Polyhedral compound

The best known is the compound of two tetrahedra called the stella octangula, discovered by Kepler. The vertices define a cube and the intersection of the two an octahedron, which shares the same face-planes as the compound. Thus it is a stellation of the octahedron, and in fact, the only stellation thereof.

The stella octangula is one of only five compounds that are vertex-, edge-, and face-uniform, called regular compounds:

  Components       Vertices           Face-planes              Symmetry group
  - - - - -        - - - -            - - - - - -              - - - - - - - 
  2 tetrahedra     Cube               Octahedron               Oh
  5 tetrahedra     Dodecahedron       Icosahedron              I
 10 tetrahedra     Dodecahedron       Icosahedron              Ih
  5 cubes          Dodecahedron       Rhombic triacontahedron[?]  Ih
  5 octahedra      Icosidodecahedron  Icosahedron              Ih

The compound of 5 tetrahedra actually comes in two enantiomorphic versions, which together make up the compound of 10 tetrahedra. Each of the tetrahedral compounds is self-dual, and the compound of 5 cubes is dual to the compound of 5 octahedra.

The stella octangula can also be regarded as a compound of a tetrahedron with its dual polyhedron, inscribed in a common sphere so that the vertices of one line up with the face centres of the other. The corresponding cube-octahedron and dodecahedron-icosahedron compounds are the first stellations of the cuboctahedron and icosidodecahedron, respectively.