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# Cube 2-Compound

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

 (6)

Cube, Cube 3-Compound, Cube 4-Compound, Cube 5-Compound, Cube 6-Compound, Cube 7-Compound, Cube 10-Compound, Cube 20-Compound, Polyhedron Compound

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## References

Brill, D. "Double Cube." Brilliant Origami: A Collection of Original Designs. Tokyo: Japan Pub., pp. 9 and 90-95, 1996.Escher, M. C. "Stars." Wood engraving. 1948. http://www.mcescher.com/Gallery/back-bmp/LW359.jpg.Forty, S. M. C. Escher. Cobham, England: TAJ Books, 2003.Hart, G. "Compound of Two Cubes." http://www.georgehart.com/virtual-polyhedra/vrml/cubes_D6_D3.wrl.Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 213, 1999.Verheyen, H. F. Symmetry Orbits. Boston, MA: Birkhäuser, 2007.

## Cite this as:

Weisstein, Eric W. "Cube 2-Compound." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Cube2-Compound.html