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 5compound is the dual of the cube 5compound. The illustration on the right above
shows a paper sculpture of the octahedron 5compound.
It is implemented in the Wolfram Language
as PolyhedronData["OctahedronFiveCompound"].
Constructing the octahedra as the duals of the cube
5compound where the cubes have unit edge lengths give a solid with edge lengths
Note that, since the octahedron has unit edge lengths,

(4)

The surface area of the compound is

(5)

The convex hull of the octahedron 5compound is the
icosidodecahedron.
See also
Cube 5Compound,
CubeOctahedron 5Compound,
Escher's Solid,
Icosahedron
Stellations,
Icosidodecahedron,
Octahedron,
Octahedron 3Compound,
Octahedron
4Compound,
Octahedron 6Compound,
Octahedron 10Compound,
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. 137138, 1989.Hart,
G. "Compound of Five Octahedra." http://www.georgehart.com/virtualpolyhedra/vrml/compound_of_5_octahedra.wrl.Hart,
G. "Compound of Five Octahedra (Five Colors)." http://www.georgehart.com/virtualpolyhedra/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 5Compound."
From MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/Octahedron5Compound.html
Subject classifications