There are a number of attractive polyhedron compounds consisting of five cubes. The first of these (left figures) consists of the arrangement
of five cubes in the polyhedron
vertices of a dodecahedron (or the centers of
the faces of the icosahedron). The first 5-compound
compound can be generated by starting with an initial cube
centered at the origin and oriented along the axes, then adding four more cubes obtained
from the initial cube by rotations through angles 
about the axis 
for 
, 2, 3, 4. The cube 5-compound is the dual of the octahedron
5-compound, and is one of the rhombic
triacontahedron stellations (Kabai 2002, p. 185).
A second compound (right figures) can be constructed from the vertices of the first cube 4-compound (E. Weisstein, Sep. 19, 2023).
These cube 5-compounds are illustrated above together with their octahedron 5-compound duals and common midspheres.
The cube 5-compounds are implemented in the Wolfram Language as PolyhedronData[
"CubeFiveCompound",
n
]
for 
,
2.
For the first compound, the common solid is a rhombic triacontahedron (Steinhaus 1999, pp. 199 and 209; Ball and Coxeter 1987) and the convex hull is a regular dodecahedron. For the second, the common solid is an unnamed solid illustrated above and the convex hull is a chamfered cube.
The beautiful figures above show the results of starting with the interior of the compound and including successively larger portions of the space enclosed by its stellations (M. Trott, pers. comm., Feb. 10, 2006). They are therefore rhombic triacontahedron stellations.
The first cube 5-compound can be inscribed on the vertices of an augmented dodecahedron, (first) cube 4-compound, cube-octahedron 5-compound, augmented dodecahedron, deltoidal hexecontahedron, disdyakis triacontahedron, dodecahedron, echidnahedron, great rhombic triacontahedron, great stellated dodecahedron, metabiaugmented dodecahedron, parabiaugmented dodecahedron, pentagonal hexecontahedron, pentakis dodecahedron, rhombic enneacontahedron, rhombic hexecontahedron, rhombic triacontahedron, small triambic icosahedron, spikey, triakis icosahedron, and triaugmented dodecahedron (E. Weisstein, Dec. 25-28, 2009).
The vertices of the first five-cube compound are included among those of the dodecahedron-icosahedron compound, ditrigonal dodecadodecahedron, great ditrigonal icosidodecahedron, and great stellated dodecahedron.
In the above figure, let 
be the length of a cube polyhedron
edge. Then
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(1)
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(2)
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(3)
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(4)
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(5)
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The compound can be constructed using pieces like those illustrated above (Cundy and Rollett 1989).
Nets are shown above for constructing the first compound such that each cube can be made a different color. For cubes of unit edge lengths, the resulting edge lengths are
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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The surface area of the compound is
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(13)
|
compared to 
for each of the five constituent cubes.
The circumradius for the first compound composed of unit cubes is
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(14)
|
and the surface area and volume are
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(15)
|
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(16)
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