Many concentric icosidodecahedra
A rhombitruncated icosidodecahedron can be built on top of another two, attaching one another by decagonal faces. Here, the two lower bodies are built around a reinforced rhombicosidodecahedron, as shown in Concentric icosidodecahedra. The upper one is built around a non reinforced, normal rhombicosidodecahedron:
The construction can be extended to three base objects and two raised ones. In this case we have added a reinforced base (version 2, explained in More concentric icosidodecahedra) to the former, as it eases their handling and increases their strength.
In fact, we could continue to a full circle with five base bodies and five raised ones, each five in the vertices of a regular pentagon, sort of a monstrous, non uniform pentagonal antiprism. If we had enough pieces available, of course.
The position of both raised rhombitruncated icosidodecahedra allows placing another one in a new third level. There is just a small problem: a portion of the space required by the new object is already taken by the corresponding first-level object; we explain below how this overlap can be solved. In principle, a full circle with 15 bodies in three levels could also be built.
Three pictures of the construction follow:
Both the body in the third level, and the central one in the first level have a square face parallel to each other. The distance between both faces is nearly 12 mm, but inside the objects, that is, both objects overlap. We avoid this overlap by leaving the decagonal faces at both sides of the square face incomplete in one of the objects (the upper one in this case), as the following picture shows. We remove five balls, twelve rods, and four pentagons in total: