The cube-octahedron compound is a polyhedron compound composed of a cube and its dual polyhedron, the octahedron.
It is implemented in the Wolfram Language as PolyhedronData["CubeOctahedronCompound"].
A cube-octahedron compound appears in the upper left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).
For a cube of edge length 1, the 14 vertices are located at (
,
,
),
(
,
0, 0), (0, 
,
0), (0, 0, 
).
Since the edges of the cube and octahedron bisect each other, the resulting solid
has side lengths 1/2 and 
and surface area and
volume given by
 |  |  |
(1)
|
 |  |  |
(2)
|
The convex hull of the cube-octahedron compound is a rhombic dodecahedron.
|
|
|
The solid common to both the cube and octahedron (left figure) in a cube-octahedron compound is a cuboctahedron (middle figure). The edges intersecting in the points plotted above are the diagonals of rhombuses, and the 12 rhombuses form a rhombic dodecahedron (right figure; Ball and Coxeter 1987).
A net for the hull of the compound is given above.
The figure above shows the pieces needed for construction of the compound, where
 |  |  |
(3)
|
 |  |  |
(4)
|
