 |  |
A compound also called Bakos' compound having the symmetry of the cube which 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). Let the first cube 
consists of a cube in standard position rotated by 
radians around the 
-axis, then the other three cubes are obtained by rotating
around the 
-axis (z-axis) by 
, 
, and 
radians, respectively. The illustration at right above shows
a paper sculpture of the cube 4-compound.
The net of the cube 4-compound is illustrated above for cubes with unit edges lengths. The indicated lengths are given by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
The surface area of the compound is
 |
(7)
|
compared to 
for each of the four constituent cubes. Rather surprisingly, the surface area of
this compound is therefore a rational number.
A second attractive 4-cube compound, illustrated above, can be obtained by taking the dual of the octahedron 4-compound obtained from the quartic vertices of the deltoidal icositetrahedron (E. W. Weisstein, Dec. 24, 2009).
These compounds are be implemented in the Wolfram Language as PolyhedronData[
"CubeFourCompound",
n
]
for 
and 2.
