Octahedra Inscribed in a Cube

The following problem was proposed by Dorman Luke, Math. Gazette, Vol. XLI, p. 194 (1957): Inscribe a regular octahedron in a cube, so that its vertices are one on each of six edges of the cube. Four such octahedra can be inscribed; what is the solid with 32 faces which is common to these four? Starting with the octahedron, how many cubes can we circumscribe to it in this way? Again, what solid formed by their vertices encases them all?