On Symmetric Structures within Symmetric Solid Objects

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

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Section XXV - [ Quad-Type Structures ] within the Icosahedron

Summary for the Section :

• In this Section , we shall identify the 10 [ Quad-Type Structures ] to be found within the Icosahedron ;
  • with each [ Quad-Type Structure ] being made-up-of four (4) specifically selected [ pieces ] , so that :
    • the [ centroodal-points ] of the 4 selected [ pieces ] always form a Tetrahedron .

  Essentially , each [ Quad-Type Structure ] is in-itself-and-by-itself [ quad-symmetric ] in nature .

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The [ ALPHA Series ] vs. the [ OMEGA Series ] :

• The 10 [ Quad-Type Structures ] can then be separated into two (2) groups :
  • the [ ALPHA Series ] --- containing a set of 5 individually-different [ Quad-Type Structures ] ,
    • to be marked-off in [ orange / magenta / dark-green / cyan / purple colors ] respectively below .

  • the [ OMEGA Series ] --- containing another set of 5 individually-different [ Quad-Type Structures ] ,
    • to be marked-off in [ light-green / blue / yellow / brown / red colors ] respectively below .

• For the [ APLHA Series ] :
  • the 5 [ Quad-Type Structures ] are structurally identical ; and

  • the 20 Pieces in [ orange / magenta / dark-green / cyan / purple colors ] are mutually exclusive with respect to one-another ,

    so that :

    • by combining the 5 differently-colored [ Quad-Type Structures ] ,

      we would end-up with a 20-Piece Icosahedron in the said-[ 5-colors ] .

    And this [ Coloring Scheme ] here for the resultant 5-Colors Icosahedron :

    • shall be known as the [ ALPHA Series - Quad-Type Structure Coloring Scheme ] ,

      for identification purposes .

• For the [ OMEGA Series ] :
  • the 5 [ Quad-Type Structures ] are also structurally identical ; and

  • the 20 Pieces in [ light-green / blue / yellow / brown / red colors ] are also mutually exclusive with respect to one-another ,

    so that :

    • by combining the 5 differently-colored [ Quad-Type Structures ] ,

      we would end-up with a 20-Piece Icosahedron in the said-[ 5-colors ] .

    And this [ Coloring Scheme ] here for the resultant 5-Colors Icosahedron :

    • shall be known as the [ OMEGA Series - Quad-Type Structure Coloring Scheme ] ,

      for identification purposes .

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The [ ALPHA-Series ] [ Quad-Type Structures ] :

• Let us now present the [ ALPHA Series - Quad-Type Structure Coloring Scheme ] for the 20-Piece Icosahedron , and also :
  • identify the 5 [ Quad-Type Structures ] in [ orange / magenta / dark-green / cyan / purple colors ] ,
    • in the set of 15 diagrams to follow , below .

  

  

  

  

  

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The [ OMEGA-Series ] [ Quad-Type Structures ] :

• Let us now present the [ OMEGA Series - Quad-Type Structure Coloring Scheme ] for the 20-Piece Icosahedron , and also :
  • identify the 5 [ Quad-Type Structures ] in [ light-green / blue / yellow / brown / red colors ] ,
    • in the set of 15 diagrams to follow , below .

  

  

  

  

  

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Why are they [ Quad-Symmetric ] in nautre :

• It is important to understand why the [ Quad-Type Structures ] are indeed [ quad-symmetric ] , and :
  • we shall now arbitrarily pick the 4 [ pieces ] in [ magenta-color ] for our analysis purposes here .

  

  And we have marked-off the 4 [ magenta-color pieces ] as [ Piece A ] , [ Piece B ] , [ Piece C ] and [ Piece D ] respectively ,

  • as shown in the diagram above .

• Based on the concept of the { [ 1-3-6-6-3-1-Split ] on the [ relative positions ] } we have presented in the last Section ,
  • we shall now make the following 4 statements on these 4 [ magenta-color pieces ] .

  STATEMENT I :

  • If we were to select [ Piece A ] as the [ Base Piece ] in the [ 1-3-6-6-3-1-Split ] , then :
    • [ Piece B ] , [ Piece C ] and [ Piece D ] would be on the [ Opposite Layer-Of-Six ] .

  STATEMENT II :

  • If we were to select [ Piece B ] as the [ Base Piece ] in the [ 1-3-6-6-3-1-Split ] , then :
    • [ Piece B ] , [ Piece C ] and [ Piece A ] would be on the [ Opposite Layer-Of-Six ] .

  STATEMENT III :

  • If we were to select [ Piece C ] as the [ Base Piece ] in the [ 1-3-6-6-3-1-Split ] , then :
    • [ Piece D ] , [ Piece A ] and [ Piece B ] would be on the [ Opposite Layer-Of-Six ] .

