
The Stella Octangula, or compound of two tetrahedra was named by Johannes Kepler. In this arrangement the two tetrahedra are reciprocal;
the faces of one correspond to the vertices of the other and their
edges cross at right angles. Many symmetries are apparent: the two
tetrahedra have their vertices at the corners of a cube and their edges form the diagonals of the cube faces. Less obviously, the volume common to both of them is an octahedron.
In fact, the two tetrahedra together can be thought of as a variety of
octahedron; they have 8 triangular faces arranged symmetrically. 

The compound of ten tetrahedra similarly arises from an icosahedron,
which is the interior volume common to all of them: the ten tetrahedra
between them have 20 triangular faces. They can also be seen as an
arrangement of 5 Stella Octangulae, which suggests the compounds of
five cubes and five octahedra below. 

The compound of five tetrahedra
is a subset of the compound of ten, found by taking one tetrahedron
from each Stella Octangula. It has less symmetry than the interior icosahedron;
it has the rotational symmetries but not reflection symmetry.
It therefore exists in two mirrorimage forms, which is obvious from
the twisted appearance. 

The compound of five cubes is suggested by taking the exterior cubes of the five Stella Octangulae of the ten tetrahedra, or the reciprocals of the five octahedra. It is bounded by a dodecahedron, but the interior volume common to all five cubes is not a regular polyhedron. 

The compound of five octahedra is another variety of icosahedron, having 20 triangular faces; the interior common volume is an icosahedron. It is also reciprocal to the compound of five cubes. 