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)
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(2)
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The convex hull of the cube-octahedron compound is a rhombic dodecahedron.
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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)
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(4)
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