On Symmetric Structures within Symmetric Solid Objects

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

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Section XVIII - The Hexa-Symmetric Nature of the Dodecahedron

Summary for the Section :

• In this Section , we shall investigate the Hexa-Symmetric Nature of the Dodecahedron .

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The 2-colors { Hexa-Symmetric Configuration } :

• First , we present this { 2-colors Configuration } of the Dodecahedron which exhibits Hexa-Symmetry :

  

  And , in order to understand why this [ configuration ] here is indeed Hexa-Symmetric :

  • we shall now go thru the construction process for this 2-colors Dodecahedron , via :
    • first identifying an { Axis-Of-Tri-Symmetry } ,

      same as before in Section VI , when we investigated the Tri-Symmetric nature of the Dodecahedron .

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Identifying the { Axis-Of-Tri-Symmetry } :

• To set-up the { Axis-Of-Tri-Symmetry } , we shall first identify any pair of 2 [ nodal-points ] that are directly-opposite to one-another ,
  • as marked-off as the points [ Point Q1 ] and [ Point Q2 ] in the diagrams below :

  

  • We then construct a straight-line passing-thru the two (2) points , [ Point Q1 ] and [ Point Q2 ] respectively , and :
    • this then is the [ Axis-Of-Tri-Symmetry ] for our current { 2-colors Configuration } .

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The 4 Layers of [ Tri-Symmetric Pieces ] :

• For better viewing purposes , let us now put the [ Axis-Of-Tri-Symmetry ] in an upright ( quasi-vertical ) position ,
  • to arrive at this set of 3 diagrams below .

  

• Let us now regroup the 12 [ pieces ] of the Dodecahedron into 4 layers of [ Tri-Symmetric Pieces ] :
  • with each layer containing 3 [ pieces ] that are tri-symmetric with respect to one-another .

  
  
  
  

• Let us now flip-over the 2-colors Dodecahedron so that :
  • [ Point Q2 ] now becomes [ Point Q1 ] , and
  • [ Point Q1 ] now becomes [ Point Q2 ] .

  

  And a small-rotation about the [ Axis-of-Tri-Symmetry ] then yields us this configuration :

  

  • And this [ 2-colors configuration ] here is exactly the same as the one we had before the Flip-Over .

• Essentially , via the Flipping-Over process , we were able to :
  • exchange the positions of [ Layer 1 ] and [ Layer 4 ] , and
  • exchange the positions of [ Layer 2 ] and [ Layer 3 ] ;

    but without effecting any changes to the overall [ color configuration ] of the Dodecahedron .

  Thus , we have now achieved Hexa-symmetry for both the [ orange-color ] and the [ yellow-color ] ,

  • via doubling-up on the [ Tri-Symmetric Layers ] .

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Where is the Hexagon :

• But where is the Hexagon ?

  Let us try the [ 6 mid-points ] on the [ 6 edges ] common to each pair of { adjacent yellow-color pieces } :

  

  • Interesting , eh !
    • Benzene Ring ?

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A Quick Question :

• Again we ask this question :
  • How many different and distinct [ Axis-Of-Tri-Symmetry ] are there ?
    • And the answer here is again Ten (10) .

  Very simply , there are twenty (20) [ nodal-points ] to a Dodecahedron , so that :

  • there are exactly 10 pairs of [ directly-opposite nodal-points ] available for the formation of an individual [ Axis-Of-Tri-Symmetry ] .

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go to the next section : Section XIX - Bi-Polar Structures within the Icosahedron

go to the last section : Section XV - The Quad-Symmetric Nature of the Dodecahedron

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31