compound polyhedron

A compound polyhedron is an assemblage of two or more polyhedra, usually interpenetrating and having a common center. There are two types: a combination of a solid with its dual and an interpenetrating set of several copies of the same polyhedron. The simplest example of a compound polyhedron is the compound of two tetrahedra, known as the stella octangula and first described by Johannes Kepler. This shape is unique in that it falls under both of the above classes, because the tetrahedron is the only self-dual uniform polyhedron; the edges of the two tetrahedra form the diagonals of the faces of a cube in which the stella octangula can be inscribed. Another example of a compound follows from an important Platonic relationship: a cube can be inscribed within a dodecahedron. There are five different positions for a cube within a dodecahedron; superimposing all five gives the compound known as the rhombic triacontahedron.