On Symmetric Structures within Symmetric Solid Objects

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

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Section XI - The Bi-Symmetric Nature of the Octahedron

Summary for the Section :

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

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First Try - the { 4-colors Bi-Polar Coloring Scheme } :

• Let us now use a { 4-colors painting scheme } to color the 8 faces of the Octahedron , so that :
  • each pair of faces directly oppoiste to one-another are painted the same color .

  

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

  

• 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 [ red-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 ] , marked-off with the letter [ C ] , are green-cyan-purple in color , and :
    • the [ circular counterclockwise rotation-order ] for the 3 pieces is [ green - cyan - purple ] .

  • the 3 [ pieces ] adjacent to [ Piece B ] , marked-off with the letter [ D ] , are green-purple-cyan in color , and :
    • the [ circular counterclockwise rotation-order ] for the 3 pieces is then [ green - purple - cyan ] .

  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 { 4-colors Octahedron } into the air and it happpens to land on a [ red-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 { 4-colors Octahedron } cannot possibly be Bi-Symmetric in nature .

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Second Try - the { 4-colors Semi-Bi-Polar Coloring Scheme } :

• Let us now use another { 4-colors painting scheme } to color the 8 faces of the Octahedron , so that :
  • 1 pair of [ directly-oppoiste faces ] is painted the same color , ( red )
  • the adjacent pair of [ directly-oppoiste faces ] is painted the another color , ( cyan ) ; and
  • each of the remaining 2 pairs of [ adjacent pairs ] is each-pair painted a different color . ( green and purple )

  

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

  

• Let us now also present the [ Front View ] , [ Transparent View ] and [ Back View ] , as before :

  

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Tossing the { 4-colors Octahedron } :

• Let us now toss the { 4-colors Octahedron } into the air and let it roll on the table as if it were a dice :
  • until it finally comes to a stop and rests on one of its 8 faces on the table .

  There are four possibilities here , the { 4-colors Octahedron } either :

  • landed on a [ red-color face ] , or
  • landed on a [ cyan-color face ] , or
  • landed on a [ green-color face ] , or
  • landed on a [ purple-color face ] .

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The Question on Symmetry :

• The question on symmetry here is quadruple-folded :
  • If the { 4-colors Octahedron } landed on a { red-color face } , can we tell which of the 2 [ red-color faces ] it landed on ?
  • If the { 4-colors Octahedron } landed on a { cyan-color face } , can we tell which of the 2 [ cyan-color faces ] it landed on ?
  • If the { 4-colors Octahedron } landed on a { green-color face } , can we tell which of the 2 [ green-color faces ] it landed on ?
  • If the { 4-colors Ocvahedron } landed on a { purple-color face } , can we tell which of the 2 [ purple-color faces ] it landed on ?

  For the sake of clarity and precision , we shall list out all four (4) cases here .

• Case One --- Landed on a [ Red-Color Piece ] :

  Suppose that the { 4-colors Octahedron } did land on a [ red-color piece ] , then :

  • the [ piece ] directly opposite to the [ landed-on piece ] , at the top , is always a [ red-color piece ] ;

  • the three [ pieces ] adjacent to the [ landed-on piece ] is always [ cyan-purple-green ] , and :
    • the [ circular couterclockwise rotation-order ] for these 3 [ pieces ] is always [ cyan - purple - green ] ;

  • For these 3 [ adjacent pieces ] just afore-mentioned :
    • the next piece in-between the [ cyan-color piece ] and the [ purple-color piece ] is always a [ green-color piece ] ,
    • the next piece in-between the [ purple-color piece ] and the [ green-color piece ] is always a [ cyan-color piece ] , and
    • the next piece in-between the [ green-color piece ] and the [ cyan-color piece ] is always a [ purple-color piece ] .

    And the configuration here holds true for either of the two (2) [ red-color pieces ] .

  As such , it is impossible to tell which of the 2 [ red-color pieces ] the [ 4-colors Octahedron ] landed on .

• Case Two --- Landed on a [ Cyan-Color Piece ] :

  Suppose that the { 4-colors Octahedron } did land on a [ cyan-color piece ] , then :

  • the [ piece ] directly opposite to the [ landed-on piece ] , at the top , is always a [ cyan-color piece ] ;

  • the three [ pieces ] adjacent to the [ landed-on piece ] is always [ red-green-purple ] , and :
    • the [ circular couterclockwise rotation-order ] for these 3 [ pieces ] is always [ red - green - purple - green ] ;

  • For these 3 [ adjacent pieces ] just afore-mentioned :
    • the next piece in-between the [ red-color piece ] and the [ green-color piece ] is always a [ purple-color piece ] ,
    • the next piece in-between the [ green-color piece ] and the [ purple-color piece ] is always a [ red-color piece ] , and
    • the next piece in-between the [ purple-color piece ] and the [ red-color piece ] is always a [ green-color piece ] .

    And the configuration here holds true for either of the two (2) [ cyan-color pieces ] .

  As such , it is impossible to tell which of the 2 [ cyan-color pieces ] the [ 4-colors Octahedron ] landed on .

• Case Three --- Landed on a [ Green-Color Piece ] :

  Suppose that the { 4-colors Octahedron } did land on a [ green-color piece ] , then :

  • the [ piece ] directly opposite to the [ landed-on piece ] , at the top , is always a [ purple-color piece ] ;

  • the three [ pieces ] adjacent to the [ landed-on piece ] is always [ green-red-cyan ] , and :
    • the [ circular couterclockwise rotation-order ] for these 3 [ pieces ] is always [ green - red - cyan ] ;

  • For these 3 [ adjacent pieces ] just afore-mentioned :
    • the next piece in-between the [ green-color piece ] and the [ red-color piece ] is always a [ cyan-color piece ] ,
    • the next piece in-between the [ red-color piece ] and the [ cyan-color piece ] is always a [ purple-color piece ] , and
    • the next piece in-between the [ cyan-color piece ] and the [ green-color piece ] is always a [ red-color piece ] .

    And the configuration here holds true for either of the two (2) [ green-color pieces ] .

  As such , it is impossible to tell which of the 2 [ green-color pieces ] the [ 4-colors Octahedron ] landed on .

• Case Four --- Landed on a [ Purple-Color Piece ] :

  Suppose that the { 4-colors Octahedron } did land on a [ purple-color piece ] , then :

  • the [ piece ] directly opposite to the [ landed-on piece ] , at the top , is always a [ green-color piece ] ;

  • the three [ pieces ] adjacent to the [ landed-on piece ] is always [ purple-cyan-red ] , and :
    • the [ circular couterclockwise rotation-order ] for these 3 [ pieces ] is always [ purple - cyan - red ] ;

  • For these 3 [ adjacent pieces ] just afore-mentioned :
    • the next piece in-between the [ purple-color piece ] and the [ cyan-color piece ] is always a [ red-color piece ] ,
    • the next piece in-between the [ cyan-color piece ] and the [ red-color piece ] is always a [ green-color piece ] , and
    • the next piece in-between the [ red-color piece ] and the [ purple-color piece ] is always a [ cyan-color piece ] .

    And the configuration here holds true for either of the two (2) [ purple-color pieces ] .

  As such , it is impossible to tell which of the 2 [ purple-color pieces ] the [ 4-colors Octahedron ] landed on .

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The Bi-Symmetric Nature of the Octahedron :

• As such , we can now come to the conclusion that the { 4-colors Octahedron } does exhibit Bi-Symmetry .

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go to the next section : Section XII - The Quad-Symmetric Nature of the Octahedron

go to the last section : Section X - The Tri-Symmetric Nature of the Cube

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31