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,
 |
(4)
|
The surface area of the compound is
 |
(5)
|
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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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
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