SSO - Duality [ Dodecahedron vs. Icosahedron ]

On Symmetric Structures within Symmetric Solid Objects

Section XXXVII - Duality : Dodecahedron vs. Icosahedron

In this Section , we shall take a look at the Duality of :
- the Dodecahedron vs. the Icosahedron .

Let us now bring-in again the [ Dodecahedral Sphere ] :

Let us now identify the [ centroidal points ] for each of the 12 [ curved pentagonal faces ] :
- as marked-off via the 12 [ blue-color dots ] in the set of diagrams below .
We note here that the 12 [ blue-color dots ] are in fact the 12 [ nodal-points ] for the [ Icosahedral Sphere ] :

Let us now identify the [ centroidal points ] for each of the 20 [ curved triangular faces ] :
- as marked-off via the 20 [ red-color dots ] in the set of diagrams below .
We note here that the 20 [ red-color dots ] are in fact the 20 [ nodal-points ] for the [ Dodecahedral Sphere ] :

Very simply , the [ Dodecahedral Sphere ] and the [ Icosahedral Sphere ] are related :
- via the [ centroidal points ] of their [ curved faces ] .
In other words :
- The 12 [ centroidal points ] of the [ Dodecahedral Sphere ] are the 12 [ nodal points ] for the [ Icosahedral Sphere ] , and :
- The 20 [ centroidal points ] of the [ Icosahedral Sphere ] are the 20 [ nodal points ] for the [ Dodecahedral Sphere ] .