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 n=1, 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 X_(48) 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
The hull of this compound has surface area
| S=171sqrt(2)-(1626)/7 approx 9.54, |
(14)
|
compared to S=6 for each of the six constituent cubes.
See also
Cube, Cube-Octahedron Compound, Polyhedron CompoundExplore with Wolfram|Alpha
More things to try:
References
Hart, G. "Compounds of Cubes." https://www.georgehart.com/virtual-polyhedra/compound-cubes-info.html.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.Referenced on Wolfram|Alpha
Cube 6-CompoundCite this as:
Weisstein, Eric W. "Cube 6-Compound." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Cube6-Compound.html