On Symmetric Structures within Symmetric Solid Objects

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

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Section XII - The Quad-Symmetric Nature of the Octahedron

Summary for the Section :

• In this Section , we shall take a quick look at the Quad-Symmetric nature of the Octahedron :
  • via 3 different-and-distinct { 2-colors coloring schemes } , namely :
    • the { Checkers-Board Coloring Scheme } ,
    • the { Water-Mellon-Slice Coloring Scheme } , and
    • the { Half-On-Half Coloring Scheme } .

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The { Checkers-Board Coloring Scheme } :

• Let us now use a { 2-colors painting scheme } to color the 8 faces of the Octahedron , so that :
  • [ adjacent-faces ] are always [ of-a-different-color ] .

  We then have the resultant { 2-colors Octahedron } as shown below :

  

  And we also bring-in the [ Front View ] , the [ Transparent View ] and the [ Back View ] , as before :

  

  And this then is known as the [ Checkers-Board Coloring Scheme ] for the Octahedron .

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The { Water-Mellon-Slice Coloring Scheme } :

• Let us now use a { 2-colors painting scheme } to color the 8 faces of the Octahedron , as follows :
  • we first paint a pair ( any pair ) of [ adjacent-faces ] the first-color ( orange ) ;
  • then we paint the pair of [ adjacent-faces ] directly-oppoiste to the [ original-adjacent-pair ] above the same first-color ( orange ) ; and
  • then we paint the remaining 4 faces the other-color ( yellow ) .

  

  We then have the resultant { 2-colors Octahedron } as shown below :

  

  And we also bring-in the [ Front View ] , the [ Transparent View ] and the [ Back View ] , as before :

  

  The core coloring-operation here is then likened unto :

  • Cutting a water-mellon into 4 equal slices and painting each pair of [ opposite slices ] the same color .

    And hence the name [ Water-Mellon-Slice Coloring Scheme ] .

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The { Half-On-Half Coloring Scheme } :

• Let us now use a { 2-colors painting scheme } to color the 8 faces of the Octahedron , so that :
  • one-half of the Octahedron is painted the first-color ( orange ) , and
  • the other-half of the Octahedron is painted the other-color ( yellow ) .

  We then have the resultant { 4-colors Octahedron } as shown below :

  

  And we also bring-in the [ Front View ] , the [ Transparent View ] and the [ Back View ] , as before :

  

  And this then is known as { Half-On-Half Coloring Scheme } for the Octahedron .

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Tossing the { 2-Colors Octahedron } :

• Let us now take any of the three (3) { 2-colors Octahedra } above , i.e. :
  • the [ Checkers-Board Coloring Scheme ] { 2-colors Octahedron } ,
  • the [ Water-Mellon-Slice Coloring Scheme ] { 2-colors Octahedron } , or
  • the [ Half-On-Half Coloring Scheme ] { 2-colors Octahedron } ;

    and toss it into-the-air and let it land on any one of its 8 faces .

• There are only two (2) possibilities here , the { 2-colors Octahedron } either :
  • landed on an [ orange-color piece ] , or
  • landed on a [ yellow-color piece ] .

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A simple Statement now made :

• We now make this simple statement :
  • If the { 2-colors Octahedron } did land on an [ orange-color piece ] ,
    • we would not be able to tell which of the 4 [ orange-color pieces ] it landed-on ;

  • If the { 2-colors Octahedron } did land on an [ yellow-color piece ] ,
    • we would not be able to tell which of the 4 [ yellow-color pieces ] it landed-on ;

• This should be easy enough to derive and :
  • we shall let the reader do his / her own confirmation work here .

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The Quad-Symmetric Nature of the Octahedron :

• As such , we can now come to the conclusion that the { 2-colors Octahedron } does exhibit Quad-Symmetry .

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go to the next section : Section XIII - Brief Summary on the Tetrahedron / Cube / Octahedron

go to the last section : Section XI - The Bi-Symmetric Nature of the Octahedron

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31