Exercise: In the compound
of five octahedra, won't there be 40=5*8 faces for the interior? So
how could the interior be an icosahedron,
which we know has only 20 faces ?
Answer: Each icosahedron face is part of two octahedra. This
is dual to the fact that in the compound
of five cubes, each dodecahedron vertex is shared with two cubes.
Similarly, in the compound
of ten tetrahedra, the 40 faces come in 20 coplanar pairs, and the
interior is an icosahedron. These are both examples of stellations
of the icosahedron.