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# Cube 3-Compound

There are several attractive polyhedron compounds consisting of three cubes. The first (left figures) arises by joining three cubes such that each shares two axes (Holden 1991, p. 35; right figure). In other words, it consists of three cubes, each rotated by 1/8 of a turn about the line joining the centroids of opposite faces. A second (middle figures) rotates two cubes about a axis by 1/8 of a turn relative to one another, producing a dodecagrammic prism (middle figures). A third compound (right figure) can be constructed by rotating two cubes about a axis with respect to each other.

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

The first compound is depicted atop the left pedestal in M. C. Escher's woodcut Waterfall (Bool et al. 1982, p. 323).

These cube 3-compounds are illustrated above together with their octahedron 3-compound duals and common midspheres.

For the first compound, the common solid is a chamfered cube while the convex hull is a truncated octahedra with non-regular hexagonal faces. For the second, the common solid and convex hull are dodecahedral prisms. For the third, the common solid is a 9-trapezohedron and the convex hull is a gyroelongated nonagonal dipyramid.

The Escher compound divides the three component cubes into 67 individual cells (Hoeflin 1985). Whether another configuration of three intersecting cubes can yield more cells is an unsolved problem.

Escher's 3-cube compound can be constructed to produce cubes with unit edge lengths using pieces as illustrated above, where

 (1) (2) (3) (4) (5)

The surface area of the compound hull is

 (6)

Cube, Escher's Solid, Octahedron 3-Compound, Polyhedron Compound

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## References

Bool, F. H.; Kist, J. R.; Locher, J. L.; and Wierda, F. M. C. Escher: His Life and Complete Graphic Work. New York: Abrams, 1982.Escher, M. C. "Waterfall." Lithograph. 1961. http://www.mcescher.com/Gallery/recogn-bmp/LW439.jpg.Hart, G. "The Compound of Three Cubes." http://www.georgehart.com/virtual-polyhedra/vrml/cubes_S4_D4.wrl.Hoeflin, R. K. "Problem 36 in Mega Test." Omni 7, 129, Apr. 1985.Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991.Verheyen, H. F. Symmetry Orbits. Boston, MA: Birkhäuser, 2007.

## Cite this as:

Weisstein, Eric W. "Cube 3-Compound." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Cube3-Compound.html