A number of attractive polyhedron compounds consisting of three octahedra. The first (left figues) is the polyhedron dual of the cube 3-compound.
These compounds are implemented in the Wolfram Language as PolyhedronData[
"OctahedronThreeCompound",
n
]
for 
,
2, 3.
A hollow version of the octahedron 3-compound with beveled edges like that illustrated above (where the pair of horned lizards inside the compound have been omitted) appears at the central image in M. C. Escher's "Stars" wood engraving (Forty 2003, Plate 43). One of the lesser stars in the lower right corner of this lithograph is a solid octahedron 3-compound.
These octahedron 3-compounds are illustrated above together with their cube 3-compound duals and common midspheres.
The common solids and convex hulls of these compounds are illustrated above. The first compound has interior with the connectivity of the
tetrakis hexahedron and convex
hull of a square-augmented cuboctahedron. The
interior and convex hull of the second are both (different)
-dipyramids. The third has interior that is a polyhedral
realization of the 
-generalized
Petersen graph and convex hull that is a (non-equilateral)
9-antiprism.
The first octahedron 3-compound with unit edge lengths can be constructed using the net shown above with lengths given by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
The surface area of the compound is
 |
(5)
|
