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

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

See also Cube ,

Cuboctahedron ,

Octahedron ,

Polyhedron
Compound
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References Ball, W. W. R. and Coxeter, H. S. M. Mathematical
Recreations and Essays, 13th ed. New York: Dover, p. 137, 1987. Coxeter,
H. S. M. Introduction
to Geometry, 2nd ed. New York: Wiley, p. 158, 1969. Cundy,
H. and Rollett, A. "Cube Plus Octahedron." §3.10.2 in Mathematical
Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 130, 1989. Escher,
M. C. "Stars." Wood engraving. 1948. http://www.mcescher.com/Gallery/back-bmp/LW359.jpg . Forty,
S. M.C. Escher.
Cobham, England: TAJ Books, 2003. Kasahara, K. "Combining the Cube
and the Regular Octahedron." Origami
Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, pp. 212-213,
1988. Wenninger, M. J. "Compound of a Cube and Octahedron."
§43 in Polyhedron
Models. New York: Cambridge University Press, p. 68, 1989.
Cite this as:
Weisstein, Eric W. "Cube-Octahedron Compound."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/Cube-OctahedronCompound.html

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