TOPICS

# 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 (, , ), (, 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)

Cube, Cuboctahedron, Octahedron, Polyhedron Compound

## Explore with Wolfram|Alpha

More things to try:

## 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