On Symmetric Structures within Symmetric Solid Objects

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

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Section XXXIV - Duality Linkage via the [ TFS ( 24-Split ) Sphere ]

Saummary for the Section :

• In this Section , we shall take a look at the [ Tetrahedron vs. Tetrahedron ] Duality :
  • with the [ TFS ( 24-Split ) Sphere ] here acting as the missing-link .

  ( TFS is then the abbreviation for the [ Twenty-Four-Split ] ) .

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A First Look at the [ TFS Sphere ] :

• We now present this first look for the [ TFS Sphere ] with a [ 24 Split ] :

  

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Construction of the [ TFS Sphere ] :

• To construct the [ TFS Sphere ] , we simply take a [ Tetrahedral Sphere ] and :
  • split each of 4 [ curved triangular faces ] into six (6) equal parts .

  The procedure is as follows :

  • We first identify a [ curved triangular face ] for splitting ,
    • as marked-off in [ white-color ] in the diagrams below .

  

  • We then identify the 3 [ mid-points ] of the 3 [ Great-Circle Arc ] serving as the 3 edges ,
    • as marked-off via the 3 [ blue-color dots ] in the diagrams above .

  • We then construct 3 [ Great-Cirlce Arcs ] joining each [ mid-point ] with its opposite vertex .

  

  • as per the set of 3 diagrams above .

    This then divides the [ curve triangular face ] into six (6) equal parts .

• Let us now take a quick look at this version of the resultant [ TFS Sphere ] :

  

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Linkage via the [ TFS Sphere ] :

• We then provide this set of Linkage Diagrams for the [ Tetrahedron vs. Tetrahedron ] Duality :

  For the Front View , we have :

  

  And for the [ Transparent View ] , we have :

  

  [ Tetrahedral Sphere I ] and [ Tetrhedral Sphere II ] are therefore linked via the [ TFS ( 24-Split ) Sphere ] .

• Rather simple and straight-forward !

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A Quick Observation :

• We note here that :
  • the [ 1st Tetrahedral Sphere ] has 4 [ nodal-points ] , and
  • the [ 2nd Tetrahedral Sphere ] also has 4 [ nodal-points ] ;

    so that there are 8 [ nodal-points ] in total for the 2 [ Tetrahedral Spheres ] .

  These 8 [ nodal-points ] are in fact the 8 [ nodal-points ] for a [ Quasi-Cubic Sphere ] :

  For the [ Front View ] , we have :

  

  And for the [ Transparent View ] , we have :

  

• Isn't that Interesting !

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go to the next section : Section XXXV - Duality : Cube vs. Octahedron

go to the last section : Section XXXIII - Duality : Tetrahedron vs. Tetrahedron

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31