-Six ] and the [ Opposite Layer-Of-Six ] , we shall :
    • merge the 3 [ cyan-color pieces ] and the 3 [ cornflower-blue-color pieces ] into-the-same-color ,
      • and we shall use the [ cornflower-blue-color ] here for these 6 [ pieces ] ;

    • merge the 3 [ light-green-color pieces ] and the 3 [ dark-green-color pieces ] into-the-same-color ,
      • and we shall use the [ light-green-color ] here for these 6 [ pieces ] .

    This last operation here then creates the last 2 [ hexa-symmetries ] , and at-the-same-time :

    • permits the previous [ bi-symmetry ] and [ hexa-symmetry ] to be maintained throughout .

  We then have this 4-color [ 2-6-6-6-Symmetry Configuration ] for the Icoshedron as shown below :

  

• Wasn't that simple !

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Where is the Hexagaon :

• But where is the Hexagon ?

  Let us try :

  • the [ 3 mid-points ] on the [ 3 edges ] common to the 3 pairs of [ adjacent light-green-color pieces ] ;

    plus :

  • the [ 3 mid-points ] on the [ 3 edges ] common to the 3 pairs of [ adjacent cornflower-blue-color pieces ] .

  

• Interesting , eh !

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go to the next section : Section XXV - [ Quad-Type Structures ] within the Icosahedron

go to the last section : Section XXIII - The Deci-Symmetric Nature of the Icosahedron

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31