Dodecahedron-Icosahedron Compound
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A Polyhedron Compound of a Dodecahedron and Icosahedron which is most easily constructed by adding 20triangular Pyramids, constructed as above, to an Icosahedron. In the compound, theDodecahedron and Icosahedron are rotated 
radians with respect to each other, and the ratio of theIcosahedron to Dodecahedron edges lengths are the Golden Ratio 
.
The above figure shows compounds composed of a Dodecahedron of unit edge length and Icosahedrahaving edge lengths varying from 
(inscribed in the dodecahedron) to 2 (circumscribed about the dodecahedron).
The intersecting edges of the compound form the Diagonals of 30 Rhombusescomprising the Triacontahedron, which is the Dual Polyhedron of the Icosidodecahedron (Ball andCoxeter 1987). The dodecahedron-icosahedron is the first Stellation of the Icosidodecahedron.
See also Dodecahedron, Icosahedron, Icosidodecahedron, Polyhedron Compound
References
Cundy, H. and Rollett, A. Mathematical Models, 2nd ed. Stradbroke, England: Tarquin Pub., p. 131, 1989. Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 76, 1989.
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