There are a number of attractive polyhedron compounds consisting of five octahedra. The first octahedron 5-compound is a polyhedron compound composed of five octahedra occupying the 30 polyhedron vertices of a regular icosidodecahedron (Ball and Coxeter 1987). It is one of the icosahedron stellations (Wenninger 1983). The second compound is the dual of the second cube 5-compound as well as the rotation compound of the octahedron with an additional base octahedron.
These compounds are implemented in the Wolfram Language as PolyhedronData[
"OctahedronFiveCompound",
n
]
for 
,
2.
The vertices of the first octahedron 5-compound correspond to those of the uniform polyhedra having the regular icosidodecahedron as their convex hull, including the dodecadodecahedron, great dodecahemidodecahedron, great dodecahemicosahedron, great icosihemidodecahedron, icosidodecahedron, small dodecahemidodecahedron, and small icosihemidodecahedron.
These octahedron 5-compounds are illustrated above together with their cube 5-compound duals and common midspheres.
The common solids and convex hulls of these compounds are illustrated above. For the first compound, the interior is a regular icosahedron and the convex hull is a regular icosidodecahedron. The interior of the second compound is a square-augmented cuboctahedron.
The first compounds can be constructed from the dual of the cube 5-compound where the cubes have unit edge lengths, which gives a solid with edge lengths
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(1)
|
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(2)
|
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(3)
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Note that, since the base octahedron has unit edge lengths,
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(4)
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The surface area of the first compound hull is
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(5)
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