Not too sure I fully understand. 

I think the idea is filling triangular faces with smaller triangles, 6 along each edge ([example used in the wiki][1]) 

So creating a 2D approximation, flattening out the icosahedron we get a net of 20 triangular faces, 10 squares in a 5x5 box


[![enter image description here][2]][2]

we can get the pattern in 2D

[![enter image description here][3]][3]

Then wrap it around a sphere ([using the code from this post][4])

[![enter image description here][5]][5]

You can adjust the shape for truncated icosahedron.
Or are you are looking for a manipulation in 3D?

 [1]: https://en.wikipedia.org/wiki/Geodesic_polyhedron
 [2]: https://i.sstatic.net/bNS1I.png
 [3]: https://i.sstatic.net/PmTHm.png
 [4]: https://mathematica.stackexchange.com/questions/27493/how-can-i-put-an-image-on-a-surface
 [5]: https://i.sstatic.net/EztU9.png