A cube 2-compound is a polyhedron compound of two cubes. Several attractive examples are illustrated above. The first (left
figures) is obtained by allowing two cubes to share opposite
polyhedron vertices then rotating one a sixth
of a turn about a 
axis (Holden 1991, p. 34). A second (middle figures) combines two cubes rotated
with respect to one another
along a 
axis. A third (right figure) consists of two cubes rotated by 
with respect to each other around a common 
axis.
These compounds are implemented in the Wolfram Language as PolyhedronData[
"CubeTwoCompound",
n
]
for 
,
2, 3.
A cube 2-compound obtained by taking two cubes from the first cube 5-compound is used to construct the supercube. The six protruding tetrahedra of one cube are reflected inward across the faces of the other and subtracted from it (Reiner 2026).
The 
compound appears twice (in the lower left as a beveled wireframe and in the lower
center as a solid) in M. C. Escher's 1948 wood engraving "Stars"
(Forty 2003, Plate 43).
These cube 2-compounds are illustrated above together with their octahedron 2-compound duals and common midspheres.
|
|
The left illustration above shows an origami cube 
2-compound (Brill 1996, pp. 90-92).
The right illustration shows the net of one pyramid of the compound. Each pyramidal
portion is composed of two doms (1-2 right triangles) and
one isosceles right triangle. If the original cube has edge lengths 1, then edge
lengths of the net are given by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
The surface area of the hull of the first compound is
 |
(5)
|
compared to 
for each of the two original cubes. Rather surprisingly, the surface area of the
compound is therefore a rational number.
For the first compound, the common solid is a hexagonal dipyramid and the convex hull is an elongated hexagonal dipyramid, while for the second, the common solid and convex hull are both octahedral prisms.
If a second cube is rotated about a 
axis with respect to a fixed cube, then the edges indicated
in black above remain intersecting throughout the entire 1/3 turn. The 
-position of the intersection as a function of rotation angle
is given by the complicated expression
![x=(12costheta-3cos(2theta)-sqrt(3)[4sintheta+sin(2theta)])/(2[6+2costheta+cos(2theta)-2sqrt(3)(costheta-1)sintheta]).](/images/equations/Cube2-Compound/NumberedEquation2.svg) |
(6)
|
