The points of the icosahedron are known. Coordinates can be found for each point. From these the coordinates for the truncated icosahedron can be derived. From these the points on the edges where the pearls can be placed can be derived. The resulting polyhedron depends on the properties of the points placed on the edges. Is there a single point bisecting the edge. Are there two points symmetrically placed about the midpoint of the edge? If so how. Once this choice is made the resulting points coordinates can be determined and from this the lengths of the sides which ought to give enough information for construction.

The points of the icosahedron are known. Coordinates can be found for each point. From these the coordinates for the truncated icosahedron can be derived. From these the points on the edges where the pearls can be placed can be derived. The resulting polyhedron depends on the properties of the points placed on the edges. Is there a single point bisecting the edge. Are there two points symmetrically placed about the midpoint of the edge? If so how. Once this choice is made the resulting points coordinates can be determined and from this the lengths of the sides which ought to give enough information for construction.

In regards to the question about the easiest way to build a truncated icosahedron in the comments. I would use the fact that all if its faces are equilateral hexagons and pentagons with two pentagons and one hexagon at each vertex. If I could get rigid hexagons and pentagons I could fit them together and I would be able to get the polyhedron relatively easily. It is much easier to construct this because all the faces are regular polygons so they could be constructed first and the rest of the construction is relatively easy.