Back in time last May.

I have experimented with the opposition of identical modules.

Specifically here for each construction, two identical and simple Lobel frames (a central polygon (triangle, square, pentagon, hexagon, heptagon, octagon) surrounded by 1, 3 or 5 rows of triangles).

Opposed and rotated slightly to align the vertices of the central polygon with the sides of the central polygon of the opposite Lobel frame.

For the one with a triangular base, if we only put one row of triangles, that makes the basic icosahedron

The observations I made:

The fewer sides the central polygon has, the higher the shape obtained will be.

Conversely, the more sides the central polygon has, the more the final shape will be large and flattened, and therefore the more the angles of the join between the two Lobel frames will be closed.