Octahedron 5-Compound

Octahedron5CompoundPaper sculpture of the 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


Note that, since the octahedron has unit edge lengths,


The surface area of the compound is

 S=6sqrt(15(9-4sqrt(5))) approx 5.49.

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

See also

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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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.", G. "Compound of Five Octahedra (Five Colors).", S. "Compound of 5 Octahedra.", J. "Jim Plank's Origami Page (Modular).", 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.

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