On Symmetric Structures within Symmetric Solid Objects

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

Top Of Page

Section XXXVII - Duality Linkage via the [ OHATS ( 120-Split ) Sphere ]

Saummary for the Section :

• In this Section , we shall take a look at the [ Dodecahedron vs. Icosahedron ] Duality :
  • with the [ OHATS ( 120 split ) Sphere ] here acting as the missing-link .

  ( OHATS is then the abbreviation for the [ One-Hundred-And-Twenty-Split ] ) .

go to Top Of Page

A First Look at the [ OHATS Sphere ] :

• We now present this first look for the [ OHATS Sphere ] with a [ 120 Split ] :

  

go to Top Of Page

Construction of the [ OHATS Sphere ] :

• To construct the [ OHATS Sphere ] , we simply take an [ Icosahedral Sphere ] and :
  • split each of 20 [ 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 [ OHATS Sphere ] :

  

go to Top Of Page

Linkage via the [ OHATS Sphere ] :

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

  For the Front View , we have :

  

  And for the [ Transparent View ] , we have :

  

  The [ Dodecahedral Sphere ] and the [ Icosahedral Sphere ] are therefore linked via the [ OHATS ( 120-Split ) Sphere ] .

• Interesting indeed !

go to Top Of Page

go to the next section : Section XXXIX - Identifying [ 2 Quad-Type Structures ] for Our Analysis

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

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31