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.
These dodecahedron 5-compounds are illustrated above together with their icosahedron 5-compound duals and common midspheres.
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
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
The compound hull has surface area
 |
(11)
|
