On Symmetric Structures within Symmetric Solid Objects

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

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Section XIX - Bi-Polar Structures within the Icosahedron

Summary for the Section :

• In this Section , we shall investigate the Bi-Polar Structures found within the Icosahedron .

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Identifying 10 { Bi-Polar Pairs } :

• We take note , at this point , that the 20 faces of the Icosahedron can be divided into ten (10) pairs of [ Bi-Polar Faces ] :
  • with each pair of faces being directly-opposite to one-another .

  
  
  

  And we have marked-off each to the ten (10) [ Bi-Polar Pairs ] in a different color , i.e. :

  • the colors [ cyan ] , [ purple ] , [ yellow ] , [ brown ] , [ light-green ] , [ red ] , [ dark-green ] , [ orange ] , [ cornflower-blue ] and [ magenta ] respectively .

• We then have this view below for the [ re-combined structure ] of 10-colors :

  

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Bringing-in the [ Back View ] :

• Let us now sneak-around to the back-side to take a look at the [ Back View ] , and we present this set of 3 diagrams below :
  • the diagram on-the-left is the [ Front View ] ,
  • the diagram in-the-middle is the [ Transparent View ] , and
  • the diagram on-the-right is the [ Back View ] .

  

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

• We notice at this point that :
  • the 3 [ pieces ] adjacent to [ Piece A ] , are red-purple-lightgreen in color , and :
    • the [ circular counterclockwise rotation-order ] for the 3 pieces is [ red - purple - lightgreen ] .

  • the 3 [ pieces ] adjacent to [ Piece B ] , are red-lightgreen-purple in color , and :
    • the [ circular counterclockwise rotation-order ] for the 3 pieces is then [ red - lightgreen - purple ] .

  We see here immediately that :

  • the { [ circular counterclockwise rotation-order ] for the [ 3 adjacent pieces ] } are different for [ Piece A ] and [ Piece B ] .

• Thus , if we were to toss this { 10-colors Icosahedron } into the air and it happpens to land on an [ cyan-color piece ] :
  • we would be able to tell [ Piece A ] from [ Piece B ] because of the color-configuration difference in their [ 3 adjacent pieces ] .

  As such , this { 10-colors Icosahedron } cannot possibly be Bi-Symmetric in nature .

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Question on Bi-Symmetry :

• The Question then arises as to whether this [ Bi-Polar Coloring Scheme ] for the { 10-colors Icosahedron } is Bi-Symmetric ?
  • And the anwser is simply [ NO ] .

  Thus , we must now look for another { 6-colors Coloring Scheme } that might give us a { Bi-Symmetric Icosahedron } .

  • And we shall do so , next .

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go to the next section : Section XX - The Bi-Symmetric Nature of the Icosahedron

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

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31