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# Octahedron 5-Compound

A polyhedron compound composed of five octahedra occupying the 30 polyhedron vertices of an icosidodecahedron (Ball and Coxeter 1987). The solid is one of the icosahedron stellations (Wenninger 1983). The octahedron 5-compound is the dual of the cube 5-compound. The illustration on the right above shows a paper sculpture of the octahedron 5-compound.

It is implemented in the Wolfram Language as PolyhedronData["OctahedronFiveCompound"].

Constructing the octahedra as the duals of the cube 5-compound where the cubes have unit edge lengths give a solid with edge lengths

 (1) (2) (3)

Note that, since the octahedron has unit edge lengths,

 (4)

The surface area of the compound is

 (5)

The convex hull of the octahedron 5-compound is the icosidodecahedron.

Cube 5-Compound, Cube-Octahedron 5-Compound, Escher's Solid, Icosahedron Stellations, Icosidodecahedron, Octahedron, Octahedron 3-Compound, Octahedron 4-Compound, Octahedron 6-Compound, Octahedron 10-Compound, Polyhedron Compound, Stella Octangula

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## References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 135 and 137, 1987.Cundy, H. and Rollett, A. "Five Octahedra About in Icosahedron." §3.10.7 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 137-138, 1989.Hart, G. "Compound of Five Octahedra." http://www.georgehart.com/virtual-polyhedra/vrml/compound_of_5_octahedra.wrl.Hart, G. "Compound of Five Octahedra (Five Colors)." http://www.georgehart.com/virtual-polyhedra/vrml/compound_of_5_octahedra_(5_colors).wrl.Kabai, S. "Compound of 5 Octahedra." http://www.kabai.hu/poly/D1/poly004.htm.Plank, J. "Jim Plank's Origami Page (Modular)." http://www.cs.utk.edu/~plank/plank/pics/origami/origami.html.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, p. 55, 1983.Wenninger, M. J. "Compound of Five Octahedra." §23 in Polyhedron Models. New York: Cambridge University Press, p. 43, 1989.

## Cite this as:

Weisstein, Eric W. "Octahedron 5-Compound." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Octahedron5-Compound.html