On Symmetric Structures within Symmetric Solid Objects

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

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Section XVII - The Quad-Symmetric Nature of the Dodecahedron

Summary for the Section :

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

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The 3-colors { Quad-Symmetric Configuration } :

• First , we present this { 3-colors Configuration } of the Dodecahedron which exhibits Quad-Symmetry :

  

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

  • we shall now identify a set of 3 { Bi-Symmetric Axis } to guide us thru the analysis , below .

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Identifying the set of 3 { Bi-Symmetric Axis } :

• First , we identify the [ BLUE Axis ] , as follows :

  We first identify the [ mid-point ] on the common edge of each individual-pair of the 2 pairs of { adjacent [ cyan-color pieces ] } :

  • as marked-off as [ Point Q1 ] and [ Point Q2 ] respectively in the diagrams below :

  

  We then set up a straight-line passing thru the 2 said points , and :

  • this is our [ BLUE Axis ] .

• Secondly , we identify the [ RED Axis ] , as follows :

  We first identify the [ mid-point ] on the common edge of each individual-pair of 2 pairs of { adjacent [ orange-color pieces ] } :

  • as marked-off as [ Point Q3 ] and [ Point Q4 ] respectively in the diagrams below :

  

  We then set up a straight-line passing thru the 2 said points , and :

  • this is our [ RED Axis ] .

• Thirdly , we identify the [ GREEN Axis ] , as follows :

  We first identify the [ mid-point ] on the common edge of each individual-pair of 2 pairs of { adjacent [ light-green-color pieces ] } :

  • as marked-off as [ Point Q5 ] and [ Point Q6 ] respectively in the diagrams below :

  

  We then set up a straight-line passing thru the 2 said points , and :

  • this is our [ GREEN Axis ] .

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2 Important Properties of the set of 3 { Bi-Symmetric Axis } :

• Let us now take note of 2 important properties for the set of 3 { Bi-Symmetric Axis } :

  

  First Property :

  • The 3 [ Axis ] do intersect at a single point , namely the [ centroid ] of the Dodecahedron .

    This is because each pair of directly-opposite edges are always [ bi-polar ] in nature , so that :

    • the straight-line joining their mid-points must necessarily pass thru the [ centroid ] of the Dodecahedron .

    As such , the [ centroid ] is common to all 3 Axis and is , in fact , the [ point-of-intersection ] for the 3 [ Axis ] .

  Second Property :

  • The 3 Axis are orthogonal / perpendicular to one-another .

    This is because the Dodecahedron is [ 12-Symmetric ] and our 3-colors { coloring scheme } here does allow for :

    • the 3 colors to be exchanged freely and randomly without effecting any changes to the [ structural shape ] of the [ 3-Axis Frame ] .

    As such , each pair of [ axis ] must necessarily intersect one-another at [ right-angles ( 90-Degrees-Angles ) ] .

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The Analysis on Quad-Symmetry :

• For our analysis purposes here , let us now set up a [ Base Position ] for the [ 3-colors Configuration ] as shown below :

  

  • And we have now marked-off the 4 [ cyan-color pieces ] as [ Piece A ] , [ Piece B ] , [ Piece C ] and [ Piece D ] respectively .

• Let us now try any one of the 3 scenarios listed below , for our analysis :
  • Scenario One :

    Let us now do a [ 180 degrees Rotation ] on the { [ 3-colors Dodecahedron ] in the [ Base Position ] } ,

    • using the [ BLUE Axis ] as the [ axis-of-rotation ] .

    And by doing so , we would have :

    • exchanged the positions of [ Piece A ] and [ Piece B ] , but :
      • the overall { color configuration } of the resultant Dodecahedron would remain the same as before .

  • Scenario Two :

    Let us now do a [ 180 degrees Rotation ] on the { [ 3-colors Dodecahedron ] in the [ Base Position ] } :

    • using the [ RED Axis ] as the [ axis-of-rotation ] .

    And by doing so , we would have :

    • exchanged the positions of [ Piece A ] and [ Piece C ] , but :
      • the overall { color configuration } of the resultant Dodecahedron would remain the same as before .

  • Scenario Three :

    Let us now do a [ 180 degrees Rotation ] on the { [ 3-colors Dodecahedron ] in the [ Base Position ] } :

    • using the [ GREEN Axis ] as the [ axis-of-rotation ] .

    And by doing so , we would have :

    • exchanged the positions of [ Piece A ] and [ Piece D ] , but :
      • the overall { color configuartion } of the resultant Dodecahedron would remain the same as before .

  Thus , we can now conclude that the 4 [ cyan-color pieces ] are indeed freely interchangeable in nature .

• Essentially , if we were to toss the { 3-colors Dodecahedron } in the air and it happens to land on a [ cyan-color piece ] :
  • we would not be able to tell which of the four (4) [ cyan-color pieces ] it landed-on .

• We can then perform the exact-same kind of analysis on the other 2 remaining colors ,
  • to come to the conclusion that :
    • the 4 [ orange-color pieces ] are also freely interchangeable , and
    • the 4 [ light-green-color pieces ] are also freely interchangeable .

• We have therefore achieved [ Quad-Symmentry ] for all three (3) colors for this particular { 3-colors configuration } of the Dedecahedron .

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A Quick Question :

• We now ask this question :
  • How many different and distinct sets of 3 [ Axis-Of-Bi-Symmetry ] are there ?
    • And the answer here is Five (5) .

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

  • there are exactly 5 sets of [ 3 pairs of directly-opposite edges ] available for the formation of the [ 3-Axis-Set ] .

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go to the next section : Section XVIII - The Hexa-Symmetric Nature of the Dodecahedron

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

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31