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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"].

Octahedron5-CompoundNet

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

s_1=1/(10)(5-sqrt(5))
(1)
s_2=3/5sqrt(5)-1
(2)
s_3=1/2(3-sqrt(5)).
(3)

Note that, since the octahedron has unit edge lengths,

 s_1+s_2+s+3=1.
(4)

The surface area of the compound is

 S=6sqrt(15(9-4sqrt(5))) approx 5.49.
(5)
Octahedron5CompoundHull

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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