On Symmetric Structures within Symmetric Solid Objects

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

Top Of Page

Section IX - The Bi-Symmetric Nature of the Cube

Summary for the Section :

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

go to Top Of Page

The { 3-colors Bi-Polar Coloring Scheme } :

• Let us now use { 3-colors painting scheme } to color the 6 faces of the Cube , so that :
  • each pair of faces directly oppoiste to one-another are painted the same color .

  We then have the resultant { 3-colors Cube } as shown below :

  

go to Top Of Page

Tossing the { 3-colors Cube } :

• Let us now toss the { 3-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 three possibilities here , the { 3-colors Cube } either :

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

go to Top Of Page

The Question on Symmetry :

• The question on symmetry here is triple-folded :
  • If the { 3-colors Cube } landed on a { red-color face } , can we tell which of the 2 [ red-color faces ] it landed on ?
  • If the { 3-colors Cube } landed on a { green-color face } , can we tell which of the 2 [ green-color faces ] it landed on ?
  • If the { 3-colors Cube } landed on a { cyan-color face } , can we tell which of the 2 [ cyan-color faces ] it landed on ?

  Our analysis here on Symmetry must necessarily begin with one of the three (3) colors as a starting point :

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

• Let us then suppose that the { 3-colors Cube } did land on a [ red-color face / piece ] , then :
  • the [ face / piece ] directly opposite to the [ landed-on face / piece ] , at the top , is always a [ red-color face / piece ] ;

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

      ( the [ axis-of-rotation ] for the counterclockwise direction is always the [ axis ] perpendicular-to and pertruding-out-of the [ landed-on face ] ) .

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

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

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

    • it is also impossible to tell the difference if the { 3-colors Cube } landed on either of the 2 [ cyan-color faces / pieces ] .

go to Top Of Page

The Bi-Symmetric Nature of the Cube :

• As such , we can now come to the conclusion that the { 3-colors Cube } does exhibit Bi-Symmetry .

  But let us also bring-in this { 3-colors configuration } of the Cube for a tantilizing stint :

  

  • This configuration is also Bi-Symmetric !

go to Top Of Page

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

go to the last section : Section VIII - The Bi-Symmetric Nature of the Tetrahedron

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31