There are a number of attractive polyhedron compounds involving four cubes, several of which are illustrated above. The first (left figures),
also known as Bakos' compound, has the symmetry of the cube
and arises by joining four cubes such that each 
axis falls along the 
axis of one of the other cubes
(Bakos 1959; Holden 1991, p. 35). In particular, let the first cube 
consists of a cube in standard position be rotated by 
radians around the 
-axis, then the other three cubes are obtained by rotating
around the 
-axis (z-axis) by 
, 
, and 
radians, respectively.
A second attractive 4-cube compound (second from the left) can be obtained by taking the dual of the octahedron 4-compound obtained from the quartic vertices of the deltoidal icositetrahedron. A number of other 4-compounds (right three figures) can be obtained as the polyhedron duals of octahedron 4-compounds.
These compounds are implemented in the Wolfram Language as PolyhedronData[
"CubeFourCompound",
n
]
for 
,
..., 5.
These cube 4-compounds are illustrated above together with their octahedron 4-compound duals and common midspheres.
For the first compound, the common solid is a small triakis octahedron and the convex hull is a chamfered cube. For the second, the common solid is a chamfered cube and the convex hull has the connectivity of a hexagon-augmented truncated octahedron. The common solid of the third compound is an elongated 8-dipyramid. The common solid of the fourth compound is a 12-dipyramid and the convex hull is an elongated 12-dipyramid. For the fifth compound, the interior and convex hull are (different) 16-prisms.
The net of the first cube 4-compound (Bakos' compound) is illustrated above for cubes with unit edges lengths. The indicated lengths are given by
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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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(6)
|
The surface area of the hull of the first compound is
 |
(7)
|
