On Symmetric Structures within Symmetric Solid Objects

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

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Section XXXXI - The Rotational Orientation of the [ Curved Triangular Pieces ]

Summary for the Section :

• In this Section , we shall take a look at the [ Rotational Orientation ] of the [ curved triangular pieces ] ,
  • attached to the 4 [ nodal-points ] of each of the 2 [ Quad-Type Structures ] .

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Analysis of the [ Magenta-Color Quad-Type Structure ] :

• Let us now mark-off the 4 [ nodal-points ] of the [ Magenta-Color Tetrahedron ] :
  • as the points [ Point A ] , [ Point B ] , [ Point C ] and [ Point D ] respectively ,
    • as per the diagram on-the-left below .

  

  Let us also draw-in the [ great-circle arcs ] for [ Points A-B / A-C / A-D ] respectively ,

  • as marked-off in [ dark-green color ] in the diagram in-the-middle above .

  We then identify [ Point P ] as the [ centroid ] of the [ Sphere ] and :

  • draw-in the straight-lines [ P-A / P-B / P-C / P-D ] , respectively ,
    • as marked-off in [ light-green color ] in the diagram on-the-right above .

  We have therefore set-up a [ structural frame in dark-green / light-green colors ] ,

  • within our [ Magenta-Color Quad-Type Structure ] here .

• Let us now take a quick look of this [ structural frame ] ,
  • in relation to the original [ ALPHA Series ] [ Icosahedral Sphere ] .

  

• Our analysis here is rather simple :

  

  IF :

  • we were to start-off with the [ curved trianglar piece ] such that :
    • its 3 [ nodal-points ] are resting on the 3 dark-green-color [ great-circle arcs ] ,

      ( as per the diagram on-the-left above ) .

  THEN :

  • we would have to do a counter-clockwise rotation of [ 37.76 degrees ] , in order to :
    • get the said-[ curved triangular piece ] into its proper position ,

      as per the diagram on-the-right , above .

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Analysis of the [ Red-Color Quad-Type Structure ] :

• Let us now mark-off the 4 [ nodal-points ] of the flipped-over [ Red-Color Tetrahedron ] :
  • as the points [ Point A ] , [ Point B ] , [ Point C ] and [ Point D ] respectively ,

    as per the diagram on-the-left below .

  

  Let us also draw-in the [ great-circle arcs ] for [ Points A-B / A-C / A-D ] respectively ,

  • as marked-off in [ blue color ] in the diagram in-the-middle above .

  We then identify [ Point P ] as the [ centroid ] of the [ Sphere ] and :

  • draw-in the straight-lines [ P-A / P-B / P-C / P-D ] , respectively ,
    • as marked-off in [ cyan color ] in the diagram on-the-right above .

  We have therefore set-up a [ structural frame in blue / cyan colors ] ,

  • within our [ Red-Color Quad-Type Structure ] here .

• Let us now take a quick look of this [ structural frame ] ,
  • in relation to the original flipped-over [ OMEGA Series ] [ Icosahedral Sphere ] .

  

• Our analysis here is rather simple :

  

  IF :

  • we were to start-off with the [ curved trianglar piece ] such that :
    • its 3 [ nodal-points ] are resting on the 3 blue-color [ great-circle arcs ] ,

      ( as per the diagram on-the-left above ) .

  THEN :

  • we would have to do a clockwise rotation of [ 37.76 degrees ] , in order to :
    • get the said-[ curved triangular piece ] into its proper position ,

      as per the diagram on-the-right , above .

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The Finer Difference :

• We simply note here that the finer difference between the 2 types of [ Quad-Type Structures ] simply arises from :
  • the different [ rotational orientations ] of the [ curved triangular pieces ] ,

    with respect to the [ frame ] of the host-[ Tetrahedron ] .

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go to the next section : Section XXXXII - A Special Feature of the [ Quad-Type Structure Coloring Schemes ]

go to the last section : Section XXXX - A First Look at the 2 [ Quad-Type Structures ]

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31