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Cube 6-Compound

A number of attractive cube 6-compounds can be constructed. A first (left figures) is obtained by combining six cubes, each rotated by 1/6 of a turn about the line joining the centroids of opposite faces of an initial cube. A second compound, illustrated at right, is obtained by combining six cubes, each rotated by 1/8 of a turn about the line joining the centroids of opposite faces of an initial cube.

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

These cube 6-compounds are illustrated above together with their octahedron 6-compound duals and common midspheres.

For the first compound, the common solid is an unnamed polyhedron illustrated above and the convex hull is a polyhedral realization of the graph denoted by Li et al. (2023; E. Weisstein, Sep. 21, 2023). For the second, the common solid is an unnamed solid illustrated above and the convex hull is nonregular solid with the connectivity of the great rhombicuboctahedron.

A net for constructing the first compound is illustrated above, where

 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

The hull of this compound has surface area

 (14)

compared to for each of the six constituent cubes.

Cube, Cube-Octahedron Compound, Polyhedron Compound

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References

Hart, G. "Cube Six-Compound." http://www.georgehart.com/virtual-polyhedra/vrml/cubes_S4_D2.wrl.Li, H.; Ponomarenko, I.; and Zeman, P. "On the Weisfeiler-Leman Dimension of Some Polyhedral Graphs." 26 May 2023. https://arxiv.org/abs/2305.17302.Verheyen, H. F. Symmetry Orbits. Boston, MA: Birkhäuser, 2007.

Cite this as:

Weisstein, Eric W. "Cube 6-Compound." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Cube6-Compound.html