On Symmetric Structures within Symmetric Solid Objects

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

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Section X - The Tri-Symmetric Nature of the Cube

Summary for the Section :

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

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The { 2-colors Adjacent-Pieces Coloring Scheme } :

• Let us now use a { 2-colors painting scheme } to color the 6 faces of the Cube , so that :
  • one set of { 3 mutually-adjacent faces } is painted one color , and
  • the remaining set of { 3 mutually-adjacent faces } is painted the other color .

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

  

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Tossing the { 2-colors Cube } :

• Let us now toss the { 2-colors Cube } 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 6 faces on the table .

  There are only two possibilities here , the { 2-colors Cube } 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 Cube } landed on a { orange-color face } , can we tell which of the 3 [ orange-color faces ] it landed on ?
  • If the { 2-colors Cube } landed on a { yellow-color face } , can we tell which of the 3 [ yellow-color faces ] it landed on ?

  Our analysis here on Symmetry must necessarily begin with one of the two (2) colors as a starting point :

  • and we shall now arbitrarily select the [ orange-color ] here at the first instance .

• Let us then suppose that the { 2-colors Cube } did land on a [ orange-color face / piece ] , then :
  • the [ face / piece ] directly opposite to the [ landed-on face / piece ] , at the top , is always a [ yellow-color face / piece ] ;

  • the four [ faces / pieces ] adjacent to the [ landed-on face / piece ] is always [ 2-orange's ] and [ 2-yellow's ] , and :
    • the [ circular couterclockwise rotation-order ] for these 4 [ faces / pieces ] is always [ orange - orange - yellow - yellow ] .

    And the configuration here holds true for any of the three (3) [ orange-color faces / pieces ] .

  As such , it is impossible to tell which of the 3 [ orange-color faces / pieces ] the [ 2-colors Cube ] landed on .

• And we can now go thru the same analysis procedure for the [ yellow-color ] , to come to the conclusion that :
  • it is also impossible to tell the difference if the { 2-colors Cube } landed on any of the 3 [ yellow-color faces / pieces ] .

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The Tri-Symmetric Nature of the Cube :

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

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

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

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31