 |  |
A polyhedron compound consisting of the arrangement of five cubes in the polyhedron vertices of a dodecahedron (or the centers of the faces of the icosahedron). The illustration at right above shows a paper sculpture of the cube 5-compound.
The cube 5-compound can be inscribed on the vertices of an augmented dodecahedron, (first) cube 4-compound, cube-octahedron 5-compound, augmented dodecahedron, deltoidal hexecontahedron, disdyakis triacontahedron, dodecahedron, echidnahedron, great rhombic triacontahedron, great stellated dodecahedron, metabiaugmented dodecahedron, parabiaugmented dodecahedron, pentagonal hexecontahedron, pentakis dodecahedron, rhombic enneacontahedron, rhombic hexecontahedron, rhombic triacontahedron, small triambic icosahedron, spikey, triakis icosahedron, and triaugmented dodecahedron (E. Weisstein, Dec. 25-28, 2009).
The cube 5-compound is implemented in the Wolfram Language as PolyhedronData["CubeFiveCompound"].
The compound can be generated by starting with an initial cube centered at the origin and oriented along the axes, then adding four more cubes obtained
from the initial cube by rotations through angles 
about the axis 
for 
, 2, 3, 4. The cube 5-compound is the dual of the octahedron
5-compound, and is one of the rhombic
triacontahedron stellations (Kabai 2002, p. 185).
The cube 5-compound has the 30 facial planes of the rhombic triacontahedron (Steinhaus 1999, pp. 199 and 209; Ball and Coxeter 1987).
In the above figure, let 
be the length of a cube polyhedron
edge. Then
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
The compound can be constructed using pieces like those illustrated above (Cundy and Rollett 1989).
Nets are shown above for constructing the compound such that each cube can be made a different color. For cubes of unit edge lengths, the resulting edge lengths are
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
The surface area of the compound is
 |
(13)
|
compared to 
for each of the five constituent cubes.
The circumradius for the compound composed of unit cubes is
 |
(14)
|
and the surface area and volume are
 |  |  |
(15)
|
 |  |  |
(16)
|
Its convex hull is a dodecahedron, and its interior is a rhombic triacontahedron. The beautiful figures above show the results of starting with the interior of the compound and including successively larger portions of the space enclosed by its stellations (M. Trott, pers. comm., Feb. 10, 2006). They are therefore rhombic triacontahedron stellations.
