Not quite sure if I understand you completely, but it seems to me that the underlying problem is this:
It is impossible to cover a sphere with undistorted squares, equilateral triangles, hexagons etc. E.g. with a subdivided cube, there will be 8 corners (the original corners of the cube) with 3 edges, but all other corners have 4 edges. If you start with an icosahedron and subdivide with triangles, you will have corners with 5 and corners with 6 edges.