Icosidodecahedron as a Common Element
The internal vertices of 60 tetrahedra meet the 30 vertices of an icosidodecahedron (ID) in pairs. Other polyhedra that fit with the ID in some way may be associated with the cluster of tetrahedra and with one another. The relationships are often evident, and sometimes not so evident; for example, that the vertices of 60 tetrahedra meet the middle of the edges of a cluster of 30 tetrahedra. There is also a near miss here, namely, the edges of 30 tetrahedra almost coincide with the edges of a pentagonal hexecontahedron. Many more relationships could be explored based on the ID as an intermediate element.
From the many possible polyhedra that could be associated with the ID, this Demonstration shows icosahedron (ico), pentagonal hexecontahedron (ph), snub dodecahedron (sd), cluster of 30 tetrahedra (tet30), cluster of 30 rhombic dodecahedra (rd30) and small rhombicosidodecahedron (srd).
 S. Kabai. "30 Cubes on a Rhombic Triacontahedron," in Proceedings of Bridges Pécs: Mathematics, Music, Art, Architecture, Culture, Hungary, 2010 (G. W. Hart and R. Sarhangi, eds.), Phoenix, AZ: Tessellations Publishing. (Jun 9, 2021) archive.bridgesmathart.org/2010/bridges2010-317.pdf.