Cube 2-Compound

There are a number of attractive polyhedron compounds of two cubes. The first (left figures) 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). A second (middle figures) combines two cubes rotated with respect to one another along a axis. A third (right figure) consists of two cubes rotated by with respect to each other around a common axis.

These compounds are implemented in the Wolfram Language as PolyhedronData["CubeTwoCompound", n] for , 2, 3.

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

These cube 2-compounds are illustrated above together with their octahedron 2-compound duals and common midspheres.

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 the hull of the first 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.

For the first compound, the common solid is a hexagonal dipyramid and the convex hull is an elongated hexagonal dipyramid, while for the second, the common solid and convex hull are both octahedral prisms.

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

(6)