The dodecahedron-icosahedron compound is a polyhedron compound consisting of a dodecahedron and its
dual the icosahedron. In the compound, the dodecahedron
and icosahedron are rotated 
radians with respect to each other, and the ratio of the
icosahedron to dodecahedron
edges lengths are the golden ratio 
.
It is implemented in the Wolfram Language as PolyhedronData["DodecahedronIcosahedronCompound"].
Its hull is most easily constructed by adding 20 triangular pyramids to an icosahedron. If the dodecahedron is chosen to have unit edge length, the resulting compound has side lengths
 |  |  |
(1)
|
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(2)
|
The surface area and volume of the resulting hull are
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(3)
|
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(4)
|
The interior is an icosidodecahedron, making the faces of the dodecahedron-icosahedron compound a stellation of the icosidodecahedron. The intersecting edges of the compound form the polyhedron diagonals of the 30 rhombuses constituting the rhombic triacontahedron (Ball and Coxeter 1987), which is the convex hull of the compound.
