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Cube 6-Compound


Cube6Compounds

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.

Cube6CompoundsAndDuals

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

Cube6CompoundsInteriorsAndConvexHulls

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.

Cube6-CompoundNetC3

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

s_1=sqrt((79)/(784)-(13)/(98sqrt(2)))
(1)
s_2=1/2sqrt(215-152sqrt(2))
(2)
s_3=1/2(3sqrt(2)-4)
(3)
s_4=1/2sqrt(51-36sqrt(2))
(4)
s_5=1/2sqrt(51-36sqrt(2))
(5)
s_6=1/2sqrt(95-64sqrt(2))
(6)
s_7=1/4sqrt(255-180sqrt(2))
(7)
s_8=sqrt(9/8-3/(2sqrt(2)))
(8)
s_9=1/(14)sqrt(15)
(9)
s_(10)=sqrt(7/(16)-1/(2sqrt(2)))
(10)
s_(11)=1/2(2-sqrt(2))
(11)
s_(12)=1/7(3sqrt(2)-2)
(12)
s_(13)=1.
(13)

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 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

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