On Symmetric Structures within Symmetric Solid Objects

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

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Section VIII - The Bi-Symmetric Nature of the Tetrahedron

Summary for the Section :

• In this Section , we shall take a quick look at the Bi-Symmetric nature of the Tetrahedron .

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The { 2-colors Coloring Scheme } :

• Let us now use a { 2-colors painting scheme } to color the 4 faces of the Tetrahedron , so that :
  • one pair of [ adjacent faces ] is painted one color , and
  • the remaining pair of [ adjacent faces ] is painted the other color .

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

  

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Tossing the { 2-colors Tetrahderon } :

• Let us now toss the { 2-colors Tetrahedron } into the air :
  • and let it land on one of its 4 [ faces ] .

  There are only two possibilities here , the { 2-colors Tetrahedron } either :

  • landed on a [ orange-color face ] , or
  • landed on a [ yellow-color face ] .

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The Question on Symmetry :

• The question on symmetry here is two-folded :
  • If the { 2-colors Tetrahedron } landed on a { orange-color face } , can we tell which of the 2 [ orange-color faces ] it landed on ?
  • If the { 2-colors Tetrahedron } landed on a { yellow-color face } , can we tell which of the 2 [ yellow-color faces ] it landed on ?

  The answer is simply [ NO ] : we would not be able to tell the difference .

• And the reasoning here is as follows :
  • If the [ 2-colors Tetrahedron ] landed on a { orange-color face } , then :
    • the 3 [ adjacent faces ] showing-up-for-view are then [ 1-orange ] and [ 2-yellow 's ] , and :
      • the [ circular rotation-order ] for the 3 [ faces / pieces ] is then either :
        • [ orange - yellow - yellow ] , or
        • [ yellow - yellow - orange ] , or
        • [ yellow - orange - yellow ] ;

          which is , in fact , one-and-the-same [ circular rotation-order ] .

    Thus, it is impossible to tell which of the two (2) [ orange-color faces ] the { 2-colors Tetrahedron } landed on .

  • If the [ 2-colors Tetrahedron ] landed on a { yellow-color face } , then :
    • the 3 [ adjacent faces ] showing-up off-the-ground are then [ 1-yellow ] and [ 2-orange's ] , and :
      • the [ circular rotation-order ] for the 3 [ faces / pieces ] is then :
        • [ yellow - orange - orange ] , or
        • [ orange - yellow - orange ] , or
        • [ orange - orange - yellow ] ;

          which is in fact one-and-the-same [ circular rotation-order ] .

    Thus, it is impossible to tell which of the two (2) [ yellow-color faces ] the { 2-colors Tetrahedron } landed on .

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The Bi-Symmetric Nature of the Tetrahedron :

• We simply state here that the { 2-colors Tetrahedron } does exhibit Bi-Symmetry .

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go to the next section : Section IX - The Bi-Symmetric Nature of the Cube

go to the last section : Section VII - Basic Properties of the Icosahedron

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31