A polyhedron compound of two cubes. The compound at left (also called the double cube) is obtained by allowing two cubes
to share opposite polyhedron vertices, then
rotating one a sixth of a turn about a axis (Holden 1991, p. 34). The compound at right combines
two cubes, one rotated
with respect to the other along a axis, producing an octagrammic prism.

The
compound appears twice (in the lower left as a beveled wireframe and in the lower
center as a solid) as polyhedral "stars" in M. C. Escher's 1948
wood engraving "Stars" (Forty 2003, Plate 43).

The left illustration above shows an origami cube 2-compound (Brill 1996, pp. 90-92).
The right illustration shows the net of one pyramid of the compound. Each pyramidal
portion is composed of two doms (1-2 right triangles) and
one isosceles right triangle. If the original cube has edge lengths 1, then edge
lengths of the net are given by

(1)

(2)

(3)

(4)

The surface area of this compound is

(5)

compared to
for each of the two original cubes. Rather surprisingly, the surface area of the
compound is therefore a rational number.

The solid common to the two cubes in the compound is a hexagonal
dipyramid (left figure), and the convex hull is
an elongated hexagonal dipyramid (right figure).

If a second cube is rotated about a axis with respect to a fixed cube, then the edges indicated
in black above remain intersecting throughout the entire 1/3 turn. The -position of the intersection as a function of rotation angle
is given by the complicated expression