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Tetrahedron 6-Compound


Tetrahedron6Compounds

A number of attractive tetrahedron 6-compounds can be constructed. The first compound (left figures) is obtained by combining three stella octangula. A second can be obtained by arranging six regular tetrahedra about a common C_2 copies. A third can be obtained by combining two oppositely-oriented tetrahedron 3-compounds.

Tetrahedron6CompoundsAndDuals

These tetrahedron 6-compounds are illustrated above together with their duals and common midspheres.

Tetrahedron6CompoundsIntersectionsAndConvexHulls

The common solids and convex hulls are illustrated above. For the first compound, the interior has the connectivity of a tetrakis hexahedron and the convex hull has the connectivity of the truncated octahedron. For the second, the interior is a 12-dipyramid and the convex hull is a (non-equilateral) 12-prism. For the third, the interior is a truncated 9-trapezohedron (with the connectivity of the (18,2)-generalized Petersen graph) and the convex hull is a gyroelongated 9-dipyramid.

Tetrahedron6-CompoundNet

A net for the hull of the first compound is illustrated above, with

s_1=1/6(2-sqrt(2))
(1)
s_2=sqrt(7/(24)-7/(18sqrt(2)))
(2)
s_3=1/4(2-sqrt(2))
(3)
s_4=sqrt(5/(12)-5/(9sqrt(2)))
(4)
s_5=1/2(sqrt(2)-1)
(5)
s_6=1/2(2-sqrt(2))
(6)
s_7=sqrt((15)/8-5/(2sqrt(2))).
(7)

See also

Polyhedron Compound, Regular Tetrahedron

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References

Hart, G. "Compound of Six Tetrahedra." http://www.georgehart.com/virtual-polyhedra/vrml/compound_of_six_tetrahedra.wrl.Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, p. 129, 2002.Skilling, J. "Uniform Compounds of Uniform Polyhedra." Math. Proc. Cambridge Phil. Soc. 79, 447-457, 1976.

Cite this as:

Weisstein, Eric W. "Tetrahedron 6-Compound." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Tetrahedron6-Compound.html

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