On Symmetric Structures within Symmetric Solid Objects

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

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Section XXXVI - Duality Linkage via the [ FES ( 48-Split ) Sphere ]

Saummary for the Section :

• In this Section , we shall take a look at the [ Cube vs. Octahedron ] Duality :
  • with the [ FES ( 48 split ) Sphere ] here acting as the missing-link .

  ( FES is then the abbreviation for the [ Forty-Eight-Split ] ) .

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A First Look at the [ FES Sphere ] :

• We now present this first look for the [ FES Sphere ] with a [ 48 Split ] :

  

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Construction of the [ FES Sphere ] :

• To construct the [ FES Sphere ] , we simply take an [ Octahedral Sphere ] and :
  • split each of 8 [ 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 diagram on-the-left 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 diagram in-the-middle above .

  • We then construct 3 [ Great-Cirlce Arcs ] joining each [ mid-point ] with its opposite vertex .
    • as per the diagram on-the-right 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 [ FES Sphere ] :

  

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Linkage via the [ FES Sphere ] :

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

  For the Front View , we have :

  

  And for the [ Transparent View ] , we have :

  

  The [ Quasi-Cubical Sphere ] and the [ Octahedral Sphere ] are therefore linked via the [ FES ( 48-Split ) Sphere ] .

• Pretty neat , eh !

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go to the next section : Section XXXVII - Duality --- the Dodecahedron vs. the Icosahedron

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

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Original dated 2010-3-31