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. In this paper, we discussed the relationship of edges and volumes between the core and the components of the corresponding compound.

Abstract 2
Chapter 1. Introduction 4
Basic Definition 4
1. Polyhedral Compound 4
2. Core 4
Chapter 2. Compounds of Two Platonic Solids 5
Edge Length and Volume 5
1. Octahedron as the Intersection of Two Tetrahedra 6
2. Cuboctahedron as the Intersection of Cube and Octahedron 7
3. Icosidodecahedron as the Intersection of Dodecahedron and Icosahedron 8
4. Truncated Tetrahedron as the Intersection of Two Tetrahedra 9
5. Truncated Cube as the Intersection of Cube and Octahedron 11
6. Truncated Octahedron as the Intersection of Octahedron and Cube 12
7. Truncated Dodecahedron as the Intersection of Dodecahedron and Icosahedron 14
8. Truncated Icosahedron as the Intersection of Icosahedron and Dodecahedron 17
Chapter 3. Compounds of Five Platonic Solids 20
Edge Length and Volume 20
1. Small Rhombicuboctahedron as the Intersection of Four Cubes and an Octahedron 20
2. Great Rhombicuboctahedron as the Intersection of Four Cubes and an Octahedron 22
3. Icosahedron as the Intersection of Five Tetrahedra 24
4. Icosahedron as the Intersection of Five Octahedra 25
5. Rhombic Triacontahedron as the Intersection of Five Cubes 26
Chapter 4. Compounds of Seven Platonic Solids 28
Edge Length and Volume 28
1. Small Rhombicosidodecahedron as the Intersection of a Dodecahedron, an Icosahedron, and Five Cubes 28
2. Great Rhombicosidodecahedron as the Intersection of a Dodecahedron, an Icosahedron, and Five Cubes 31
Chapter 5. Reference 34

Weisstein, Eric W. Cube. <http://mathworld.wolfram.com/Cube.html>.
—. Decagon. <http://mathworld.wolfram.com/Decagon.html>.
—. Dodecahedron. <http://mathworld.wolfram.com/Dodecahedron.html>.
—. Great Rhombicosidodecahedron. <http://mathworld.wolfram.com/GreatRhombicosidodecahedron.html>.
—. Great Rhombicuboctahedron. <http://mathworld.wolfram.com/GreatRhombicuboctahedron.html>.
—. Icosahedron. <http://mathworld.wolfram.com/Icosahedron.html>.
—. Octagon. <http://mathworld.wolfram.com/Octagon.html>.
—. Pentagon. <http://mathworld.wolfram.com/Pentagon.html>.
—. Platonic Solid. <http://mathworld.wolfram.com/PlatonicSolid.html>.
—. Rhombic Triacontahedron. <http://mathworld.wolfram.com/RhombicTriacontahedron.html>.
—. Small Rhombicosidodecahedron . <http://mathworld.wolfram.com/SmallRhombicosidodecahedron.html>.
—. Small Rhombicuboctahedron. <http://mathworld.wolfram.com/SmallRhombicuboctahedron.html>.
—. Tetrahedorn. <http://mathworld.wolfram.com/Tetrahedron.html>.
—. Truncated Dodecahedron. <http://mathworld.wolfram.com/TruncatedDodecahedron.html>.
—. Truncated Icosahedron. <http://mathworld.wolfram.com/TruncatedIcosahedron.html>.
—. Truncated Octahedron. <http://mathworld.wolfram.com/TruncatedOctahedron.html>.
Wolfram. Rhombic Triacontahedron. <http://www.wolframalpha.com/input/?i=Rhombic+Triacontahedron>.