Attractive compounds of four octahedra can be constructed as the duals of the cube 4-compounds.
These compounds are implemented in the Wolfram Language as PolyhedronData[
"OctahedronFourCompound",
n
]
for 
,
2, ..., 5.
These octahedron 4-compounds are illustrated above together with their cube 4-compound duals and common midspheres.
The common solids and convex hulls of these compounds are illustrated above. The first compound has interior of a square-augmented cuboctahedron and convex hull with the connectivity of the truncated cube. The second compound has convex hull corresponding to a square-augmented cuboctahedron. The fourth compound has convex hull that is a (non-equilateral) 12-prism. The interior and convex hull of the fifth are (different) 16-dipyramids.
The figure above gives nets for constructing the first octahedron 4-compound. For a unit octahedron, the net edge lengths are
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
The surface area of the hull of the first compound is
 |
(6)
|
