This support structure is amazingly close (TODO: how close). It consists of five intersecting tetrahedra, connected to the spheres in the truncated icosahedron with valency 6 (there are twenty of them). The length of the tetrahedral edges is 5 rods. The edges of the tetrahedra should be 143.8035 mm, and they are 146.84 mm actually. That is 2.11% too long, but the excess is easily absorbed by the hexagons.