Cube 3-Compound


There are several compounds of three cubes. The illustrations above show 3-compounds produced by rotating cubes about a C_3 axis (left figure), a C_4 axis producing a dodecagrammic prism (middle figure), and with the symmetry of the cube which arises by joining three cubes such that each shares two C_2 axes (Holden 1991, p. 35; right figure). The latter solid is depicted atop the left pedestal in M. C. Escher's woodcut Waterfall (Bool et al. 1982, p. 323).

It is implemented in the Wolfram Language as PolyhedronData["CubeThreeCompound"].

Paper sculpture of the cube 3-compound

The illustration above shows Escher's cube 3-compound.


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


The surface area of the compound is

 S=72-45sqrt(2) approx 8.36,

compared to S=6 for each of the three constituent cubes.


The solid common to the three cubes in Escher's compound (left figure) and the convex hull (right figure) are illustrated above.

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.

See also

Cube, Cube 2-Compound, Cube 4-Compound, Cube 5-Compound, Cube 6-Compound, Cube 7-Compound, Cube 10-Compound, Cube 20-Compound, Escher's Solid, Octahedron 3-Compound, Polyhedron Compound

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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., G. "The Compound of Three Cubes.", 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.

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