I'm working on an applied problem that requires designing symmetrical structures. I'd like to form an arrangement with full icosahedral symmetry (point group Ih in 3D) out of building blocks with Cs symmetry. What number of building blocks can be arranged in this way? Clearly only some numbers work, but what does the whole sequence look like?
There are some obvious constructions: an icosahedron has 20 (triangular) faces, 30 edges and 12 (pentagonal) vertices. Putting one Cs building block at the midpoint of each edge with the symmetry plane perpendicular to the edge and passing through and center of the icosahedron is a fully Ih symmetrical arrangement (n=30).
Arranging three building blocks into a triplet with C3 symmetry and then placing one of those triplets at a midpoint of a face with symmetry axis pointing normal to the face is another Ih arrangement (n=3*20=60). Similarly, arranging five building blocks into a C5 symmetry and placing that at a vertex with the symmetry axis passing through the center of the icosahedron should work (n=5*12=60). I'm not sure if every "C3 at a face" can be constructed as some "C5 at a vertex" - it seems they can; on the other hand "Cs at an edge" can be seen as a special case of "C5 at a vertex", when the building blocks from connected vertices are at the exact midpoint of the edge and so can be the same building block (halving the total needed).
Putting building block on both edges and faces, both edges and vertices, or all three, gives n=90 and n=150. Also, any integer multiple of 30 or 60 works, by placing building blocks in n nested icosahedrons, or by placing n building blocks into a Cn arrangement (which is also Cs) and using that as a bigger building block.
From that argument, the sequence of Cs building blocks that can be placed into an arrangement with icosahedral symmetry is 30, 60, 90, 120, ... or essentially all multiples of 30. Is this all the possible numbers and arrangements?
More generally, what is this problem called? (build a solid with higher symmetry out of building blocks with lower symmetry) Are there any general results about what arrangements are possible, and what numbers of building blocks are in each arrangement?