On Symmetric Structures within Symmetric Solid Objects

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

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Section XXI - The Quad-Symmetric Nature of the Icosahedron

Summary for the Section :

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

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The 5-colors { Quad-Symmetric Configuration } :

• First , we present this { 5-colors Configuration } of the Icosahedron 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 Icosahedron .

    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 Icosahedron .

    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 Icosahedron is [ 20-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 interesect 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 [ 5-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 { [ 5-colors Icosahedron ] 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 Icosahedron would remain the same as before .

  • Scenario Two :

    Let us now do a [ 180 degrees Rotation ] on the { [ 5-colors Icosahedron ] 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 Icosahedron would remain the same as before .

  • Scenario Three :

    Let us now do a [ 180 degrees Rotation ] on the { [ 5-colors Icosahedron ] 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 { 5-colors Icosahedron } 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 for the [ orange-color pieces ] and the [ light-green-color pieces ] ,
  • 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 also notice , at this point , that via the { [ Scenario One ] / [ Scenario Two ] / [ Scenario Three ] 180-Degrees Rotations } :
  • the positions of the 4 [ dark-khaki-color pieces ] can also be freely exchanged , and
  • the positions of the 4 [ yellow-color pieces ] can also be freely exchanged .

• As such , we have now achieved [ Quad-Symmentry ] for all five (5) colors for this particular { 5-colors configuration } of the Icosahedron .

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Identifying the Cube within the Icosahedron :

• Let us now take a quick look at the [ centroids ] for 8 particular [ pieces ] , namely :
  • the 4 [ dark-khaki-color pieces ] , and
  • the 4 [ yellow-color pieces ] .

    as identified via the 8 [ magenta-color dots ] in the diagrams below .

  

  We notice here that these 8 [ magenta-color dots ] are in fact the 8 [ nodal-points ] for a Cube :

  

• And of course :
  • the 4 [ centroidal points ] for the 4 [ dark-khaki-color pieces ] would form one [ tetrahedron ] , and
  • the 4 [ centroidal points ] for the 4 [ yellow-color pieces ] would form the other [ tetrahedron ] ;

    both hidden inside the above-said Cube .

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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 Icosahedron , 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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A Quick Comment here :

• For each set of { 3 Orthogonal Axis } , there is exactly one and only one [ Cube ] associated with the set :
  • so that we can actually identify five (5) different-and-unique [ Cubes ] for the 5 sets of { 3 orthogonal Axis } just mentioned above .

  Since there are 8 [ nodal-points ] to a [ cube ] , we are therefore looking at a total of 40 [ nodal-points ] for the 5 different-and-unique [ Cubes ] .

• But there are only 20 [ triangular pieces ] to a Icosahedron and therefore 20 [ centroidal points ] :
  • And this essentially would mean that each of the 20 [ centroidal points ] must serve on two (2) different-and-unique [ Cubes ] .

  We will provide more details on this matter later in Section XXV :

  • when we look at the [ Quad-Type Structures ] found within the Icosahedron .

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

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

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31