SSO - Duality [ Cube vs. Octahedron ]

On Symmetric Structures within Symmetric Solid Objects

Section XXXV - Duality : Cube vs. Octahedron

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

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

Let us now identify the [ centroidal points ] for each of the 6 [ curved 4-sided faces ] :
- as marked-off via the 6 [ blue-color dots ] in the set of diagrams below .
We note here that the 6 [ blue-color dots ] are in fact the 6 [ nodal-points ] for the [ Octahedral Sphere ] :

Let us now identify the [ centroidal points ] for each of the 8 [ curved triangular faces ] :
- as marked-off via the 8 [ red-color dots ] in the set of diagrams below .
We note here that the 8 [ red-color dots ] are in fact the 8 [ nodal-points ] for the [ Quasi-Cubical Sphere ] :

Very simply , the [ Quasi-Cubical Sphere ] and the [ Octahedral Sphere ] are related :
- via the [ centroidal points ] of their [ curved faces ] .
In other words :
- The 6 [ centroidal points ] of the [ Quasi-Cubical Sphere ] are the 6 [ nodal points ] for the [ Octahedral Sphere ] , and :
- The 8 [ centroidal points ] of the [ Octahedral Sphere ] are the 8 [ nodal points ] for the [ Quasi-Cubical Sphere ] .