  STATEMENT IV :

  • If we were to select [ Piece D ] as the [ Base Piece ] in the [ 1-3-6-6-3-1-Split ] , then :
    • [ Piece A ] , [ Piece B ] and [ Piece C ] would be on the [ Opposite Layer-Of-Six ] .

• The combined effect of the 4 Statements here is that :
  • we would be able to establish that :
    • the [ relative positions ] of the 4 said-[ pieces ] with respect to one-another will always be the same .

  And as such , the 4 [ pieces ] together do always form a [ quad-symmetric structure ] .

• Let us now bring-in this set of 12 diagrams related to the above 4 Statemenrts , with :
  • the [ Base Piece ] and the [ Opposite Piece ] as marked-off in Bold-Letters ,
  • the [ 1st Layer-Of-Three ] as marked-off in [ white-color ] ,
  • the [ Opposite Layer-Of-Three ] as marked-off also in [ white-color ] , and
  • the [ 1st Layer-Of-Six ] as marked-off in [ brown-khaki-color ] .

  

  

  

  

• And because :
  • the analysis procedure we have here is in fact applicable to { all 10 colors } ;

  we can now come to the conclusion that :

  • all 10 differently-colored [ Quad-Type Structures ] are indeed all [ quad-symmetric ] in nature .

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A Quick Observation here :

• We noticed here that :
  • the [ piece ] directly-opposite to any one of the 4 [ magenta-color pieces ] is always of-a-different color .

  i.e. :

  • the [ piece ] directly-opposite to [ Piece A ] is [ Piece H ] in [ cyan-color ] ;
  • the [ piece ] directly-opposite to [ Piece B ] is [ Piece J ] in [ dark-green-color ] ;
  • the [ piece ] directly-opposite to [ Piece C ] is [ Piece K ] in [ orange-color ] ; and
  • the [ piece ] directly-opposite to [ Piece A ] is [ Piece L ] in [ purple-color ] .

• And this is true of both the [ ALPHA-Series ] and [ OMEGA Series ] [ Quad-Type Coloring Schemes ] ,
  • all of the 10 differently-colored [ Quad-Type Structures ] exhibit this particular feature here .

  And the direct consequence of this is that :

  • If we were to toss either [ ALPHA-Series ] or [ OMEGA-Series ] Icosahedron into the air and let it land on one of its 20 [ pieces ] :
    • we would be able to tell which of the [ 4 same-color pieces ] it landed-on , simply via :
      • looking at the [ color ] of the [ piece ] directly-opposite to the [ landed-on piece ] .

  Very simply :

  • When we put the 5 differently-colored [ Quad-Type Structures ] together to form an Icosahedron ,

    the resultant [ strutcture ] may not necessarily be [ quad-symmetric ] in nature !

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Cube Formation via 2 Tetrahedra :

• Let us now bring-in the [ Magenta-Color Tetrahedron ] and the [ Red-Color Tetrahedron ] from above to form a [ Cube ] :

  

• And in fact , we can form 5 [ cubes ] via the matching Tetrahedras in this table below :

  	Tetrahedron from the ALPHA Series	Tetrahedron from the OMEGA Series
  ***********	*************************	*************************
  1st Cube	Orange Color Tetrahedron	Brown Color Tetrahedron
  2nd Cube	Magenta Color Tetrahedron	Red Color Tetrahedron
  3rd Cube	Dark-Green Color Tetrahedron	Light-Green Color Tetrahedron
  4th Cube	Cyan Color Tetrahedron	Blue Color Tetrahedron
  5th Cube	Purple Color Tetrahedron	Yellow Color Tetrahedron

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An Outstanding Issue here :

• We noticed here that :
  • while :
    • the 5 [ Quad-Type Structures ] in the [ APLHA Series ] are structurally identical , and

    • the 5 [ Quad-Type Structures ] in the [ OMEGA Series ] are also structurally identical .

    the 2 types of [ Quad-Type Structures ] are in fact different .

  The finer difference here being that :

  • the [ rotational orientation ] of the 4 [ triangular pieces ] with respect to the [ frame ] of the Tetrahedron are not-the-same .

• And we shall take a closer look at this finer difference later in Part VIII of this paper ;
  • but we need to , first , bring-in the Spherical Solid Objects ,
    • and we shall do so , next , in the up-coming Part VI of this paper .

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go to the next section : Section XXVI - Defining Great-Circles , Great Circle-Planes and Great-Circle-Arcs

go to the last section : Section XXIV - The [ 2-6-6-6-Symmetry ] co-existing within the Icosahedron

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31