This Demonstration summarizes others in which various methods are used to explore clusters of polyhedra or spheres associated with decagonal rings. You can cluster rhombic triacontahedra (rt), rhombic dodecahedra (rd), icosahedra, dodecahedra, cubes, or octahedra.
The Demonstration "Rings of Five and Ten Polyhedra" explores the possibility of arranging various polyhedra in a ring. In another Demonstration, "Cluster of 30 Cubes", the polyhedra are placed on the faces of a rhombic triacontahedron (RT). (Here, the placement method is based on the fact that the compound of five cubes encloses a space corresponding to an RT.) The same cluster can be produced by rotating five rings as shown in "Six Decagons and Six Great Circles of a Sphere". Using 12 such rings, aligned in parallel with the faces of a dodecahedron, produces further clusters of polyhedra with facial, edgewise or vertex connection, such as in "120 Rhombic Triacontahedra". "Cluster of 120 Spheres" explores the same geometrical features with the help of spheres.