Cube 4-Compound

Cube4-CompoundPaper 4-cube model by Emily Herrstrom

A compound also called Bakos' compound having the symmetry of the cube which arises by joining four cubes such that each C_3 axis falls along the C_3 axis of one of the other cubes (Bakos 1959; Holden 1991, p. 35). Let the first cube c_1 consists of a cube in standard position rotated by pi/3 radians around the (1,1,1)-axis, then the other three cubes are obtained by rotating c_1 around the (0,0,1)-axis (z-axis) by pi/2, -pi/2, and pi radians, respectively. The illustration at right above shows a paper sculpture of the cube 4-compound.


The net of the cube 4-compound is illustrated above for cubes with unit edges lengths. The indicated lengths are given by


The surface area of the compound is

 S=(687)/(77) approx 8.92,

compared to S=6 for each of the four constituent cubes. Rather surprisingly, the surface area of this compound is therefore a rational number.


A second attractive 4-cube compound, illustrated above, can be obtained by taking the dual of the octahedron 4-compound obtained from the quartic vertices of the deltoidal icositetrahedron (E. W. Weisstein, Dec. 24, 2009).

These compounds are be implemented in the Wolfram Language as PolyhedronData[{"CubeFourCompound", n}] for n=1 and 2.

See also

Cube, Cube 2-Compound, Cube 3-Compound, Cube 5-Compound, Cube 6-Compound, Cube 7-Compound, Cube 10-Compound, Cube 20-Compound, Octahedron 4-Compound, Polyhedron Compound

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Bakos, T. "Octahedra Inscribed in a Cube." Math. Gaz. 43, 17-20, 1959.Hart, G. "The Compound of Four Cubes.", 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 4-Compound." From MathWorld--A Wolfram Web Resource.

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