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There are several compounds of three cubes. The illustrations above show 3-compounds produced by rotating cubes about a 
axis (left figure), a 
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 
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"].
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
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(1)
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(2)
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(3)
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(4)
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(5)
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The surface area of the compound is
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(6)
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compared to 
for each of the three constituent cubes.
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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.
