A compound of five regular dodecahedra with the symmetry of the icosahedron (Wenninger 1983, pp. 145-147) can be constructed by taking a dodecahedron
with top and bottom vertices aligned along the z-axis
and one vertex oriented in the direction of the -axis, rotating about the y-axis
by an angle

(1)

and then rotating this solid by angles radians for , 1, ..., 4. A number of other attractive compounds can also
be constructed, as illustrated above.

The dodecahedron 5-compounds illustrated above are implemented in the Wolfram Language as PolyhedronData["DodecahedronFiveCompound",
n]
for ,
2, 3.

For the first compound, the common solid has the connectivity of the deltoidal hexecontahedron and the convex hull is the unnamed
polyhedron illustrated above.

Nets for the dodecahedron 5-compound are shown above, where the lengths are given by