On Symmetric Structures within Symmetric Solid Objects

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

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Section XVI - The Tri-Symmetric Nature of the Dodecahedron

Summary for the Section :

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

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The 4-colors { Tri-Symmetric Configuration } :

• First , we present this { 4-colors Configuration } of the Dodecahedron which exhibits Tri-Symmetry :

  

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

  • we shall now go thru the Construction Process for the construction of this particular [ configuration ] , next :
    • for better insights and to highlight some of the issues involved .

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The Construction Process for the { 4-colors Configuration } :

• STEP 1 :

  We first set up the [ Axis-Of-Tri-Symmetry ] for our [ 4-colors Configuration ] here , via :

  • first identifying 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 { 4-colors Configuration } .

• STEP 2 :

  We then color the three (3) [ pieces ] touching on [ Point Q1 ] the [ 1st Color ] ( magenta ) , and

  • we also color the three (3) [ pieces ] touching on [ Point Q2 ] the [ 2nd-Color ] ( orange ) .

  

• STEP 3 :

  In order to maintain the Tri-Symmetric nature of the 3 [ magenta-color pieces ] , we must now necessarily :

  • color the 3 [ pieces ] { each adjacent to a pair of 2 adjacent [ magenta-color pieces ] } , the [ 3rd Color ] ( green ) .

  And , in order to maintain the Tri-Symmetric nature of the 3 [ orange-color pieces ] , we must now necessarily :

  • color the 3 [ pieces ] { each adjacent to a pair of 2 adjacent [ orange-color pieces ] } , the [ 4th-Color ] ( yellow ) .

  

  This then completes our Construction Process here .

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The Analysis on Tri-Symmetry :

• Let us now bring back the [ 4-colors Configuration ] we have above :

  

  • And we have now marked-off the 3 [ magenta-color pieces ] as [ Piece A ] , [ Piece B ] , and [ Piece C ] respectively .

  Let us now do a [ 120 degrees Rotation ] on the [ 4-colors Dodecahedron ] ,

  • using the [ Axis-Of-Tri-Symmetry ] as the [ axis-of-rotation ] ;

    to arrive at this configuration below .

  

  • We simply note here that the [ overall 4-colors configuration ] have remained the same / unchanged after the Rotation .

  Let us now do another [ 120 degrees Rotation ] on the [ 4-colors Dodecahedron ] ,

  • using again the [ Axis-Of-Tri-Symmetry ] as the [ axis-of-rotation ] ;

    to arrive at this configuration below .

  

  • We simply note here that the [ overall 4-colors configuration ] have also remained the same / unchanged after the Rotation .

  We have therefore exchanged the positions of the 3 [ magenta-color pieces ] , i.e. [ Piece A ] , [ Piece B ] , and [ Piece C ] :

  • in a { circular-rotationary-order } ,
    • but the [ 4-colors configuration ] for the { 4-colors Dodecahedron } have always remained the same / unchanged .

  Essentially , if we were to toss the { 4-colors Dodecahedron } into the air and it happens to land on a [ magenta-color piece ] :

  • we would not be able to tell which of the three (3) [ magenta-color pieces ] it landed-on .

• We also take note , at this point , that via the two (2) { 120 Degrees Rotations } above :
  • the positions of the 3 [ orange-color pieces ] have also been exchanged in a { circular-rotationary-order } ;
  • the positions of the 3 [ green-color pieces ] have also been exchanged in a { circular-rotationary-order } ; and
  • the positions of the 3 [ yellow-color pieces ] have also been exchanged in a { circular-rotationary-order } ;

    but the [ overall 4-colors configuration ] for the { 4-colors Dodecahedron } have always remained the same .

  As such , if we were to toss the { 4-colors Dodecahedron } into the air and it happens to land on any one of these 3 colors :

  • we also would not be able to tell which of the 3 [ same-color-pieces ] the Dodecahedron landed-on .

• We have therefore achieved [ Tri-Symmetry ] for all four (4) colors for this particular { 4-colors configuration } of the Dodecahedron .

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A Quick Question :

• We now ask this question :
  • How many different and distinct [ Axis-Of-Tri-Symmetry ] are there ?
    • And the answer here is 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 XVII - The Quad-Symmetric Nature of the Dodecahedron

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

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31