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,

(4)

The surface area of the compound is

(5)

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

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More things to try:

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.