SSO - Icosahedral Sphere ( 20-Split )

On Symmetric Structures within Symmetric Solid Objects

Section XXXI - The Icosahedral Sphere ( 20-Split )

In this Section , we shall introduce the Icosahedral Sphere :
- a sphere with its outer-surface divided into 20 [ equal parts ] .

Let us recall that the Icosahedron is twenty-faced Symmetric Solid Object with exactly twelve (12) [ nodal-points ] , and :
- the 12 [ nodal-points ] are all equi-distant from the [ centroid ] of Symmetric Solid Object .
This then permits us to put the 12 [ nodal-points ] onto the surface of a Sphere , so that :
- the 12 [ nodal-points ] are now all equi-distant from the [ center ] of the Sphere .
Let us now join the twelve (12) [ nodal-points ] via [ Great-Circle Arcs ] and we have this Spherical Solid Object below :
- and this then is our { Icosahedral Sphere } with a [ 20-Split ] .
Let us now take a quick look at this { Icosahedral Sphere } in a 10-colors [ Bi-Polar pattern ] ,
- for a slightly better feel of the Spherical Solid Object .

We simply note here that the [ surface area ] of the Sphere is given by :
- [ Surface Area ] = 4 * [ pi ] * [ R-Square ]
The [ surface area ] of an [ Individual Piece ] here is then :
- { 4 * [ pi ] * [ R-Square ] } divided by [ 20 ] ,
or simply :
- [ pi ] * [ R-Square ] * [ 1 / 5 ] .