On Symmetric Structures within Symmetric Solid Objects

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

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Section XXIII - The Deci-Symmetric Nature of the Icosahedron

Summary for the Section :

• In this Section , we shall investigate the Deci-Symmetric Nature of the Icosahedron .

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The 2-colors { Deci-Symmetric Configuration } :

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

  

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

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

      same as before in the last Section ( Section XXII ) , when we investigated the Penta-Symmetric nature of the Icosahedron .

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

• To set-up the { Axis-Of-Penta-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 [ Penta-Symmetric Pieces ] :

• We note here that our [ Axis-Of-Penta-Symmetry ] here in already in an upright ( quasi-vertical ) position ,
  • and we can start identifying the { penta-symmetric groups } immediately .

• Let us now regroup the 20 [ pieces ] of the Icosahedron into 4 layers of [ Penta-Symmetric Pieces ] :
  • with each layer containing 5 [ pieces ] that are penta-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 an [ 180-Degrees Rotation ] about the [ Axis-of-Penta-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 Icosahedron .

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

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

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A Quick Question :

• Again we ask this question :
  • How many different and distinct [ Axis-Of-Penta-Symmetry ] are there ?
    • And the answer here is again Six (6) .

  Very simply , there are twelve (12) [ nodal-points ] to a Icosahedron , so that :

  • there are exactly 6 pairs of [ directly-opposite nodal-points ] available for the formation of an individual [ Axis-Of-Penta-Symmetry ] .

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go to the next section : Section XXIV - The [ 2-6-6-6-Symmetry ] co-existing within the Icosahedron

go to the last section : Section XXII - The Penta-Symmetric Nature of the Icosahedron

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31