A polyhedral compound is an arrangement of polyhedra sharing a common center. Given a compound, consequently there’s a core, i.e. the intersection of the compound. Conversely, if we have the core and a collection of platonic solids, the arrangement of the compound is uniquely determined. Compounds of platonic solids are especially interesting.
There are 15 well-known compounds of platonic solids. Their cores inclues 2 Platonic Solids, 11 Archimedean Solids, and 1 Cataland Solid.
In this paper, we will discuss the relationship of edges and volumes between the core and the components of its compound.