On Symmetric Structures within Symmetric Solid Objects

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

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Section XXII - The Penta-Symmetric Nature of the Icosahedron

Summary for the Section :

• In this Section , we shall investigate the Penta-Symmetric Nature of the Icosahedron .

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The 4-colors { Penta-Symmetric Configuration } :

• First , we present this { 4-colors Configuration } of the Icosahedron which exhibits Penta-Symmetry :
  

  And , in order to understand why this [ configuration ] here is indeed Penta-Symmetric :

  • we shall now go thru the Construction Process for the construction of this particular [ configuration ] , next :
    • for better insights and to highlight some of the issues involved .

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The Construction Process for the { 4-colors Configuration } :

• STEP 1 :

  We first set up the [ Axis-Of-Penta-Symmetry ] for our [ 4-colors Configuration ] here , via :

  • first identifying any pair of 2 [ nodal-points ] that are directly-opposite to one-another ,
    • as marked-off as the points [ Point Q1 ] and [ Point Q2 ] in the diagrams below :

  

  • We then construct a straight-line passing-thru the two (2) points , [ Point Q1 ] and [ Point Q2 ] respectively , and :
    • this then is the [ Axis-Of-Penta-Symmetry ] for our current { 4-colors Configuration } .

• STEP 2 :

  We then color the five (5) [ pieces ] touching on [ Point Q1 ] the [ 1st Color ] ( magenta ) , and

  • we also color the five (5) [ pieces ] touching on [ Point Q2 ] the [ 2nd-Color ] ( green ) .

  

• STEP 3 :

  In order to maintain the Penta-Symmetric nature of the 5 [ magenta-color pieces ] , we must now necessarily :

  • color the 5 [ pieces ] { each adjacent to a [ magenta-color piece ] } the [ 3rd Color ] ( yellow ) .

  And , in order to maintain the Penta-Symmetric nature of the 5 [ green-color pieces ] , we must now necessarily :

  • color the 5 [ pieces ] { each adjacent to a [ green-color piece ] } , the [ 4th-Color ] ( orange ) .

  

  This then completes our Construction Process here .

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The Analysis on Penta-Symmetry :

• Let us now bring back the [ 4-colors Configuration ] we have above :

  

  • And we have now marked-off the 5 [ magenta-color pieces ] as [ Piece A ] , [ Piece B ] , [ Piece C ] , [ Piece D ] and [ Piece E ] respectively .

  Let us now do a successive of five (5) [ 72-Degrees Rotations ] on the [ 4-colors Icosahedron ] ,

  • using the [ Axis-Of-Penta-Symmetry ] as the [ axis-of-rotation ] ;

• We simply note here that , via the [ 72-Degrees Rotations ] :
  • we would be able to exhange [ Piece A ] with any one of the other 4 [ magenta-color pieces ] ,
    • but the [ overall 4-colors configuration ] will remain the same / unchanged after the Rotation .

  Thus , we have now achieved Penta-Symmetry for the 5 [ magenta-color pieces ] .

  Essentially , if we were to toss the { 4-colors Icosahedron } into the air and it happens to land on a [ magenta-color piece ] :

  • we would not be able to tell which of the five (5) [ magenta-color pieces ] it landed-on .

• We also take note , at this point , that via the series of { 72-Degrees Rotations } above :
  • the positions of the 5 [ green-color pieces ] may also be freely exchanged ,
  • the positions of the 5 [ yellow-color pieces ] may also be freely exchanged , and
  • the positions of the 5 [ orange-color pieces ] may also be freely exchanged ;

    but the [ overall 4-colors configuration ] for the { 4-colors Icosahedron } would have remained the same .

  As such , if we were to toss the { 4-colors Icosahedron } into the air and it happens to land on any one of these 3 colors :

  • we also would not be able to tell which of the 5 [ same-color-pieces ] the Icosahedron landed-on .

• We have therefore achieved [ Penta-Symmetry ] for all four (4) colors for this particular { 4-colors configuration } of the Icosahedron .

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A Quick Question :

• We now ask this question :
  • How many different and distinct [ Axis-Of-Penta-Symmetry ] are there ?
    • And the answer here is Six (6) .

  Very simply , there are twelve (12) [ nodal-points ] to a Icosahedron , so that :

  • there are exactly 6 pairs of [ directly-opposite nodal-points ] available for the formation of an individual [ Axis-Of-Penta-Symmetry ] .

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go to the next section : Section XXIII - The Deci-Symmetric Nature of the Icosahedron

go to the last section : Section XXI - The Quad-Symmetric Nature of the Icosahedron

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31