On Symmetric Structures within Symmetric Solid Objects

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

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Section XV - The Bi-Symmetric Nature of the Dodecahedron

Summary for the Section :

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

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The 6-colors { Bi-Symmetric Configuration } :

• First , we present this { 6-colors Configuration } of the Dodecahedron which exhibits Bi-Symmetry :

  

  And , in order to understand why this [ configuration ] here is indeed Bi-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 { 6-colors Configuration } :

• STEP 1 :

  We first identify any pair of 2 adjacent [ pieces ] and color these the [ 1st-Color ] ( cyan ) , and :

  • we also color the two (2) [ pieces ] directly-opposite to these 2 [ pieces ] the [ 2nd-Color ] ( blue ) .

  

  Let us now also take this opportunity to identify the [ Axis-Of-Bi-Symmetry ] for our later use below , via :

  • First identifying the [ mid-point ] on the edge common to the 2 [ cyan-color pieces ] as the point [ Point Q1 ] ;
  • Secondly , identifying the [ mid-point ] on the edge common to the 2 [ blue-color pieces ] as the point [ Point Q2 ] .

  

  • 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-Bi-Symmetry ] for our current { 6-colors Configuration } .

• STEP 2 :

  In order to maintain the Bi-Symmetric nature of the 2 [ cyan-color pieces ] , we must now necessarily :

  • color the 2 [ pieces ] { each adjacent to both of the 2 [ cyan-color pieces ] } , the [ 3rd Color ] ( orange ) .

  And , in order to maintain the Bi-Symmetric nature of the 2 [ blue-color pieces ] , we must now necessarily :

  • color the 2 [ pieces ] { each adjacent to both of the 2 [ blue-color pieces ] } , the [ 4th-Color ] ( magenta ) .

  

• Step 3 :

  For the 4 remaining [ pieces ] , we shall now use a [ Bi-Polar coloring scheme ] on each of the 2 pairs of [ bi-polar pieces ] , as follows :

  • For the first pair of [ bi-polar pieces ] , we simply color these the [ 5th-Color ] ( green ) , and
  • For the other pair of [ bi-polar pieces ] , we simply color these the final [ 6th-Color ] ( purple ) .

  

  This then permits for :

  • the [ bi-symmetric nature ] of the [ 2 cyan-color pieces ] to be maintained ; and
  • the [ bi-symmetric nature ] of the [ 2 blue-color pieces ] to be maintained , too .

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The Analysis on Bi-Symmetry :

• Let us now bring back the [ 6-colors Configuration ] we have above :

  

  • And we have now marked-off the 2 [ cyan-color pieces ] as [ Piece A ] and [ Piece B ] respectively .

  Let us now do a [ 180 degrees Rotation ] on the [ 6-colors Dodecahedron ] ,

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

    to arrive at this configuration below .

  

  We have therefore exchanged the positions of the 2 [ cyan-color pieces ] , i.e. [ Piece A ] and [ Piece B ] :

  • but the [ 6-colors configuration ] for the { 6-colors Dodecahedron } remains the same .

  Essentially , if we were to toss the { 6-colors Dodecahedron } into the air and it happens to land on a [ cyan-color piece ] :

  • we would not be able to tell which of the two (2) [ cyan-color pieces ] it landed-on .

• We also take note , at this point , that via the { 180 Degrees Rotation } above :
  • the positions of the 2 [ blue-color pieces ] have also been exchanged ;
  • the positions of the 2 [ orange-color pieces ] have also been exchanged ;
  • the positions of the 2 [ magenta-color pieces ] have also been exchanged ;
  • the positions of the 2 [ green-color pieces ] have also been exchanged ; and
  • the positions of the 2 [ purple-color pieces ] have also been exchanged .

    but the [ overall 6-colors configuration ] for the { 6-colors Dodecahedron } remained the same .

  As such , if we were to toss the { 6-colors Dodecahedron } into the air and it happens to land on any one of these 5 colors :

  • we also would not be able to tell which of the 2 [ same-color-pieces ] the Dodecahedron landed-on .

• We have therefore achieved [ Bi-Symmentry ] for all six (6) colors for this particular { 6-colors configuration } of the Dedecahedron .

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A Quick Question :

• We now ask this question :
  • How many different and distinct [ Axis-Of-Bi-Symmetry ] are there ?
    • And the answer here is Fifteen (15) .

  Very simply , there are thirty (30) edges to a Dodecahedron , so that :

  • there are exactly 15 pairs of [ directly-opposite edges ] available for the formation of individual [ Axis-Of-Bi-Symmetry ] .

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go to the next section : Section XVI- The Tri-Symmetric Nature of the Dodecahedron

go to the last section : Section XIV - The Bi-Polar Nature of the Dodecahedron

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31