Cube-Octahedron Compound
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 (+/-1/2, +/-1/2, +/-1/2), (+/-1, 0, 0), (0, +/-1, 0), (0, 0, +/-1). Since the edges of the cube and octahedron bisect each other, the resulting solid has side lengths 1/2 and sqrt(2)/2 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 CompoundExplore with Wolfram|Alpha
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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. https://mcescher.com/wp-content/uploads/2019/04/LW-359.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.
Referenced on Wolfram|Alpha
Cube-Octahedron CompoundCite this as:
Weisstein, Eric W. "Cube-Octahedron Compound." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Cube-OctahedronCompound.html